New Trends in the Theory of Hyperbolic Equations (Operator Theory: Advances and Applications, 159) 3764372834, 9783764372835

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Operator Theory Advances and Applications Vol. 159

New Trends in the Theory of Hyperbolic Equations Michael Reissig Bert-Wolfgang Schulze Editors

Birkhäuser

Advances in Partial Differential Equations

Operator Theory: Advances and Applications Vol. 159 Editor: I. Gohberg

Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) G. Heinig (Chemnitz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam)

H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Halmos (Santa Clara) T. Kailath (Stanford) P. D. Lax (New York) M. S. Livsic (Beer Sheva)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze Institut für Mathematik Universität Potsdam 14415 Potsdam Germany

Sergio Albeverio Institut für Angewandte Mathematik Universität Bonn 53115 Bonn Germany

Michael Demuth Institut für Mathematik Technische Universität Clausthal 38678 Clausthal-Zellerfeld Germany

Elmar Schrohe Institut für Mathematik Universität Hannover 30060 Hannover Germany

New Trends in the Theory of Hyperbolic Equations

Michael Reissig Bert-Wolfgang Schulze Editors

Advances in Partial Differential Equations

Birkhäuser Verlag Basel . Boston . Berlin

Editors: Michael Reissig TU Bergakademie Freiberg Fakultät für Mathematik und Informatik Institut für Angewandte Analysis 09596 Freiberg Germany e-mail: [email protected]

Bert-Wolfgang Schulze Institut für Mathematik Universität Potsdam 14415 Potsdam Germany e-mail: [email protected]

2000 Mathematics Subject Classification 35L05, 35L10

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at

ISBN 3-7643-7283-4 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7283-4 ISBN-13: 978-3-7643-7283-5 987654321

www.birkhauser.ch

Contents Editorial Preface

xi

Wave Maps and Ill-posedness of their Cauchy Problem Piero D’Ancona and Vladimir Georgiev

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6

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational motivation of the wave maps equations . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Harmonic maps and special harmonic maps on the sphere . 2.3 Equivariant wave maps and construction of special solutions Local existence result for equivariant wave maps . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Localization in time . . . . . . . . . . . . . . . . . . . . . . 3.3 Estimates for the homogeneous problem . . . . . . . . . . . 3.4 Estimates for the non-homogeneous problem . . . . . . . . 3.5 Bilinear estimates for the homogeneous problem in H s,δ . . 3.6 Bilinear estimates in H s,δ for the inhomogeneous problem . Concentration of the local energy . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Construction of the solutions . . . . . . . . . . . . . . . . . 4.3 Higher regularity of the solution . . . . . . . . . . . . . . . 4.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-uniqueness result in the subcritical case . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Equivariant and self-similar solutions . . . . . . . . . . . . . 5.3 Low regularity self-similar solutions . . . . . . . . . . . . . 5.4 Appendix A: The self-similar ODE . . . . . . . . . . . . . . 5.5 Appendix B: Some technical lemmas . . . . . . . . . . . . . Ill-posedness in the critical case (Fourier analysis approach) . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Well-posedness of the Cauchy problem for semilinear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 4 6 12 17 17 20 24 26 31 36 40 40 42 46 52 55 55 57 60 66 72 77 77 79

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6.3 The wave map system in stereographic projection . . . . . 6.4 Conclusion of the proof of Theorem 6.1 . . . . . . . . . . 6.5 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . Ill-posedness in the critical case (fundamental solution approach) 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . 7.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 80 . 83 . 86 . 90 . 90 . 92 . 102 . 109

On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations Hideo Kubo and Masahito Ohta 1 2

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Introduction . . . . . . . . . . . . . Single wave equation . . . . . . . . 2.1 Blow-up . . . . . . . . . . . 2.2 Small data global existence 2.3 Almost global existence . . 2.4 Self-similar solution . . . . 2.5 Asymptotic behavior . . . . Semilinear system, I . . . . . . . . 3.1 Blow-up . . . . . . . . . . . 3.2 Small data global existence 3.3 Self-similar solution . . . . 3.4 Asymptotic behavior . . . . Semilinear system, II . . . . . . . . 4.1 Small data global existence 4.2 Self-similar solution . . . . 4.3 Generalization . . . . . . . Semilinear system, III . . . . . . . 5.1 Blow-up . . . . . . . . . . . Small data global existence 5.2 Nonlinear system . . . . . . . . . . 6.1 Blow-up . . . . . . . . . . . 6.2 Null condition . . . . . . . Appendix . . . . . . . . . . . . . . References . . . . . . . . . . . . . .

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Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains Mitsuhiro Nakao 1 2

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113 117 119 125 133 137 140 156 157 163 166 167 168 170 173 176 178 179 183 187 192 196 202 205

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

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Local energy decay . . . . . . . . . . . . . . . . . . . . . . . . . . 218 3.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 218 3.2 Proof of Theorem 3.1. . . . . . . . . . . . . . . . . . . . . . 219 3.3 Proof of Corollary 3.1. . . . . . . . . . . . . . . . . . . . . . 223 Total Energy decay for the wave equation with a localized dissipation226 4.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 226 4.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . 227 4.3 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . 232 Linear equations with variable coefficients; Unique continuation property and a basic inequality . . . . . . . . . . . . . . . . . . . . . . . 233 5.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 233 5.2 Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . 235 5.3 Proof of Theorems 5.1 and 5.2 . . . . . . . . . . . . . . . . 237 5.4 Proof of Proposition 5.2 . . . . . . . . . . . . . . . . . . . . 239 Lp estimates for the wave equation in exterior domains . . . . . . . 241 6.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 241 6.2 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . 243 Semilinear wave equations . . . . . . . . . . . . . . . . . . . . . . . 249 7.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 249 7.2 Proof of Theorems 7.1 and 7.2 . . . . . . . . . . . . . . . . 251 7.3 Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . 253 Quasilinear wave equations . . . . . . . . . . . . . . . . . . . . . . 259 8.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 259 8.2 Energy decay for the quasilinear equation . . . . . . . . . . 261 8.3 Estimation of higher-order derivatives of solutions . . . . . 265 8.4 Proof of Theorems 8.2 and 8.3. . . . . . . . . . . . . . . . . 272 The wave equation with a half-linear dissipation . . . . . . . . . . 280 9.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 280 9.2 A basic inequality . . . . . . . . . . . . . . . . . . . . . . . 283 9.3 Proof of Theorem 9.1 . . . . . . . . . . . . . . . . . . . . . 286 9.4 Proof of Theorems 9.2 and 9.3 . . . . . . . . . . . . . . . . 289 Some open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 293 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Global Existence in the Cauchy Problem for Nonlinear Wave Equations with Variable Speed of Propagation Karen Yagdjian 301 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Counterexamples to the global existence . . . . . . . . Blow-up for the problem with large potential energy of Parametric resonance and wave map type equations .

. . . . . . . . . . . . . . nonlinearity . . . . . . .

301 303 320 324

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6 7 8 9 10 11 12

Proof 5.1 5.2 5.3

of Theorem 4.1: Parametric resonance . . . . . . . . . . . . . Some properties of the Hill’s equation . . . . . . . . . . . . Borg’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . Construction of an exponentially increasing solution to Hill’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Construction of blow-up solutions . . . . . . . . . . . . . . Coefficient stabilizing to a periodic one. Parametric resonance dominates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.1: Perturbation theory . . . . . . . . . . . . . . Nonexistence for equations with permanently restricted domain of influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global existence for a model equation with a polynomially growing coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example with an exponentially growing coefficient . . . . . . . Fast oscillating coefficients: no resonance ?! . . . . . . . . . . . . . Linear wave equations with oscillating coefficients . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

On the Nonlinear Cauchy Problem Massimo Cicognani and Luisa Zanghirati 1 2

3

4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Well-posedness in C ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Function and symbol spaces . . . . . . . . . . . . . . . . . . 2.2 Levi conditions . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The linear problem . . . . . . . . . . . . . . . . . . . . . . . 2.5 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The equivalent quasilinear system . . . . . . . . . . . . . . 2.7 Local C ∞ solutions . . . . . . . . . . . . . . . . . . . . . . . 2.8 Analytic regularity . . . . . . . . . . . . . . . . . . . . . . . Well-posedness in Gevrey classes . . . . . . . . . . . . . . . . . . . 3.1 The linear problem . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gevrey-Levi conditions . . . . . . . . . . . . . . . . . . . . 3.3 Factorization under Gevrey-Levi conditions . . . . . . . . . 3.4 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The equivalent quasilinear system in Gevrey spaces . . . . . 3.6 Local Gevrey solutions and propagation of the analytic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strictly hyperbolic equations with non-Lipschitz coefficients and C ∞ solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Log-Lipschitz coefficients or unbounded derivatives . . . . . 4.2 The linear problem with non-regular coefficients . . . . . . 4.3 The map u
→ v . . . . . . . . . . . . . . . . . . . . . . . . .

327 327 334 338 342 344 345 354 358 362 370 373 382

387 387 392 392 394 397 398 403 407 409 410 416 416 419 422 423 427 428 430 430 432 436

Contents 5

Holder ¨ coefficients and Gevrey Solutions . . . . 5.1 Gevrey well-posedness . . . . . . . . . . 5.2 From the factorization to the quasilinear References . . . . . . . . . . . . . . . . . . . . .

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Sharp Energy Estimates for a Class of Weakly Hyperbolic Operators Michael Dreher and Ingo Witt 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Well-posedness of the Cauchy problem . . . . . . . . . 1.2 Degenerate differential operators . . . . . . . . . . . . 1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Formulation of the results . . . . . . . . . . . . . . . . . . . . 2.1 Motivation and plan of the paper . . . . . . . . . . . . 2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . 3 A model case . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Taniguchi–Tozaki’s example . . . . . . . . . . . . . . . 3.2 Conversion into a 2 × 2 system . . . . . . . . . . . . . 3.3 Estimation of the fundamental matrix . . . . . . . . . 3.4 Function spaces: An approach via edge Sobolev spaces 3.5 Establishing energy estimates . . . . . . . . . . . . . . 3.6 Summary of Section 3 . . . . . . . . . . . . . . . . . . 4 Symbol classes and function spaces . . . . . . . . . . . . . . . 4.1 The symbol classes S m,η;λ . . . . . . . . . . . . . . . . 4.2 The symbol classes S˜m,η;λ . . . . . . . . . . . . . . . . m,η;λ 4.3 The symbol classes S+ for η ∈ Cb∞ (Rn ; R) . . . . . 4.4 Function spaces: An approach via weight functions . . 4.5 Summary of Section 4 . . . . . . . . . . . . . . . . . . 5 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . 5.1 Improvement of G˚ ˚ arding’s inequality . . . . . . . . . . 5.2 Symmetric-hyperbolic systems . . . . . . . . . . . . . 5.3 Symmetrizable-hyperbolic systems . . . . . . . . . . . 5.4 Higher-order scalar equations . . . . . . . . . . . . . . 5.5 Local uniqueness . . . . . . . . . . . . . . . . . . . . . 5.6 Sharpness of energy estimates . . . . . . . . . . . . . . A Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . B Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Editorial Preface Hyperbolic partial differential equations describe phenomena of material or wave transport in the applied sciences. Despite of considerable progress in the past decades, the mathematical theory still faces fundamental questions concerning the influence of nonlinearities or multiple characteristics of the hyperbolic operators or geometric properties of the domain in which the evolution process is considered. The current volume is dedicated to modern topics of the theory of hyperbolic equations such as evolution equations – multiple characteristics – propagation phenomena – global existence – influence of nonlinearities. It is addressed both to specialists and to beginners in these fields. The contributions are to a large extent self-contained. The first contribution is written by Piero D’Ancona and Vladimir Georgiev. Piero D’Ancona graduated in 1982 from Scuola Normale Superiore of Pisa. Since 1997 he is full professor at the University of Rome 1. Vladimir Georgiev graduated in 1981 from the University of Sofia. Since 2000 he is full professor at the University of Pisa. The first part of the paper treats the existence of low regularity solutions to the local Cauchy problem associated with wave maps. This introductory part follows the classical approach developed by Bourgain, Klainerman, Machedon which yields local well-posedness results for supercritical regularity of the initial data. The nonuniqueness results are established by the authors under the assumption that the regularity of the initial data is subcritical. The approach is based on the use of self-similar solutions. The third part treats the ill-posedness results of the Cauchy problem for the critical Sobolev regularity. The approach is based on the effective application of the properties of a special family of solutions associated with geodesics on the target manifold. The second contribution is written by Hideo Kubo and Masahito Ohta. Hideo Kubo graduated from Hokkaido University in 1996. Since 2003 he is associate professor at Osaka University. Masahito Ohta graduated from the University of Tokyo in 1996. Since 2003 he is associate professor at Saitama University. Initially they consider in their contribution wave equations with small nonlinear perturbation. The problems of interest are local well-posedness in time, blow-up, asymptotic behaviour and existence of self-similar solutions. The main topic of their contribution is how the theory of wave equations is transferred to systems of nonlinear wave equations with different propagation speeds. They explain, in particular, how the fact of different propagation speeds can be utilized to the advantage of a detailed analysis of initial value problems. The systematic study of such systems is based on pointwise decay estimates for solutions of the Cauchy problem for inhomogeneous wave equations in L∞ spaces with hyperbolic

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weights. The blow-up results are established by deriving lower estimates of this type. Further fields of interest of H. Kubo are associated with the study of the system of elastic equations and the Maxwell system. Among other things, M. Ohta is also interested in issues of stability and instability of standing wave solutions for nonlinear Klein-Gordon equations. The third contribution is written by Mitsuhiro Nakao. He received his doctoral degree (Doctor of Science) in 1977 from Kyushu University and is full professor at the same university since 1976. He has been strongly interested in decay problems for the wave equation with various types of dissipations in bounded domains. He developed his own techniques, which many authors use today (“Nakao’s inequality”). Since 1998 he focused his interest on the Cauchy problem and on initial-boundary value problems in exterior domains. His strategy is to combine the energy method with the geometry of the exterior domain. For non-trapping domains a restricted localized effective dissipation is employed. He has derived decay results with algebraic rates for local and total energies. Moreover, he found the critical order of nonlinearities in his models. Here the property of stability comes in. In the future M. Nakao wants to apply his knowledge of decay properties for nonlinear damped wave equations to the problems concerning global attractors. He also plans to consider related problems for nonlinear degenerate parabolic equations which is another field of his research interest. The fourth contribution is written by Karen Yagdjian. Karen Yagdjian has received his Doctor of Physical and Mathematical Sciences degree from Moscow State University in 1990. Since 2004 he is assistant professor at the University of Texas-Pan American. His main interests are microlocal analysis and its application to partial differential equations. The main goal of his contribution is to study the phenomenon of parametric resonance for wave map type equations. In particular, he is interested in the study of the influence of the oscillating behaviour of coefficients in t on the global existence of small data solutions. A special transformation reduces the Cauchy problem for the wave map type equations to linear Cauchy problems for the wave equation with a special constraint. To attack these Cauchy problems Floquet’ theory, especially Borg’s theorem for Hill’s equation is used. The question for stability and instability is discussed in a systematic way. Other models with growing coefficients or stabilizing coefficients are treated in a similar fashion. The fifth contribution is written by Massimo Cicognani and Luisa Zanghirati. Luisa Zanghirati graduated from Ferrara University in 1965 and is full professor there since 1985. Massiomo Cicognani graduated from Bologna University in 1983 and is full professor there since 1998. Their contribution is devoted to local in time existence of smooth solutions for nonlinear degenerate hyperbolic problems. Different kinds of degeneracies are

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explored. On the one hand they consider weakly hyperbolic Cauchy problems (characteristics of constant multiplicity), thus Levi conditions of nonlinear type come into play. On the other hand degeneracies are produced by low regularity in time of the coefficients. The goal of the authors is to present a unified approach for these problems. The main tools of the approach are an effective diagonalization procedure with regularized characteristic roots, an effective representation of commutators, a sharp G˚ ˚ arding’s inequality for systems and a suitable transformation containing the loss of derivatives. The loss of derivatives is characteristic for degenerate hyperbolic problems. Moreover, they describe the propagation of analytic regularity of solutions. Further interests of L. Zanghirati are questions concerning hypoellipticity, and M. Cicognani is interested in microlocal methods to describe the propagation of singularities for pseudodifferential operators. The sixth contribution is written by Michael Dreher and Ingo Witt. Michael Dreher graduated from the University of Freiberg in 1999. Since 2004 he is assistant professor at the University of Konstanz. Ingo Witt graduated from the University of Bonn in 1995. Since 2004 he has a DFG fellowship at the Imperial College London. The goal of the authors is to derive sharp energy estimates for weakly hyperbolic Cauchy problems with finite time degeneracy of the coefficients at t = 0. From such estimates they obtain the precise loss of regularity that depends on the spatial variables. In the case of time-dependent coefficients they show an interesting relation to the theory of edge Sobolev spaces, a tool which is used for the study of differential operators on manifolds with singularities. In the general case (where the coefficients’ depend on time–and spatial variables) the authors introduce Sobolev spaces of variable order. The main step is to find the correct class of pseudodifferential symbols and to establish a pseudodifferential calculus which contains a symmetrizer. An aparding’s inequality gives rise to a sharp energy estimate. plication of a sharp G˚ Sharpness is proved by using the method of Lyapunov functionals, where suitable estimates lead to an instability result. We would like to thank all the referees for their valuable contribution in the evaluation process. We also wish to thank Dr. Jens Wirth (TU Bergakademie Freiberg) for the considerable effort he put into producing the final layout of this volume. Last not least, the editors would like to thank all the staff of Birkh¨ auser Publishing Company, in particular, Dr. Thomas Hempfling, for the pleasant cooperation. Freiberg and Potsdam M. Reissig

B.-W. Schulze

April 2005

Operator Theory: Advances and Applications, Vol. 159, 1–111 c 2005 Birkhauser
¨ Verlag Basel/Switzerland

Wave Maps and Ill-posedness of their Cauchy Problem Piero D’Ancona and Vladimir Georgiev

Abstract. In this review article we present an introduction to the theory of wave maps, give a short overview of some recent methods used in this field and finally we prove some recent results on ill–posedness of the corresponding Cauchy problem for the wave maps in critical Sobolev spaces. The low regularity solutions for the wave map problem are studied by the aid of appropriate bilinear estimates in the spirit of ones introduced by Klainerman and Bourgain. Our approach to obtain ill–posedness in critical Sobolev norms uses suitable family of wave maps constructed via geodesic flow on the target manifold. We give two alternative proofs of the ill–posedness: the first approach is based on the application of Fourier analysis tools, while the second proof is based on the application of the classical fundamental solution representation for the free wave equation. For the case of subcritical Sobolev norms we establish non–uniqueness of the corresponding Cauchy problem. Mathematics Subject Classification (2000). 35L05, 35J10, 35P25, 35B40. Keywords. Equivariant wave maps, H s -spaces, blow-up of solution.

1. Preface Let (N, g) be a smooth n-dimensional Riemannian manifold with metric g, which with no loss of generality we can isometrically embed in Rn1 for some n1 > n. For the functions u : R × Rm → N defined on the flat Minkowski space Rt × Rm x with values in the target N , consider the functional
J(u) = ∂α u, ∂ α ug(u) dtdx, R×Rm

The authors are partially supported by Research Training Network (RTN) HYKE, financed by the European Union, contract number : HPRN-CT-2002-00282.

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Piero D’Ancona and Vladimir Georgiev

where summation over α = 0, 1, . . . , m is intended, with ∂t , ∂x1 , . . . , ∂xm ), (∂0 , . . . , ∂m ) = (∂

(∂ 0 , . . . , ∂ m ) = (∂ ∂t , −∂ ∂x1 , . . . , −∂ ∂xm )

as usual, while ·, ·g is the product in the metric g. The critical points of the functional J are called wave maps. If we choose a system of coordinates on N , then locally smooth wave maps satisfy the equation
u + Γbc (u)∂α ub ∂ α uc = 0,

 = 1, . . . , n,

(1.1)

where Γjk denote the Christoffel symbols on N in the chosen coordinates. The natural problem for this system of wave equations is clearly the Cauchy problem with data at t = 0 u(0, x) = u0 , ut (0, x) = u1 ; (1.2) the usual space for the data are Sobolev spaces (u0 , u1 ) ∈ H s (Rm , N ) × H s−1 (Rm , T N ) for suitable values of s ∈ R. Here we used the space H s (Rm ; N ) ≡ {v ∈ H s (Rm ; Rn1 ), v(Rm ) ⊆ N },

s∈R

(1.3)

with the induced norm; notice that H s (Rm ; N ) = ∅ if 0 ∈ N , but it causes no loss of generality to assume that 0 ∈ N after a translation in the ambient space. An alternative description of the wave map system, which usually gives a simpler expression in presence of symmetry of the target is the following: a wave map is a function u : R × Rm → Rn1 such that u(t, x) ∈ N,


u ⊥ N

for all (t, x).

A good introduction on this subject with comprehensive references f can be found in [30]. The Cauchy problem for wave maps has been extensively studied in recent years, starting with the work of Ginibre and Velo [15]. Not many general results for the Cauchy problem are known. One can unify the existing results in few groups. • Global existence of weak H 1 -solutions when the target is compact and has dimension 2 (several authors, see, e.g., [25], [29], [47]). • Local existence for data in H s , s > m/2. This is classical if s is large enough, but for s close to the critical value s = m/2 it is a much more difficult result, due to Klainerman, Machedon, Selberg and obtained through careful bilinear estimates (see in particular [20] or Chapter 3 below). See also Tataru [41] for the case of Besov spaces. • Global existence for small data. Again, this result can be proved by “standard” methods in the smooth case (Y. Choquet-Bruhat), but the recent results of Tao (see in particular [40], see also [31] and [26]) show that it is sufficient to assume that the data are in H s for some s > m/2 and that they are small in the homogeneous H˙ m/2 -norm.

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3

• In the presence of symmetry and space dimension two one has sharper results; to this class belong the radial case, considered by Christodoulou, TahvildarZadeh and more recently by Struwe (see [8], [37], [38]), and the equivariant case, for which a fairly complete theory exists, due to Shatah, TahvildarZadeh, Struwe, and Grillakis (see, e.g., [13], [29], [32], [36]). However several outstanding problems remain open. In particular we mention the case of two-dimensional wave maps u : R × R2 → N, when the target N is the two-dimensional sphere in R3 . The wave map system for u can be written in this case
u + u(|ut |2 − |∇x u|2 ) = 0. It is not known if the local smooth solutions (even with equivariant symmetric data) may develop singularities in a finite time; or, equivalently, if the global weak solutions (which exist) are unique. In this case the “null form” character of the nonlinear term is particularly evident, and it is easy to check that the critical space with respect to scaling is exactly H˙ m/2 (in the scale of Sobolev spaces H s ). For this we shall call this exponent s = m/2 critical (with respect to the scaling argument). For the definition and the properties of the quantity sc = m/2 as a critical (with respect to the scaling) exponent one can see Section 1.3 in [21]. In the present review we would like to give a detailed exposition of some recent results, obtained by the authors in a series of works, concerning the problem of the behavior of the wave map system for the cases s < m/2 and s = m/2. These results are closely connected with the general well-posed conjecture formulated in section 1.3 in [21], that asserts that for a large class of wave type equations (called initial value problems for basic field theories) the scale critical exponent sc is such that • a) the corresponding Cauchy problem is (locally in time) well-posed for initial data in H s with s > sc ; • b) the Cauchy problem has a global solution provided the initial data have small H sc norm; • c) the Cauchy problem is ill-posed for s < sc . More precisely, we shall see that the third part c) of this conjecture is indeed true for the wave map problem and we can show non-uniqueness of suitable weak solutions to the wave map problem. For the case s = m/2 it is natural to ask if the Cauchy problem is well-posed. This situation is quite similar to the well-posedness problem studied by Bourgain [3], Kenig, Ponce, Vega [22] for the case of several dispersive equations, such as the nonlinear Schr¨ o¨dinger or the Korteweg-de Vries equations. The basic idea is to study the properties of the solution map “data
→ solution” and show that in the critical or subcritical case some change of behavior occurs. Indeed, for s > m/2 the bilinear estimates in [19], [20], [21] show that a contraction method works and yields local well-posedness for the corresponding Cauchy problem. Moreover, this

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contraction argument shows also that the solution map is Lipschitz continuous (and hence even smoother). A breakdown of this property may indicate an instability of the equation, and suggest that a blow up may occur in a finite time. As a minimal goal, if one can show that the solution map is not Lipschitz or even uniformly continuous, one obtains that the contraction method cannot be applied to prove well-posedness. This is indeed the case for the wave map system with data in the Sobolev space H˙ m/2 (see Section 6). The review is organized as follows. Section 2 is devoted to some background material, i.e., to show the effect that suitable symmetry assumptions have on the form of the equations, and to give a self contained proof of the local well-posedness in the supercritical case. In Section 3 we consider some bilinear estimates in the spirit of the results in [20] and show how they can improve the local existence result for the Cauchy problem for wave maps. In Section 4 we show a first instability example, by constructing smooth solutions for which the energy is concentrated in a cusp-like domain; notice that the energy cannot concentrate in a self-similar way for the 2-dimensional case as it is well known. Section 5 is devoted to the subcritical case; for the 2-D wave maps with values into the 2-D sphere one can prove (as expected) a quite strong ill-posedness result, namely the non uniqueness of weak solutions. In Section 6 we begin the study of the solution map in the critical space, which is completed in Section 7, where we prove that the solution map can never be locally uniformly continuous for any base space dimension and an arbitrary (non flat) target. The authors are grateful to A. Ivanov, N. Visciglia for the careful reading and corrections of the manuscript of this work.

2. Variational motivation of the wave maps equations 2.1. Introduction Wave maps arise in several problems of mathematical physics (see, e.g., the Higgs field model in [15], and the relativity models in [7]). To be more precise, let (N, g) be an n-dimensional manifold endowed with a Riemannian metric structure, i.e., a positive definite bilinear form g in every point of N . We call N the target manifold. Let M = Rm+1 be the Minkowski space time equipped with the flat metric h = (−1, 1, . . . , 1). A wave map is a map that satisfies the equation Dα ∂α u = 0; here

(2.1)

∂ , (xα ) = (t, x) ∈ R1+m , α = 0, 1, . . . , m, ∂xα while Dα is the covariant pull-back derivative in the bundle u∗ T N. As usual, the Greek indices α, β run from 0 to m; we use the summation convention over repeated indices. By the Nash embedding theorem, we may assume that the target N is isometrically embedded in some Rd for d large enough. So, we can consider u as ∂α =

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an Rd valued function u = (u1 , . . . , ud ). Then the intrinsic equation (2.1) can be rewritten in extrinsic form utt − ∆u − B(u)(∂α u, ∂ α u) = 0,

(2.2)

where

B(p) : Tp N × Tp N → Tp N ⊥ is the second fundamental form of N ⊂ Rd . Given any function F : (t, x) ∈ Rm+1 → F (t, x) ∈ Tu(t,x) N we shall also consider the following inhomogeneous version of (2.2) utt − ∆u − B(u)(∂α u, ∂ α u) = F.

(2.3)

In this work we study the Cauchy problem for (2.3) subject to the initial conditions u(0, x) = u0 (x) ∈ H s (Rm ; N ),

∂t u(0, x) = u1 (x) ∈ H s−1 (Rm ; T N ).

(2.4)

More precisely, we identify H = H (R ; N ), s ≥ 0, with the space of funcs

s

m

tions u(x) ∈ H s (Rm ; Rd ),  Rd is satisfying u(x) ∈ N for almost every x ∈ Rm (here the embedding N → m used in an essential way). Similarly, if U ⊂ R is an open set, then H s (U ) = H s (U ; N ), s ≥ 0, is the space of functions u(x) ∈ H s (U ; Rd ), satisfying u(x) ∈ N for almost every x ∈ U. The homogeneous case F = 0 is treated in [19], [21], [42]. From these results it follows that given any s > m/2, any data (u0 , u1 ) ∈ H s × H s−1 one can find a finite time interval [0, T ], 0 < T < T0 , such that there exists a unique solution u ∈ C([0, T ]; H s ) to the Cauchy problem (2.2), (2.4). Our first goal in this chapter is to verify that the Cauchy problem (2.3), (2.4) has a local solution u ∈ C([0, T ]; H s ) whenever the initial data satisfy (u0 , u1 ) ∈ H s × H s−1 with s > m/2 and the source term is such that F ∈ L1 ((0, T0 ); H s−1 ). After this we shall concentrate our attention to the specific case, when m = 2 and the target is one typical compact manifold namely N = S2 . The next step in

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Piero D’Ancona and Vladimir Georgiev

this chapter is the study of special types of wave maps. More precisely, we consider equivariant wave maps and study their relation with the corresponding harmonic maps on S2 of arbitrary degree. The results in [32] guarantee the existence of a global solution, when the target is S2 , the degree of the harmonic map is 1 and the initial data are small in the H 1 -norm. We shall show in particular that this result can be extended to the case of arbitrary harmonic map degree. Our approach in this introductory chapter follows the argument in [32] and relies on suitable local (in time) a priori estimates. We can apply the general local existence result, which in dimension 2 implies the local well-posedness of the Cauchy problem in H s with s > 1. We shall precise the dependence on the index s > 1 of the a priori estimate, which is the essential tool in the proof of the well-posedness, by computing the constants appearing in the estimate as a function of s. Our main observation is that these constants are uniform in s > 1 as s tends to 1. 2.2. Harmonic maps and special harmonic maps on the sphere Harmonic maps can be regarded as a special case of the above construction, when the manifold M is a m-dimensional Riemannian manifold equipped with a metric σ. The target (N, g), as above, is a n-dimensional manifold endowed with a Riemannian metric g. Thus the harmonic map satisfies the equation m 

Dj ∂j u = 0,

(2.5)

j=1

where ∂ ∂xj and xj , j = 1, . . . , m are the local coordinates on M. Moreover, Dj is the covariant pull-back derivative in the bundle u∗ T N. To introduce the energy functional we suppose that y 1 , . . . , y n are local coordinates on N provided y1, . . . , yn ∈ Y ∂j =

with Y being a small neighborhood of 0 ∈ Rn . Given any small neighborhood X of 0 ∈ Rm and any map U : x = (x1 , . . . , xm ) ∈ X → y = (y 1 , . . . , y n ) ∈ Y we can define locally the energy functional
 m n  √ E(U ) = σ jk (x)gab (y)∂ ∂j y a (x)∂k y b (x) σdx. X a,b=1 j,k=1

To simplify the notations we use the summation convention for repeated indices, so that we can write
√ σ jk (x)gab (y)∂ ∂j y a (x)∂k y b (x) σdx. (2.6) E(U ) = X

Wave Maps

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The Euler–Lagrange equation associated with this functional has the form √ √ √ −2gab ∂k σ jk σ∂ ∂j y b − 2∂c gab ∂k y c σ jk σ∂ ∂j y b + (∂a gbc ∂j y c σ jk σ∂k y b ) = 0. (2.7) Since the Laplace–Beltrami operator ∆M has local representation ∆M =

m  j,k=1

√ 1 √ ∂j σ σ jk ∂k , σ

(2.8)

where σ = det (σ σjk ) , we may write 1 gab ∆M y b + ∂c gab ∂k y c σ jk ∂j y b − (∂a gbc ∂j y c σ jk ∂k y b ) = 0. (2.9) 2 We recall the explicit expression of the Christoffel symbols: 1 γc;ab = (∂a gbc + ∂b gac − ∂c gab ). (2.10) 2 If we write 1 ∂c gab ∂k y c σ kj ∂j y b = (∂c gba ∂k y c σ kj ∂j y b + ∂b gac ∂k y c σ kj ∂j y b ) 2 and use the expression of Christoffel symbols, then we arrive at the following equation gab ∆M y b + γa;bc ∂k ub σ jk (x)∂ ∂j uc = 0. Raising the index a, we obtain a ∆M y a + γbc ∂k y b σ jk (x)∂ ∂j y c = 0.

(2.11)

By the Nash embedding theorem, we may assume that the target N is embedded in some Rd for d large enough. Hence u is a d-dimensional function u = (u1 , . . . , ud )l on the other hand, the local coordinates y 1 , . . . , y n on N enable us to parameterize the manifold N locally, and we can think of u as a function u = u(y),

u : Y ⊆ Rn → R d .

In the following we shall restrict to the simpler case d = n + 1, i.e., when N is a hypersurface in Rd . The Riemannian metric g on N is induced by the Euclidean metric on Rd which is simply gab = ∂ ∂ya u, ∂yb uRd ,

(2.12)

where ·, ·Rd is the scalar product in Rd . Then the intrinsic equation (2.11) can be rewritten in extrinsic form as ∆M u +

m 

σ jk B(u)(∂ ∂j u, ∂k u) = 0,

j,k=1

where B(p) : Tp N × Tp N → Tp N ⊥

(2.13)

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Piero D’Ancona and Vladimir Georgiev

is the second fundamental form on N ⊂ Rd and ∆M is the Laplace–Beltrami operator on the manifold M. Recall that the second fundamental form is defined by n  B(u)(v, w) = bac (u)v a wc ν(u) (2.14) a,c=1

for any two vectors v, w ∈ Tp N , with coefficients bac defined as follows bac = −∂ ∂ya ∂yc u(y), ν(u(y))Rd , a, c = 1, . . . , n,

(2.15)

where ν(u) is the unit normal at u ∈ N. In the above local representation we have used a local basis dy 1 , . . . , dy n in Tp N, which is dual to the basis of vector fields ∂y1 , . . . , ∂yn so that n n   v= vj dy j , w = wj dy j . j=1

j=1

To verify the above assertion it is sufficient to rewrite the energy functional in (2.6) as follows
√ E(U ) = σ jk (x)∂ ∂j u(y(x))∂k u(y(x))Rd σdx. (2.16) X

Taking the variation of this integral over u ∈ H 1 , under the constraint u(y) ∈ N , we obtain the Euler–Lagrange equation with Lagrange multiplier ∆M u = −µν(u),

(2.17)

where the Lagrange multiplier µ can be obtained by scalar multiplication with u, i.e., √ 1 µ = − √ ∂ ∂j σ σ jk ∂k u, νRd . σ Using the property ∂k u ∈ Tu (N ), ν(u) ∈ Tu (N )⊥ , we get µ = − σ jk ∂ya ∂yc u, νRd ∂j y a ∂k y c . Now the definition (2.15) of the second fundamental form leads to (2.13). From now on we restrict our attention to the special case when M = Sm−1 = N. Then we can assume that the standard metric on M = N is induced by the embedding Sm−1 ⊂ Rm . If ω ∈ Sm−1 , then ω = (ω1 , . . . , ωm ) ∈ Rm and |ω| = 1. We shall denote by κ = (κ1 , . . . , κm−1 ) any local coordinates on Sm−1 and by x = (x1 , . . . , xm ) the coordinates on Rm ; then the standard metric on Sm−1 can be written m−1  j,k=1

σjk (κ)dκj dκk

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9

for suitable coefficients σjk . Introducing spherical coordinates x ∈ Sm−1 , r = |x|, ω = |x| we have the following decomposition of the Laplace operator in Rm m−1 1 ∂r + 2 ∆Sm−1 . ∆x = ∂r2 + (2.18) r r Then the intrinsic form of harmonic map equation in (2.11) implies that a map κ = (κ1 , . . . , κm−1 ) −→ λ = λ(κ) = (λ1 , . . . , λm−1 ) is a (local) harmonic map if ∆M λa +

m−1  m−1 

a γbc (λ)σ jk (κ)∂κk λb ∂κj λc = 0,

(2.19)

j,k=1 b,c=1

where a = 1, . . . , m − 1. The embedding Sm−1 ⊂ Rm enables us to consider the corresponding diffeomorphism λ ∈ Rm−1 −→ θ = θ(λ) ∈ Sm−1

(2.20)

onto a small neighborhood which takes a small neighborhood of the origin in R on the sphere Sm−1 . Then the equation (2.17) shows that a map m−1

κ = (κ1 , . . . , κm−1 ) −→ θ = θ(κ) = (θ1 , . . . , θm ) ∈ Sm−1 is a (local) harmonic map if ∆Sm−1 θ = −Kθ,

(2.21)

where K > 0 is a constant. Lemma 2.1. Let κ = (κ1 , . . . , κm−1 ) −→ θ = θ(κ) = (θ1 , . . . , θm ) ∈ Sm−1 be a local C 2 solution to ∆Sm−1 θ = −Kθ,

(2.22)

for some constant K > 0. Then K=

m−1 

σ jk (κ)∂κj θ, ∂κk θRm

(2.23)

j,k=1

and K=

m−1  m−1 

σbc (λ)σ jk (κ)∂κj λb ∂κk λc ,

(2.24)

b,c=1 j,k=1

where and θ

−1

λ(κ) = θ−1 θ(κ) is the inverse diffeomorphism to (2.20).

(2.25)

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Piero D’Ancona and Vladimir Georgiev

Remark 2.1. The map (2.25) in the above lemma can be extended as a map ω ∈ Sm−1 −→ λ(ω) = (λ1 (ω), . . . λm−1 (ω)), since κ = (κ1 , . . . , κm−1 ) are local coordinates on S

m−1

(2.26)

.

Proof. Multiplying the equation (2.22) by θ, we get K = −∆Sm θ, θRm , where (see (2.8)) ∆M =

m−1  j,k=1

√ 1 √ ∂κj σ σ jk ∂κk . σ

(2.27)

From the relation θ(κ), θ(κ)Rm = 1 we obtain ∂κk θ(κ), θ(κ)Rm = 0 so that K =−

m−1 

σ jk ∂κj ∂κk θ, θRm =

j,k=1

m−1 

σ jk ∂κj θ, ∂κk θRm

j,k=1

and this proves the first relation (2.23). The second relation (2.24) follows from σbc (λ) = ∂λb θ(λ), ∂λc θ(λ)Rm and the chain rule ∂κj θ =

m−1 

∂λb θ ∂κj λb .

b=1

This completes the proof of the lemma.




To construct the solutions of equation (2.21) we can follow the idea from [17] and look for polynomial functions x = (x1 , . . . , xm ) ∈ Rm −→ P (x) = y = (y1 , . . . , yn ) ∈ Rn ,

(2.28)

such that P (x) = (P P1 (x), . . . , Pn (x)) and Pj (x) are homogeneous polynomials of order L ≥ 1, harmonic in x, i.e., ∆Rm Pj (x) = 0

(2.29)

P1 (x))2 + · · · + (P Pn (x))2 = 1. (x1 )2 + · · · + (xm )2 = 1 ⇒ (P

(2.30)

and such that By homogeneity, (2.30) is a consequence of  L Pn (x))2 = (x1 )2 + · · · + (xm )2 . (P P1 (x))2 + · · · + (P

(2.31)

Once the above problem (2.29) and (2.31) is solved, we can introduce polar coordinates x r = |x|, ω = |x|

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and set u(ω) = P (ω). Using the decomposition of the Laplace operator together with (2.29) and the relation Pj (x) = rL Pj (ω) we can rewrite (2.29) as rL−2 (L(L − 1) + (m − 1)L + ∆Sm−1 ) Pj (ω) = 0 so that u(ω) = P (ω) satisfies ∆Sm−1 u = −L(L + m − 2)u

(2.32)

and the equation (2.21) is satisfied with µ = L(L + m − 2). First, we consider the case n = m = 2. Then we can take κ ∈ [0, 2π) as a local coordinate on M = S1 while λ ∈ [0, 2π) is the local coordinate on N = S1 . We have simply ∆S1 = ∂κκ and setting u(κ) = (cos λ , sin λ), λ = λ(κ), the equation (2.32) becomes − sin λ ∂κκ λ − cos λ (∂κ λ)2 cos λ ∂κκ λ − sin λ (∂κ λ)2

= −L2 cos λ, = −L2 sin λ.

An obvious solution is λ = Lκ. An alternative approach to the solution of the system (2.29), (2.31) can be found using the embeddings S1 ⊂ R1+1 = C. If x1 , x2 are the coordinates on R2 and we can define the polynomial vector-valued function z = x1 + ix2 −→ P (z) = z L . Since ∆C = ∂z ∂z¯, we see that P (z) are harmonic polynomials of order L so that (2.29) is satisfied. Property (2.31) follows from the obvious relation |z L |2 = |z|2L . For L = 2 we obtain in particular P1 (x) = (x1 )2 − (x2 )2 , P2 (x) = 2x1 x2 .

(2.33)

Next, we consider the case m = n = 3. For L = 1 we can take Pj (x) = xj and see that (2.29) and (2.31) are satisfied. For L = 2 we use the argument of the previous case m = n = 2 and see that all polynomials (see (2.33)) (x2 )2 − (x3 )2 , (x3 )2 − (x1 )2 , (x1 )2 − (x2 )2 as well as x1 x2 , x2 x3 , x3 x1

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Piero D’Ancona and Vladimir Georgiev

are harmonic. For this we choose   P1 (x) = a (x2 )2 − (x3 )2 + b (x1 x2 + x2 x3 + x3 x1 ) ,   P2 (x) = a (x3 )2 − (x1 )2 + b (x1 x2 + x2 x3 + x3 x1 ) ,   P3 (x) = a (x1 )2 − (x2 )2 + b (x1 x2 + x2 x3 + x3 x1 ) , where a, b are suitable constants chosen so that (2.31) is fulfilled. Note that P2 (x))2 + (P P3 (x))2 (P P1 (x))2 + (P

  = 2a2 ((x1 )4 + (x2 )4 + (x3 )4 ) + (3b2 − 2a2 ) (x1 x2 )2 + (x2 x3 )2 + (x3 x1 )2 . Comparing this relation with  2 (x1 )2 + (x2 )2 + (x3 )2

  = ((x1 )4 + (x2 )4 + (x3 )4 ) + 2 (x1 x2 )2 + (x2 x3 )2 + (x3 x1 )2 ,

we see that it is sufficient to take 2a2 = 1,

3b2 − 2a2 = 2,

i.e., 1 a= √ , 2

b = 1.

With this choice we have

2  (P P1 (x))2 + (P P2 (x))2 + (P P3 (x))2 = (x1 )2 + (x2 )2 + (x3 )2 ,

so (2.31) is satisfied with L = 2. For higher-dimensional case L ≥ 3 or for n ≥ m ≥ 3 the existence of harmonic polynomial maps satisfying (2.29) and (2.31) is discussed in [17]. For our considerations concerning the concentration of local energy for two-dimensional wave maps the case alone n = m = 2 is sufficient. 2.3. Equivariant wave maps and construction of special solutions In this section we shall derive briefly the wave map equation and shall construct a special class of equivariant wave maps that solve the inhomogeneous problem (2.3). The equation (2.1) is the Euler–Lagrange equation related to the density ∂α u, ∂ α ug(u) ,

(2.34)

which in a small neighborhood of a fixed u0 ∈ N has the form ∂α u, ∂ α ug(u) = hαβ gab ∂α ua ∂β ub . Here and below the Greek indices α, β run from 0 to m, while the Latin indices a, b, c, d run from 1 to n. A summation convention for repeated indices is also used. The corresponding Lagrangian is given by:
hαβ gab ∂α ua ∂β ub . (2.35) L[u] = M

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13

Since we assumed M to be the Minkowski space R1+m with the flat metric h = diag(−1, 1, . . . , 1), we can simplify the Lagrangian to




L[u] = R1+m

gab ∂ α ua ∂α ub .

(2.36)

Then the Euler-Lagrange equations become −2∂α (gab ∂ α ub ) + ∂α uc ∂ α ub ∂a gbc = 0,

(2.37)

or equivalently 1 −gab ∂α ∂ α ub − ∂c gab ∂α uc ∂ α ub + (∂a gbc ∂α uc ∂ α ub ) = 0. 2 In terms of D’Alembertian we may write 1 gab 2ub + ∂c gab ∂α uc ∂ α ub − (∂a gbc ∂α uc ∂ α ub ) = 0, (2.38) 2 where 2 = −∂α ∂ α = ∂02 − ∂12 − · · · − ∂n2 . The Christoffel symbols are given by the expressions 1 (2.39) Γc;ab = (∂a gbc + ∂b gac − ∂c gab ). 2 If we write 1 ∂c gab ∂α uc ∂ α ub = (∂c gba ∂α uc ∂ α ub + ∂b gac ∂α uc ∂ α ub ) 2 and use the above expression of Christoffel symbols, then we arrive at the following equation gab 2ub + Γa;bc ∂α ub ∂ α uc = 0. Raising the index a, we obtain 2ua + Γabc ∂α ub ∂ α uc = 0.

(2.40)

In order to handle the inhomogeneous case, a minor modification of the density (2.34) is sufficient: ∂α u, ∂ α ug(u) + F, ug(u) ,

(2.41)

where F : x = (x0 , x1 , . . . , xm ) ∈ R1+m → F (x) ∈ Tu(x) N is the given source term. The corresponding inhomogeneous problem has the form 2ua + Γabc ∂α ub ∂ α uc = F a .

(2.42)

As in the previous section we can rewrite these equations in extrinsic form. To this purpose assume that N is a n-dimensional surface in Rn+1 with metric induced by the Euclidean metric on Rn+1 . Thus u is a d = n + 1-dimensional vector u = (u1 , . . . , ud ); on the other hand, on N we can take local coordinates y 1 , . . . , y n so that N is described locally by a chart u = u(y),

y ∈ Y ⊂ Rn .

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Piero D’Ancona and Vladimir Georgiev

The Riemannian metric g on N is induced by the Euclidean metric on Rd (see (2.12) of the previous section) gab = ∂ ∂ya u, ∂yb uRd ,

(2.43)

where ·, ·Rd is the scalar product in R . Then the wave map (locally) is a function d

x = (x0 , x1 , . . . , xm ) ∈ X ⊂ Rm+1 −→ y = y(x) ∈ Rn defined in a small neighborhood X of the origin in Rn+1 , satisfying the intrinsic equation (2.42), i.e., 2y a + Γabc ∂α y b ∂ α y c = F a . (2.44) It is easy to verify that the wave map v(x) := u(y(x)), x ∈ X satisfies the extrinsic equation m 
v + hαβ B(v)(∂α v, ∂β v) = 0,

(2.45)

α,β=0

where B(p) : Tp N × Tp N → Tp N ⊥ is the second fundamental form of N ⊂ Rd . We recall the explicit form (2.14) of the second fundamental form from the previous section: n  bac (u)v a wc ν(u) (2.46) B(u)(v, w) = a,c=1

for any two vectors v, w ∈ Tp N , and with coefficients bac defined as follows (see (2.15)) bac = −∂ ∂ya ∂yc u(y), ν(u(y))Rd , a, c = 1, . . . , n; (2.47) ν(u) denotes as usual the unit normal at u ∈ N. To verify the above claim it is sufficient to rewrite the energy functional in (2.35) as follows
L[u] = hαβ ∂α u(y(x))∂ ∂β u(y(x))Rd dx. (2.48) X

Taking the variation of this integral over u ∈ H 1 , under the constraint u(y) ∈ N we obtain the Euler-Lagrange equation with Lagrange multiplier 2u = −µν(u), (2.49) where the Lagrange multiplier µ can be obtained by scalar multiplication with u: µ = hαβ ∂α ∂β u, νRd . Using the property ∂yk u ∈ Tu (N ), ν(u) ∈ Tu (N )⊥ ,

Wave Maps

15

we get µ = hαβ  ∂ya ∂yc u, νRd ∂α y a ∂β y c . Now the definition (2.15) of the second fundamental form leads to (2.13). In the case when N = Sn , we have ν(u) = u, and equation (2.45) simplifies to m  hαβ ∂α u, ∂β Rn+1 u = 0. (2.50) 2 u− α,β=0

We now recall the equivariant wave map ansatz. Assume that N is a smooth n-dimensional rotationally symmetric, wrapped product manifold defined as N = {(φ, λ); φ ∈ (0, φ∗ ), λ ∈ Sn−1 } with metric dφ2 + g(φ)2 dσ 2 , where dσ is the standard metric on S 2

n−1

(2.51)

which we shall denote by

2

dσ = σjk (λ)dλj dλk , while (λ1 , . . . , λn−1 ) are the local coordinates on Sn−1 . In these coordinates we have gφλj = 0, gφφ = 1, gλi λj = g 2 (φ)σij (λ). (2.52) If at least two of the indices a, b, c are equal to φ, then (2.52) implies that Γa,bc = 0. If only one of indices a, b, c is φ, then Γφ,λi λj = −g  (φ)g(φ)σij and Γλi ,λj φ = g  (φ)g(φ)σij . Finally, Γλi ,λj λk = g 2 (φ)γi,jk where  1 γi,jk = ∂λj σik + ∂λk σij − ∂λi σj k , 2 are the Christoffel symbols for the metric σ. The equivariant ansatz is the following: a wave map u, expressed in the coordinates of N as u = (φ, λ), is equivariant if φ(t, x) = φ(t, r),

λj = λj (ω),

(2.53)

where j = 1, . . . , n − 1 ω ∈ Sm−1
→ λj (ω) ∈ R, is the map of (2.25). Recall that this map in the local coordinates κ = (κ1 , . . . , κm−1 ) on S

m−1

defines a solution to the equation ∆Sm−1 λj + Kλj = 0,

where K = L(L + m − 2), L ≥ 1 is an integer, and K = σbc (λ)σ jk (κ)∂κj λb ∂κk λc , due to (2.24) in Lemma 2.1. Choosing a = φ in (2.40) we obtain 2φ + Γφλb λc (u)∂α λb ∂ α λc = 0, where

Γφλb λc (u) = −g  (φ)g(φ)σbc (λ).

(2.54)

16

Piero D’Ancona and Vladimir Georgiev Note that ∂α λb ∂ α λc = σ jk (κ)

∂κj λb ∂κk λc r2

so from (2.54) we find Kg  (φ)g(φ) = 0. r2 The corresponding inhomogeneous problem is of course 2φ +

(2.55)

Kg  (φ)g(φ) = f. (2.56) r2 In the special case when the target is the two-dimensional sphere S2 , the metric on S2 has the form dφ2 + sin2 φ dλ2 . Let u : R × R2 → S2 be an equivariant wave map. Then u = (u1 , u2 , u3 ) with 2φ +

u1 = cos(φ) cos(λ), u2 = cos(φ) sin(λ), u3 = sin(φ).

(2.57)

Introducing polar coordinates (r, κ) in R2 , we have x1 = r cos κ, x2 = r sin κ; so the equivariant ansatz (2.53) shows that φ = φ(t, r) satisfies (2.55) and λ = λ(κ) is a harmonic map between S1 and S1 . The simplest possible choice of λ is clearly the identity map λ(κ) = κ, and the equation (2.56) becomes then sin(2φ) = f, (2.58) 2r2 where 2φ = (∂ ∂t2 − ∂r2 − 1r ∂r )φ. The vector-valued function u in (2.57) solves the equation   utt − ∆u + |ut |2 − |∇x u|2 u = F, (2.59) provided φ solves the inhomogeneous equation (2.58). Indeed, we have the relations 2φ +

ut = ∂t φ ∂φ u, utt =

−φ2t u

ur = ∂r φ ∂φ u, ∂r2 u

+ φtt ∂φ u,

(2.60) 2

= − (∂ ∂r φ) u + ∂rr φ∂φ u

and the representation formula 1 1 2 = ∂t2 − ∂r2 − ∂r − 2 ∂κ2 . r r From

(2.61)

|∇x u|2 = |∂ ∂r u|2 + r−2 |∂κ u|2

and (2.57) we get |∇x u|2 = |∂ ∂r φ|2 +

cos2 φ , r2

|∂ ∂t u|2 = |∂ ∂t φ|2

whence |∂ ∂t u|2 − |∇x u|2 = φ2t − φ2r −

cos2 φ . r2

(2.62)

Further, from (2.60) and (2.61) we find 2u = −φ2t u + φtt ∂φ u + φ2r u − φrr ∂φ u −

φr 1 ∂φ u − 2 ∂κκ u. r r

(2.63)

Wave Maps

17

To conclude our computation, we need the simple Lemma 2.2. The function u satisfies the identity sin(2φ) ∂κκ u = − (cos φ)2 u + ∂φ u. 2 Proof. Consider the vectors e = (cos λ, sin λ, 0) and e3 = (0, 0, 1). Then

(2.64)

u = e cos φ + e3 sin φ, ∂φ u = −e sin φ + e3 cos φ and from these relations we get immediately e = u cos φ − ∂φ u sin φ. This relation and the identity ∂κκ u = −e cos φ imply (2.64). This completes the proof.
Combining the above Lemma and (2.62), we obtain Corollary 2.1. If u is defined by (2.57), then the following relation holds:    2  sin(2φ) 2 2u + |ut | − |∇x u| u = 2φ + ∂φ u. 2r2

(2.65)

We conclude this section by a final remark. If φ solves the inhomogeneous Cauchy problem for (2.56), i.e., sin(2φ) = f, 2r2 φ(0, x) = φ0 (|x|, ∂t φ0 (0, x) = φ1 (|x|) 2φ +

(2.66)

then we get immediately a solution of the corresponding extrinsic problem   (2.67) utt − ∆u + |ut |2 − |∇x u|2 u = F, u(0, x) = u0 (x), ∂t u(0, x) = u1 (x), where u0 = u(φ0 ), u1 = u(φ1 ), F = f ∂φ u (2.68) with u = u(φ), defined according to (2.57). An analogous connection exists between the Sobolev spaces associated with these two problems (see Lemmas 5.1 and 5.2 in the Appendix to Chapter 5).

3. Local existence result for equivariant wave maps 3.1. Introduction The wave map system can be written (locally) as a semilinear problem for the wave equation utt − ∆u = Q(u, ∇u) + F. (3.1) Here n u = u(t, x), F = F (t, x) : Rt × Rm x −→ R , ∇u = (∂ ∂t u, ∂x1 u, . . . , ∂xm u),

18

Piero D’Ancona and Vladimir Georgiev

while Q(u, v) is the nonlinear term, i.e., a smooth function Q : u × v ∈ Rn × Rn(m+1) −→ Rn , satisfying the condition Q(0, 0) = 0, ∇u Q(0, 0) = 0, ∇v Q(0, 0) = 0.

(3.2)

In this chapter we study the existence of a local solution to the Cauchy problem for (3.1) with initial data u(0, x) = u0 (x), ∂t u(0, x) = u1 (x).

(3.3)

Clearly, by rewriting the wave map equation in this local form, we are dropping the geometric constraint that u should take its values on the target. This simplification is crucial for the study of the local existence for the Cauchy problem with data u0 ∈ H s , u1 ∈ H s−1 .

(3.4)

Here and below H s = H s (Rm ) is the classical Sobolev space with norm f H s = (1 + | · |)s fˆL2 for any s ∈ R, while fˆ(ξ) denotes the Fourier transform of f (x)
ˆ e−ixξ f (x)dx. f (ξ) = Rm

The linear analogue of the Cauchy problem (3.1), (3.3) is the wave equation utt − ∆u = F, u(0, x) = u0 (x), ut (0, x) = u1 (x).

(3.5)

The standard method to solve this problem in Sobolev spaces is an application of the Fourier transform with respect to space variables
e−ixξ u(t, x)dx (3.6) u˜(t, ξ) = Rm

which gives immediately the standard a priori energy estimate   uC([0,T ];H s) ≤ C u0 H s + u1 H s−1 + F L1 ((0,T );H s−1 ) .

(3.7)

This leads to the following existence result: Lemma 3.1. If s ∈ R, T > 0, and the data satisfy u0 ∈ H s , u1 ∈ H s−1

(3.8)

F ∈ L ((0, T ); H

(3.9)

1

s−1

),

then there exists a unique solution u ∈ C([0, T ]; H s ) to the Cauchy problem (3.5).

(3.10)

Wave Maps

19

Turning back to the nonlinear problem (3.1), we see that a nonlinear analogue of Lemma 3.1 should be a consequence of the following property: if R > 0, then there exists C = C(R) > 0, such that for any T, 0 < T ≤ 1, the conditions uC([0,T ];H s ) ≤ R, wC([0,T ];H s ) ≤ R

(3.11)

imply that Q(u, ∇u) − Q(w, ∇w)L1 ((0,T );H s−1 ) ≤ C(R)T δ u − wC([0,T ];H s ) ,

(3.12)

δ

for some δ > 0. Indeed, the factor T enables one to apply a contraction argument, when T > 0 is sufficiently small. Note that for the typical example, when Q(u, ∇u) = Q(∇u) is a quadratic form in ∇u, we have the following estimate Q(u, ∇u) − Q(w, ∇w)L1 ((0,T );H s−1 ) ≤ C(R)T u − wC([0,T ];H s ) provided s>1+

n 2

(3.13) (3.14)

and uC([0,T ];H s) ≤ R, wC([0,T ];H s ) ≤ R. However, in some cases a special structure of the nonlinear term allows to improve the condition (3.14) and to construct local solutions with a lower regularity. To this end, we shall follow the approach developed by Bourgain [2], [3], for the Schrodinger ¨ type equations and by Klainerman, Machedon and Selberg [19], [20], [21] for the wave type equations. This approach is based on the use of the full space-time Fourier transform

u ˆ(τ, ξ) = e−itτ −ixξ u(t, x)dxdt. R

Rm

The crucial tool of this method is the modified Sobolev space H s,θ = H s,θ (Rm+1 ) with norm

u2H s,θ = |τ | + |ξ|2s |τ | − |ξ|2θ | u(τ, ξ)|2 dξdτ, (3.15) R

where

Rm

 1/2 r = 1 + r2 .

It is clear that this norm is invariant for translations with respect to time Ta u(t, x) = u(t + a, x), or translations in space. Indeed, we can write −iτ a u (τ, ξ), T a u(τ, ξ) = e

so that

 u(τ, ξ)|. Ta u(τ, ξ) = |

20

Piero D’Ancona and Vladimir Georgiev

3.2. Localization in time In this section we follow the approach in [21]. We begin by recalling some basic properties of the spaces H s,θ . It is not difficult to see that the embedding 1 H s,θ →  C([0, T ]; H s (Rn )), s ∈ R, θ > , T > 0 2 is continuous. To verify this property we recall the relation
∞ 1 u ˜(t, ξ) = eitτ u ˆ(τ, ξ)dτ 2π −∞ so that, applying the Cauchy inequality, we find for any θ > 1/2
∞ 2 2 u(t, ξ)| ≤ C |τ | + |ξ|2s |τ | − |ξ|2θ |ˆ u(τ, ξ)| dτ ξ2s |˜

(3.16)

(3.17)

(3.18)

−∞

and this relation implies (3.16) after an integration in ξ. An essential property is that localization in time is a bounded operation. Indeed, given any function in the Schwartz class ϕ(t) ∈ S(Rt ), the multiplication operator u ∈ H s,θ
→ ϕu ∈ H s,θ is continuous. But also a stronger property holds: frequently, the localization in time is realized by means of a function   t ϕT (t) = ϕ , T where the parameter T is a positive number and ϕ(s) is a smooth compactly supported non-negative function such that

1, if |s| ≤ 1; ϕ(s) = (3.19) 0, if |s| ≥ 2. Then we have the following uniform estimate       ϕ t u   s,θ ≤ CuH s,θ , T

(3.20)

H

where C > 0 is a constant independent of the parameters T > 1, s, θ. Indeed, we have the relation
∞ ϕ u(τ, ξ) = C ϕ T (τ − τ1 ) u(ττ1 , ξ)dτ1 . T −∞

Now ϕ T (τ ) = T ϕ(τ ˆ T) and by our assumption on ϕ we have |ϕ T (τ )| ≤ T

CN τ T N

(3.21)

Wave Maps

21

for any real N ≥ 0. Then, estimate (3.20) is a consequence of the following estimate |τ | + |ξ|s |τ | − |ξ|θ τ1 | + |ξ|s |ττ1 | − |ξ|θ N1 −N |τ ≤ C(1 + T ) |τ − τ1 |T N |τ − τ1 |T N1

(3.22)

which is valid for all positive integers N, N1 and all reals s > 0 and θ > 0 such that |s + θ| ≤ N − N1 . N ≥ N1 , We shall prove inequality (3.22) by means of the the following technical Lemmas. Lemma 3.2. If a, b are real numbers with a ≥ b and a ≥ 0, then there exists a constant C = C(a, b) > 0 so that x ± yb ≤ Cxa yb

(3.23)

for any real numbers x, y. If in addition we have the inequality |b| ≤ a, then we can choose C = C(a) independent of b. Proof. We shall establish only the estimate with sign + in (3.23), since the case of opposite sign is similar. The assertion is clear when |x| ≤ 2 and |y| ≤ 2. If |x| ≤ 1 and |y| ≥ 2, then x ∼ 1, x + y ∼ y and (3.23) takes the form yb ≤ Cyb which is obvious. If |x| ≥ 2 and |y| ≤ 1, then y ∼ 1, x + y ∼ x and (3.23) takes the form xb ≤ Cxa and recalling the assumption a ≥ b we see that also in this case the inequality is true. Finally, for |x| ≥ 2 and |y| ≥ 2, we can use the inequality |x + y| ≤ C|x| |y| whence x + yb ≤ Cxb yb ≤ Cxb yb

(3.24)

and this completes the proof of the Lemma for the case 0 ≤ b ≤ a. If −a ≤ b ≤ 0, then we can use the following modification of the inequality (3.24) x + yb ≤ Cx−b yb ≤ Cxa yb (3.25) and this completes the proof of the Lemma.




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Piero D’Ancona and Vladimir Georgiev

Lemma 3.3. If the real numbers s, s1 , N, N1 satisfy the conditions N ≥ N1

s1 ≥ N1 − N,

s ≤ N − N1 ,

s − s1 ≤ 0,

(3.26)

then there exists a constant C0 = C0 (N, N1 ) > 0, such that |τ | − As τ1 | − As1 N1 −N |τ ≤ C (1 + T ) 0 |τ − τ1 |T N |τ − τ1 |T N1

(3.27)

for any T > 0 and any real numbers τ, τ1 , A. Proof. The proof is divided in a few cases. If |τ | ≤ 2 and |ττ1 | ≤ 2, then |τ | − A ∼ |ττ1 | − A ∼ A and the conditions N ≥ N1 , s1 ≥ s imply the estimate As C0 As1 ≤ N |τ − τ1 |T  |τ − τ1 |T N1 which in turn implies (3.27). If |τ | ≤ 1 and |ττ1 | ≥ 2, then |τ | − A ∼ A, |τ − τ1 | ∼ |ττ1 | ∼ ττ1  and using the chain of inequalities |τ | − As |τ − τ1 |T N

≤ ≤

1 C|τ | − As N 1 |τ − τ1 |T  |τ − τ1 |T N −N1 T N1 −N C|τ | − As , N 1 |τ − τ1 |T  ττ1 N −N1

we see that in order to prove (3.27) it is sufficient to show that (recall that s1 ≥ s) As ≤ C|ττ1 | − As . ττ1 N −N1 This last inequality follows from Lemma 3.2 and the assumptions N ≥ N1 , N − N1 ≥ s. If |τ | ≥ 2 and |ττ1 | ≤ 1, then |ττ1 | − A ∼ A, |τ − τ1 | ∼ |τ | ∼ τ  and using the inequalities |τ | − As |τ − τ1 |T N

≤ ≤

1 C|τ | − As |τ − τ1 |T N1 |τ − τ1 |T N −N1 T N1 −N C|τ | − As , N |τ − τ1 |T  1 τ N −N1

we see that (3.27) follows from (recall that s1 ≥ s) |τ | − As ≤ CAs ; τ N −N1

Wave Maps

23

now, the last inequality follows from Lemma 3.2 and the assumptions N ≥ N1 , N − N1 ≥ s. It remains to consider the case |τ | ≥ 2 and |ττ1 | ≥ 2. Here we have two additional subcases. When |τ − τ1 | ≤ 1, then |τ | − A ∼ |ττ1 | − A and from the conditions s1 ≥ s and N ≥ N1 we obtain |τ | − As |τ | − As1 ≤ C0 N |τ − τ1 |T  |τ − τ1 |T N1 so (3.27) is fulfilled. On the other hand, when |τ − τ1 | ≥ 1, then we take advantage of the inequalities |τ | − As |τ − τ1 |T N

≤ ≤

1 C|τ | − As |τ − τ1 |T N1 |τ − τ1 |T N −N1 T N1 −N C|τ | − As . |τ − τ1 |T N1 τ − τ1 N −N1

As before, (3.27) follows from |τ | − As ≤ C|ττ1 | − As , τ − τ1 N −N1 and the last inequality is a consequence of Lemma 3.2 and the assumptions N ≥ N1 , N − N1 ≥ s. The proof of the lemma is concluded.
Now we are in the position to verify the inequality (3.22). Proposition 3.1. If s, θ, N, N1 are real numbers such that N ≥ N1 ,

|s + θ| ≤ N − N1 ,

(3.28)

then there exists a positive constant C = C(N, N1 ), such that τ1 | + |ξ|s |ττ1 | − |ξ|θ |τ | + |ξ|s |τ | − |ξ|θ N1 −N |τ ≤ C(1 + T ) |τ − τ1 |T N |τ − τ1 |T N1

(3.29)

for any T > 0 and any τ, τ1 ∈ R, ξ ∈ Rn . Proof. Estimate (3.29) is an immediate consequence of estimate (3.27) of the preceding lemma.


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Piero D’Ancona and Vladimir Georgiev

3.3. Estimates for the homogeneous problem The spaces H s,θ can be regarded as Sobolev spaces “adapted” to the wave equation, as the following properties show. Consider the linear homogeneous wave equation utt − ∆u = 0, t ∈ R, x ∈ Rn u(0, x) = u0 , ut (0, x) = u1 (x).

(3.30)

The classical energy estimates, namely u(t, ·)H s ≤ C (u0 H s + u1 H s−1 ) ,

(3.31)

where 0 ≤ t ≤ T0 and C > 0, imply for any s ∈ R and T0 > 0 the existence of a solution u0 ∈ H s , u1 ∈ H s−1 ⇒ u ∈ C([0, T0 ]; H s ). (3.32) This solution, properly localized, belongs to the space H s,θ :   t u0 ∈ H s , u1 ∈ H s−1 ⇒ ϕ u ∈ H s,θ , θ ∈ (1/2, n − 1/2), T

(3.33)

provided ϕ(s) is a smooth compactly supported non-negative function satisfying (3.19), and T > 0. Indeed, the solution to the Cauchy problem (3.30) can be expressed using Dirac’s delta in the following way: u ˆ(τ, ξ) = c1 δ(τ 2 − |ξ|2 ) |ξ| u 0 (ξ) + c2 δ(τ 2 − |ξ|2 ) u 1 (ξ)

(3.34)

and this relation shows that we need the cut-off function ϕ in (3.33). More precisely, we have: Lemma 3.4. Assume s ≥ 0,

θ > 0.

Then for any T > 0 the condition u0 ∈ H s , u1 ∈ H s−1 implies that       ϕ t Dt,x u   T

≤ CT 1−N (1 + T s ) (u0 H s + u1 H s−1 ) ,

(3.35)

H s−1,θ

for all 1 N >s+θ+ . 2 Proof. Write for brevity

  t ϕT (t) = ϕ . T

Then ϕ T (τ ) = T ϕ(τ ˆ T) and we can apply estimate (3.21) for any real N ≥ 0.

(3.36)

Wave Maps

25

Consider now the case u1 = 0 (the complementary case is similar). Then the relation (3.34) (with u1 = 0) takes the form ϕ T (τ − |ξ|) + ϕ T (τ + |ξ|)) u 0 (ξ); T u(τ, ξ) = C (ϕ

(3.37)


using (3.21), we obtain immediately the claimed inequality (3.35). Combining the above estimate and (3.31) we can derive the following: Corollary 3.1. For any s≥0

θ > 0, C0∞ (R)

and for any ϕ(t) ∈ one can find a constant C = C(ϕ, s, θ), such that the solution u of the free wave equation with data u0 ∈ H s ,

u1 ∈ H s−1

satisfies ϕuH s,θ ≤ C (u0 H s + u1 H s−1 ) .

(3.38)

Proof. We can reduce the proof to the case when suppξ u 0 (ξ) ∪ suppξ u 1 (ξ) ∪ suppξ u (τ, ξ) ⊆ {|ξ| ≤ 1}. Then we have

  L2 L2 . ϕuH s,θ ∼ τ s+θ ϕu τ

Since

(3.39)


|τ |≥1

ξ

 s+θ 2 2 τ  ϕu(τ, ·)L2 dτ ≤ C ϕ∂ ∂t uH s−1,δ , ξ

an application of Lemma 3.4 implies that
2  s+θ τ  ϕu(τ, ·)L2 dτ ≤ C (u0 H s + u1 H s−1 ) . |τ |≥1

ξ

Thus, it only remains to show that
  2 ϕu(τ, ·)L2 dτ ≤ C u0 2H s + u1 2H s−1 , ξ

|τ |≤1

and this follows immediately from the Plancherel identity

2 2 ϕu(τ, ·)L2 dτ = C ϕ(t)˜ u(t, ·)L2 dt ξ

R

R

ξ

and from estimate (3.31), which under the assumption (3.39) takes the form u(t, ·)L2 ≤ C (u0 L2 + u1 L2 ) . This completes the proof.

(3.40)


26

Piero D’Ancona and Vladimir Georgiev

3.4. Estimates for the non-homogeneous problem The relation established in the preceding section between the homogeneous wave equation and the H s,θ spaces has a partial extension to the non-homogeneous case. Indeed, consider the non-homogeneous problem utt − ∆u = F,

u(0, x) = ut (0, x) = 0

(3.41)

under the assumptions F ∈ H s−1,θ−1 ,

suppt F ⊆ [−1, 1],

s > 1, θ > 1/2.

(3.42)

A first useful property is the following: Theorem 3.1. Let n ≥ 2. If F satisfies the assumptions (3.42) and u is the solution to (3.41), then for any t ≥ 0 we have u(t, ·)H s ≤ CF H s−1,θ−1 .

(3.43)

Proof. Following the idea of the proof of Corollary 3.1, we see that it is sufficient to verify the following inequality ∇u(t, ·)H s−1 ≤ CF H s−1,θ−1 . Consider the function v(t, x) defined as
t sin((t − t1 )|ξ|) ˜ F (t1 , ξ)dt1 , v˜(t, ξ) = |ξ| −∞

(3.44)

(3.45)

where v˜(t, ξ) denotes the Fourier transform with respect to space variables (see (3.6)). Then v solves the equation 2v = F. If we can verify the following inequality similar to (3.44) ∇v(0, ·)H s−1 ≤ CF H s−1,θ−1 ,

(3.46)

then we are in the position to conclude that v(0, ·) ∈ H s , ∂t v(0, ·) ∈ H s−1 . Now an application of the energy inequality will complete the proof, since the difference u − v satisfies the homogeneous wave equation 2(u − v) = 0 with initial data ∇x (u(0, ·) − v(0, ·)) ∈ H s−1 , ∂t (u0 (t, ·) − v(0, ·)) ∈ H s−1 . To prove (3.46) consider the operators
t
−ixξ e e±i(t1 −t)|ξ| F˜ (t1 , ξ)dt1 dξ. I(F )(t, x) = Rn

−∞

(3.47)

Wave Maps

27

From (3.45) it is sufficient to show that the operators

0 −ixξ I(F )(0, x) = e e±it1 |ξ| F˜ (t1 , ξ)dt1 dξ,

(3.48)

−∞

Rn

maps H s−1,θ−1 into H s−1 . Taking, e.g., the sign + (the other case is identical), we obtain
0
)(0, ξ) = c I(F eit1 |ξ| F˜ (t1 , ξ)dt1 . (3.49) −∞


1 F˜ (t1 , ξ) = eit1 τ F (τ, ξ)dτ 2π R and the well-known identity (in a suitable distributional sense)  
0 1 + πδ(τ + |ξ|) eit1 (τ +|ξ|) dt1 = −iPv τ + |ξ| −∞ From the relation

we see that it is sufficient to prove that the operators

Jj (F )(ξ) = ξs−1 vj (τ + |ξ|)F (τ, ξ)dτ = ξs−1 vj (τ )F (τ − |ξ|, ξ)dτ, j = 1, 2, R

R

(3.50)

  1 , τ are bounded operators from H s−1,θ−1 into L2 provided F satisfies the properties (3.42). We also recall that the action of the distribution Pv(1/τ ) may be written as  
∞ 1 ϕ(τ ) − ϕ(−τ ) Pv dτ. , ϕ = τ τ 0 Now, take a function ψ(t) ∈ C0∞ (R) such that ψ(t) = 1 for |t| ≤ 1. Then F (t, x) = ψ(t)F (t, x) and
 − τ1 )F (ττ1 , ξ)dτ1 .  F (τ, ξ) = ψ(τ

where

v1 (τ ) = δ(τ ),

v2 (τ ) = P v

R

Consider first the operator J1 , defined by
s−1  J1 (F )(ξ) = ξ ψ(−|ξ| − τ1 )F (ττ1 , ξ)dτ1 , j = 1, 2.

(3.51)

R

Since ψˆ is a rapidly decreasing function, we can use the inequality | − |ξ| − τ1 | ≥ ||ττ1 | − |ξ|| to find  |ψ(−|ξ| − τ1 )| ≤

CN , ∀N ∈ N. |ττ1 | − |ξ|N

Hence, by the Cauchy inequality we obtain
|J J1 (F )(ξ)|2 ≤ Cξ2s−2 |ττ1 | − |ξ|2θ−2 |F(ττ1 , ξ)|2 dττ1 . R

(3.52)

(3.53)

28

Piero D’Ancona and Vladimir Georgiev

Integrating over ξ, we arrive at the required estimate: J J1 (F )(ξ)L2 ≤ CF H s−1,θ−1 . Next, consider the operator J2 defined by (3.50), i.e.,

 − |ξ| − τ1 )F (ττ1 , ξ)dττ1 dτ. J2 (F )(ξ) = ξs−1 v2 (τ ) ψ(τ R

(3.54)

(3.55)

R

To estimate this double integral it is sufficient to verify the inequality
 − a)dτ ≤ C . v2 (τ )ψ(τ a

(3.56)

R

Indeed, if this inequality is true, we can take a = τ1 + |ξ| and we find
C  − |ξ| − τ1 )dτ ≤ v2 (τ )ψ(τ |ττ1 | − |ξ| .

(3.57)

R

From this inequality and (3.55) we easily derive
|J J2 (F )(ξ)| = ξs−1 |ττ1 | − |ξ|−1 F (ττ1 , ξ)dττ1 ,

(3.58)

so that, by the Cauchy inequality, we obtain as in (3.53)
|J J2 (F )(ξ)|2 ≤ Cξ2s−2 |ττ1 | − |ξ|2θ−2 |F(ττ1 , ξ)|2 dττ1 ,

(3.59)

R

R

for all θ > 1/2. As before, this inequality implies J J2 (F )(ξ)L2 ≤ CF H s−1,θ−1 .

(3.60)

Thus it remains to prove the estimate (3.56). To this end, we start from the following two simple properties of the distribution v2 (τ ) = P v(1/τ ): v2 (ξ) = iπ sgn ξ, C |v2 (τ )χ(τ )| ≤ τ  for any smooth function χ(τ ) ∈ L∞ such that χ(τ ) = 0 near τ = 0. Now, the proof of (3.57) can be reduced to the proof of
 − a)dτ ≤ C , v2 (τ )χ(τ )ψ(τ a

(3.61) (3.62)

(3.63)

R

where a is a real number and χ satisfies one of the following two conditions: either χ ∈ C0∞ (R), or

(3.64)

χ ∈ C ∞ (R), χ(τ ) = 0 for |τ | ≤ 1, χ(τ ) = 1 for |τ | ≥ 2. (3.65) In the case (3.64) we use the property (3.61) and see that the needed inequality (3.63) follows from the estimate
C  − a)e−iτ ξ dτ ≤ χ(τ )ψ(τ (3.66) ξN aN R

Wave Maps

29

for any integer N ≥ 1. This inequality in turn follows easily from the fact that χ and ψ are rapidly decreasing functions in the Schwartz class S(R). In the case (3.65) we use (3.62) and see that it is sufficient to apply the obvious estimate
τ −1 τ − a−N dτ ≤ C , N ≥ 2. (3.67) a R This completes the proof of (3.63) and of the theorem.




Theorem 3.2. Assume that n ≥ 2, F satisfies the assumptions (3.42) and u is a solution to utt − ∆u = F, with initial data u(0, x) = ut (0, x) = 0. Then for any ϕ(t) ∈ C0∞ (R) we have ϕu ∈ H s,θ ,

(3.68)

for all s > n/2 and 1/2 < θ < 1. Proof. As in the proof of the previous theorem we consider the function v(t, x) defined by (3.45). Further, we can localize the space Fourier transform in x so that suppξ F (t, ξ) ⊂ {|ξ| ∼ 2j }, j = 0, 1, 2, . . .

(3.69)

Then we take two smooth compactly supported functions ϕj (t), j = 1, 2, and see that it is sufficient to replace F (t, x) by ϕ2 (t)F (t, x) and show that the function defined by
t sin((t − t1 )|ξ|) v˜(t, ξ) = ϕ2 (t1 )F (t1 , ξ)dt1 , (3.70) |ξ| −∞ satisfies ϕ1 v ∈ H s,θ . (3.71) It is not difficult to show that ϕ  1 v(τ, ξ) = C


R

K(τ, τ1 , ξ)F (ττ1 , ξ)dττ1 ,

(3.72)

where (the integrals are in classical sense and ξ = 0)

sin(t − t1 )|ξ| H(t − t1 ) ϕ1 (t)ϕ2 (t1 ) e−i(tτ −t1 τ1 ) dtdt1 . (3.73) K(τ, τ1 , ξ) = |ξ| R R First, we consider the case when ξ is bounded, i.e., j is bounded in (3.69). Then we can integrate by parts in (3.73) and in this way we find   1 |K(τ, τ1 , ξ)| ≤ C ττ1 τ τ − τ1 N for any integer N ≥ 1; this estimate implies τ M |K(τ, τ1 , ξ)| ≤ Cττ1 M−2 f (τ − τ1 ),

(3.74)

30

Piero D’Ancona and Vladimir Georgiev

for any integer M ≥ 1 and any non-negative function f ∈ L1 (R). In the general case, when j ≥ 2, we shall prove the following variant: |τ | + |ξ|s |τ | − |ξ|θ |K(τ, τ1 , ξ)| ≤ C|ττ1 | + |ξ|s−1 |ττ1 | − |ξ|θ−1 f (τ − τ1 ). (3.75) By an integration by parts argument similar to the one used in the proof of (3.74), we can easily reduce the proof of (3.75) to the case when |τ | ∼ |ξ| ∼ 2j and |ττ1 | ∼ |ξ| ∼ 2j . Then we can rewrite (3.75) as |τ | − |ξ|θ |K(τ, τ1 , ξ)| ≤ C|ξ|−1 |ττ1 | − |ξ|θ−1 f (τ − τ1 ). Using (3.73), we can represent the kernel K(τ, τ1 , ξ) in the form

K(τ, τ1 , ξ) = U (t1 − t)V (t)W (t1 ) dtdt1 , R

(3.76)

(3.77)

R

where U (t) = −

sin t|ξ| H(−t), |ξ|

(3.78)

V (t) = ϕ1 (t)e−itτ ,

(3.79)

W (t) = ϕ2 (t)e−itτ1 .

(3.80)

Now the relation


 (ττ2 )V (ττ2 )W (ττ2 )dττ2 U U (t1 − t)V (t)W (t1 ) dtdt1 = C R

R

R

as well as the identities ˆ (ττ2 ) = − U

1 , (ττ2 + i0)2 − |ξ|2

(3.81)

Vˆ (ττ2 ) = ϕ 1 (ττ2 + τ ),

(3.82)

ˆ (ττ2 ) = ϕ W 2 (ττ2 + τ1 ),

(3.83)

show that it is sufficient to establish the inequality (3.76), with the kernel K replaced by
ˆ (ττ2 )χ(ττ2 ) Kχ (τ, τ1 ) = U ϕ1 (ττ2 + τ ) ϕ2 (ττ2 + τ1 ) dττ2 , (3.84) R

where χ(ττ2 ) is a smooth function. Take first χ(ττ2 ) with support close to ±|ξ|; we may assume for determinacy that the cut-off function χ has support near τ = |ξ|. Then we can take χ(ττ2 ) = χ0 (ττ2 − |ξ|) with χ0 supported near the origin and we can use the following equality (in the sense of distributions) −

1 1 Φ(ττ2 − |ξ|, ξ), χ(ττ2 ) = 2 2 (ττ2 + i0) − |ξ| τ2 + i0 − |ξ|

where Φ(τ, ξ) = −

1 χ0 (τ ) τ + 2|ξ|

(3.85)

Wave Maps

31

is a smooth compactly supported function (with support near τ = 0), such that for any integer N ≥ 0 we have C |∂ ∂τN Φ(τ, ξ)| ≤ N |ξ| with some constant C = C(N ) independent of τ, ξ. Using the change of variables τ2 → σ = τ2 − |ξ|, we get




Kχ (τ, τ1 ) =

R

1 Φ(σ, ξ) ϕ1 (σ + |ξ| + τ ) ϕ2 (σ + |ξ| + τ1 ) dσ. σ + i0

(3.86)

Now we can use a suitable variant of the estimate (3.63). More precisely, given any two smooth rapidly decreasing functions f1 , f2 and any family Φα (σ), α ∈ A, of smooth functions supported in |σ| ≤ 1, which satisfy the estimates |∂ ∂τN Φα (τ )| ≤ CN with a constant C(N ) independent of α ∈ A, we have the estimate
1 C Φα (σ)f1 (σ − a)ff2 (σ − b)dσ ≤ , N σ + i0 a bN

(3.87)

R

for any integer N ≥ 1. It remains to consider the case when χ(ττ2 ) is identically 0 near τ2 = ±|ξ|. Then we have the obvious inequality C ˆ , U (ττ2 )χ(ττ2 ) ≤ |ττ2 | − |ξ||ττ2 | + |ξ| hence (3.84) implies |Kχ (τ, τ1 )| ≤


R

1 C dττ2 |ττ2 | − |ξ||ττ2 | + |ξ| ττ2 + τ N ττ2 + τ1 N

(3.88)

and this estimate leads to the needed estimate (3.76). This completes the proof of the theorem.
3.5. Bilinear estimates for the homogeneous problem in H s,δ The first step in the iteration scheme for the non-linear problem is an estimate for the homogeneous problem utt − ∆u = 0,

u(0, x) = f,

ut (0, x) = 0.

(3.89)

More precisely, given two solutions uj , j = 1, 2, to the free wave equation having initial data uj (0, x) = fj (x), ∂t uj (0, x) = 0, the main result of this section concerns an estimate for the bilinear form n  Q(∇u1 , ∇u2 ) = ∂t u1 (t, x)∂ ∂t u2 (t, x) − ∂xk u1 (t, x)∂ ∂xk u2 (t, x). (3.90) k=1

32

Piero D’Ancona and Vladimir Georgiev

Theorem 3.3. If f1 , f2 ∈ H s , s > n/2, then Q(∇u1 , ∇u2 ) ∈ H s1 −1, θ−1 , whenever s1 + θ < s + 1/2 and θ > 1/2. Proof. f Our starting point is the following representation  Q(∇u 1 , ∇u2 )(τ, ξ)



=C

(3.91)

[(τ − τ1 )ττ1 − (ξ − ξ1 )ξ1 ] u1 (τ − τ1 , ξ − ξ1 ) u2 (ττ1 , ξ1 )dξ1 dττ1 .

Since uj is a solution to (3.89), we have the relation    δ(τ − σ|ξ|)fj (ξ). (3.92) uj (τ, ξ) = c δ(τ − |ξ|)fj (ξ) + δ(τ + |ξ|)fj (ξ) = c σ=±1

Given any ξ, ξ1 ∈ Rn , we can consider δ(τ − τ1 − σ1 |ξ − ξ1 |)δ(ττ1 − σ2 |ξ|),

σ1 , σ2 = ±1,

as a distribution on Rτ × Rτ1 defined as follows: δ(τ − τ1 − σ1 |ξ − ξ1 |)δ(ττ1 − σ2 |ξ|), ψ(τ, τ1 ) = ψ(σ2 |ξ1 | + σ1 |ξ − ξ1 |, σ2 |ξ1 |) for any test function ψ(τ, τ1 ) ∈ C0∞ (R2 ). Having in mind this property as well as the obvious relation  1 2 (τ − τ1 )ττ1 − (ξ − ξ1 )ξ1 = τ − ξ 2 − (τ − τ1 )2 + |ξ − ξ1 |2 − τ12 + |ξ1 |2 , (3.93) 2 we see that it is sufficient to verify the inequality u1 u2 H s1 , θ ≤ Cf1 H s ff2 H s . The representation formula  u 1 u2 (τ, ξ) = c




σ1 ,σ2 =±1

σ1 |ξ−ξ1 |+σ2 |ξ1 |=τ

f1 (ξ − ξ1 )f2 (ξ1 )dξ1

reduces the proof of (3.94) to the following estimate
∞
|τ | + |ξ|2s1 |τ | − |ξ|2θ |II± (τ, ξ)|2 dξdτ ≤ Cf1 2H s ff2 2H s ,

(3.94)

(3.95)

(3.96)

0




where I± (τ, ξ) =

|ξ −ξ1 |±|ξ1 |=τ

Sξ1 . f1 (ξ − ξ1 )f2 (ξ1 )dS

(3.97)

Here and below dS Sξ1 is the surface element on the surfaces S± (τ, ξ) = {ξ1 ; |ξ − ξ1 | ± |ξ1 | = τ }. Note that S+ (τ, ξ) is an ellipsoid for τ > |ξ|, and

∞
P dξ1 = P dS Sξ1 dτ Rn

for any function P ∈ L1 (Rn ).

0

S+ (τ,ξ)

(3.98)

Wave Maps The surface S− (τ, ξ) is a hyperboloid for τ < |ξ|, and
∞

P dξ1 = P dS Sξ1 dτ ξ1 ,ξ 1/2. Thus, the final step of the proof is a consequence of the following lemma: Lemma 3.5. If n ≥ 2 and s > n/2, then for |ξ| < τ we have the inequality

C ξ − ξ1 −2s dS Sξ1 + ξ1 −2s dS Sξ1 ≤ , |τ | − |ξ|2s−n+1 |ξ −ξ1 |+|ξ1 |=τ |ξ −ξ1 |+|ξ1 |=τ (3.107) while for |ξ| > τ > 0 we have

C ξ − ξ1 −2s dS Sξ1 + ξ1 −2s dS Sξ1 ≤ . 2s−n+1 |τ | − |ξ| |ξ −ξ1 |−|ξ1 |=τ |ξ −ξ1 |−|ξ1 |=τ (3.108) Proof. It is easy to show the identities

ξ − ξ1 −2s dS Sξ1 = S± (τ,ξ)

S± (±τ,ξ)

ξ1 −2s dS Sξ1 ,

where S± (τ, ξ) = {ξ1 ; |ξ − ξ1 | ± |ξ1 | = τ }. Therefore, it is sufficient to show that for τ > |ξ| we have
C L+ (τ, ξ) ≡ ξ1 −2s dS Sξ1 ≤ , 2s−n+1 |τ | − |ξ| |ξ −ξ1 |+|ξ1 |=τ while for |ξ| > τ > 0 we have
L− (τ, ξ) ≡

|ξ −ξ1 |−|ξ1 |=τ

ξ1 −2s dS Sξ1 ≤

C . |τ | − |ξ|2s−n+1

(3.109)

(3.110)

We now introduce polar coordinates ρ = |ξ1 |, ϕ ∈ (0, π), ω ∈ Sn−2 , such that ξ1 = (ρ cos ϕ, ρ sin ϕω). Then the Euclidean metric in Rn can be written (if n ≥ 3) dρ2 + ρ2 (dϕ2 + sin2 ϕdω 2 ), where dω 2 is the standard metric on Sn−2 . A trivial modification in the above relations is necessary for the special case n = 2, when ξ1 = (ρ cos ϕ, ρ sin ϕ) and the metric in R simplifies to 2

dρ2 + ρ2 dϕ2 .

Wave Maps

35

The ellipsoid S+ (τ, ξ) = {|ξ − ξ1 | + |ξ1 | = τ } in the new coordinates will have the equation (recall that S+ (τ, ξ) is an ellipsoid if and only if τ > |ξ|) ρ=

τ 2 − |ξ|2 , 2(τ − |ξ| cos ϕ)

ϕ ∈ (0, π).

(3.111)

We have further

(τ 2 − |ξ|2 )|ξ| sin ϕ . 2(τ − |ξ| cos ϕ)2 Thus ρ(ϕ) is a decreasing function and ρ (ϕ) = −

(3.112)

|τ | − |ξ| |τ | + |ξ| ≥ ρ(ϕ) ≥ ρ(π) = . 2 2 The surface element dS Sξ1 can be written  dS Sξ1 = ρ2 + |ρ |2 ρn−2 sinn−2 ϕdϕdω, ρ(0) =

so that we have




L+ (τ, ξ) = C
≤C

π

ρ−2s

 ρ2 + |ρ |2 ρn−2 sinn−2 ϕdϕ

0 π

(3.113)

ρ−2s ρn−1 sinn−2 ϕdϕ + C

0




π

ρ−2s |ρ |ρn−2 sinn−2 ϕdϕ.

0

The inequalities (3.113) imply that
π ρ−2s+n−1 sinn−3 ϕdϕ ≤ 0

By a simple change of variables we get
π
− ρ−2s ρ ρn−2 sinn−2 ϕdϕ ≤ 0

ρ(0)

C . |τ | − |ξ|2s−n+1

ρ−2s ρn−2 dρ ≤

ρ(π)

C |τ | − |ξ|2s−n+1

and hence

C . |τ | − |ξ|2s−n+1 This argument completes the proof of (3.107). The corresponding equation for the hyperboloid L+ (τ, ξ) ≤

S− (τ, ξ) = {|ξ − ξ1 | − |ξ1 | = τ } is ρ= where

|ξ|2 − τ 2 , ϕ ∈ (0, ϕ0 ) 2(|ξ| cos ϕ + τ )   τ ϕ0 = arccos − . |ξ|

(3.114)

(3.115)

Obviously, ρ ≥ C(|ξ| − τ ).

(3.116)

36

Piero D’Ancona and Vladimir Georgiev

For 0 < τ < |ξ| the denominator τ + |ξ| cos ϕ vanishes if and only if ϕ = ϕ0 . We have the following relation ρ (ϕ) =

(|ξ|2 − τ 2 )|ξ| sin ϕ . 2(τ + |ξ| cos ϕ)2

Thus, ρ(ϕ) is an increasing function and
ϕ0
−2s n−1 ρ ρ dϕ + C L− (τ, ξ) ≤ C 0

0




ϕ0

ρ−2s ρn−2 ρ dϕ.

(3.118)

0

Combining (3.116) and (3.117), we find
ϕ0 ρ−2s ρn−1 dϕ ≤ C and

ϕ0

(3.117)

ρ−2s ρn−2 ρ dϕ ≤




∞

C |τ | − |ξ|2s−n+1

ρ−2s ρn−2 dρ ≤ C

ρ(0)

0

C . |τ | − |ξ|2s−n+1

This completes the proof of (3.110) and of the lemma.




3.6. Bilinear estimates in H s,δ for the inhomogeneous problem We now consider the inhomogeneous problem utt − ∆u = F,

u(0, x) = 0,

ut (0, x) = 0,

(3.119)

where F ∈ H with s > n/2, θ > 1/2. In this section we shall consider the bilinear form (3.90), namely s,θ

Q(∇u1 , ∇u2 ) = ∂t u1 (t, x)∂ ∂t u2 (t, x) −

n 

∂xk u1 (t, x)∂ ∂xk u2 (t, x)

(3.120)

k=1

for two functions u1 , u2 ∈ H s,δ (Rt ×Rm x ). Our main goal is to establish the property Q(H s,δ , H s,δ ) →  H s−1,δ−1 .

(3.121)

The starting point is the following representation of the Fourier transform of Q(∇v, ∇w): ∇w)(τ, ξ) Q(∇v, (3.122)

=c ((τ − τ1 )ττ1 − (ξ − ξ1 )ξ1 ) v(τ − τ1 , ξ − ξ1 )w(τ  τ1 , ξ1 )dξ1 dττ1 . R

Rm

Using the identity (ττ1 + τ2 )2 − (ξ1 + ξ2 )2 − (ττ12 − ξ12 ) − (ττ22 − ξ22 ) , 2 we see that (3.121) follows from the embeddings τ1 τ2 − ξ1 ξ2 =

H s,θ · H s,θ →  H s,θ

(3.123)

H s−1,θ−1 · H s,θ →  H s−1,θ−1 .

(3.124)

and

Wave Maps

37

In the following we shall give a proof of (3.123) only; the proof of property (3.124) is similar. Theorem 3.4. If n ≥ 2, s > n/2 and θ > 1/2, then  H s,θ . H s,θ · H s,θ →

(3.125)

Proof. We start again with the following modification of (3.122):

u 1 u2 (τ, ξ) = c

u 1 (ττ2 , ξ2 ) u2 (ττ1 , ξ1 )dξ1 dτ1 ,

(3.126)

τ1 +τ2 =τ, ξ1 +ξ2 =ξ

where here and below



F (ττ1 , ξ1 , τ2 , ξ2 )dξ1 dττ1 = τ1 +τ2 =τ, ξ1 +ξ2 =ξ

Rm

F (ττ1 , ξ1 , τ − τ1 , ξ − ξ1 )dξ1 dτ1 .

We begin by proving the following elementary inequality. Lemma 3.6. If τ1 + τ2 = τ and ξ1 + ξ2 = ξ, then we have the inequality ||ττ1 + τ2 | − |ξ1 + ξ2 || ≤ ||ττ1 | − |ξ1 || + ||ττ2 | − |ξ2 || + r(ξ1 , ξ2 ), where

⎧ ⎨ ||ξ1 | + |ξ2 | − |ξ||, if τ1 τ2 ≥ 0; ||ξ1 | − |ξ2 | − |ξ||, if τ1 τ2 < 0, |ττ1 | > |ττ2 |; r(ξ1 , ξ2 ) = ⎩ ||ξ2 | − |ξ1 | − |ξ||, if τ1 τ2 < 0, |ττ1 | ≤ |ττ2 |.

(3.127)

(3.128)

Proof. If τ1 τ2 ≥ 0, then |ττ1 + τ2 | = |ττ1 | + |ττ2 |, so that |ττ1 + τ2 | − |ξ1 + ξ2 | = |ττ1 | − |ξ1 | + |ττ2 | − |ξ2 | + |ξ1 | + |ξ2 | − |ξ1 + ξ2 | and applying the triangle inequality, we deduce (3.127). In the case when τ1 τ2 < 0, |ττ1 | > |ττ2 | we have |ττ1 + τ2 | = |ττ1 | − |ττ2 |, so that |ττ1 + τ2 | − |ξ1 + ξ2 | = |ττ1 | − |ξ1 | − |ττ2 | + |ξ2 | + |ξ1 | − |ξ2 | − |ξ1 + ξ2 | and (3.127) follows again. In a similar way we treat the case τ1 τ2 < 0, |ττ1 | ≤ |ττ2 | and this completes the proof of the lemma.
Turning back to the proof of the theorem, we set Fj (τ, ξ) = |τ | + |ξ|s |τ | − |ξ|θ u j (τ, ξ),

j = 1, 2.

(3.129)

Since the integrals in (3.126) are invariant when exchanging u1 and u2 , there is no loss of generality in assuming that |ττ1 | + |ξ1 | ≥ |ττ2 | + |ξ2 |.

(3.130)

38

Piero D’Ancona and Vladimir Georgiev

Then we can use Lemma 3.6 and we find |τ | + |ξ|s |τ | − |ξ|θ u 1 u2 (τ, ξ) ≤ C(I(τ, ξ) + II(τ, ξ) + III(τ, ξ)), where

(3.131)





|ττ1 | + |ξ1 |s |ττ1 | − |ξ1 |θ F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 )dξ1 dτ1 , h(ττ1 , ξ1 )h(ττ2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ (3.132)

s θ |ττ1 | + |ξ1 | |ττ2 | − |ξ2 | F1 (ττ1 , ξ1 )F II(τ, ξ) = F2 (ττ2 , ξ2 )dξ1 dττ1 , h(ττ1 , ξ1 )h(ττ2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ (3.133)

s θ |ττ1 | + |ξ1 | r(ξ1 , ξ2 ) F1 (ττ1 , ξ1 )F III(τ, ξ) = F2 (ττ2 , ξ2 )dξ1 dττ1 , h(ττ1 , ξ1 )h(ττ2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ (3.134) and h(τ, ξ) = |τ | + |ξ|s |τ | − |ξ|θ . I(τ, ξ) =

The property (3.125) will be established if we can verify the inequality F1 L2τ,ξ F F2 L2τ,ξ . |τ | + |ξ|s |τ | − |ξ|θ u 1 u2 (τ, ξ)L2τ,ξ ≤ CF To estimate the term I we use the inequality

F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 ) dξ1 dτ1 , I(τ, ξ) ≤ h(τ τ 2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ and obtain

(3.135)

(3.136)





I

L2τ,ξ

≤ ≤

|F F2 (ττ2 , ξ2 )| dξ2 dττ2 h(ττ2 , ξ2 )   1  CF F1 L2τ ,ξ F F2 L2τ ,ξ  .  h2  2 1 1 2 2 L CF F1 

L2τ ,ξ 1 1

τ1 ,ξ1

The conditions s > n/2, θ > 1/2 imply that   1   < ∞,  h2  2 L τ1 ,ξ1

so that IL2τ,ξ ≤ CF F1 L2 F F2 L2 . As to the term II we have IIL2τ,ξ

≤ ≤

  C 

  F1  θ |ττ1 | − |ξ1 | L2

1 ξ1 Lτ1

CF F1 L2τ

1 ,ξ1

F F2 L2τ

2 ,ξ2

.

(3.137)

   F2     |ξ2 |s 

L1ξ L2τ2 2

(3.138)

Wave Maps

39

Finally, we consider the term III. Note that (3.134) implies |III(τ, ξ)|

≤

r(ξ1 , ξ2 )θ F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 )dξ1 dττ1 . τ1 | − |ξ1 |θ |ξ2 |s |ττ2 | − |ξ2 |θ τ1 +τ2 =τ, ξ1 +ξ2 =ξ |τ (3.139) It will be sufficient to study only the cases F2 ⊆ {ττ2 > 0}, Case A: suppF F1 ⊆ {ττ1 > 0}, suppF Case B: suppF F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 ) ⊆ {ττ1 > 0, τ2 < 0, τ1 + τ2 > 0}, since the other cases are similar. In Case A we make the change of variables τ1 → κ = τ1 − |ξ1 |

(3.140)

and introduce elliptic coordinates λ = |ξ1 | + |ξ − ξ1 |

(3.141)

so that an application of Lemma 3.6 gives r(ξ1 , ξ2 ) = |λ − |ξ|| and |ττ2 | − |ξ2 | = τ2 − |ξ2 | = τ − τ1 − |ξ2 | = τ − κ − λ. From (3.139) we deduce

∞
λ − |ξ|θ |III(τ, ξ)| ≤ F ∗ F ∗ dS Sξ1 dλdκ, θ s θ 1 2 0 |ξ1 |+|ξ−ξ1 |=λ κ |ξ − ξ1 | τ − κ − λ (3.142) where F1∗ = F1 (κ + |ξ1 |, ξ1 ), F2∗ = F2 (τ − κ − λ + |ξ − ξ1 |, ξ − ξ1 ). Applying the Cauchy inequality together with Lemma 3.5, we find

∞
2 |III(τ, ξ)| ≤ C |F F1∗ F2∗ |2 dS Sξ1 dλdκ (3.143) 0 |ξ1 |+|ξ−ξ1 |=λ

= C |F F1 (ττ1 , ξ1 )|2 |F F2 (ττ2 , ξ2 )|2 dξ1 dττ1 , τ1 +τ2 =τ, ξ1 +ξ2 =ξ

so integrating in τ, ξ, we obtain IIIL2τ,ξ ≤ CF F1 L2 F F2 L2 .

(3.144)

In Case B we make the change of variables τ1 → κ = τ1 − |ξ1 |

(3.145)

and introduce hyperbolic coordinates λ = |ξ1 | − |ξ − ξ1 | so that an application of Lemma 3.6 gives r(ξ1 , ξ2 ) = |λ − |ξ||

(3.146)

40

Piero D’Ancona and Vladimir Georgiev

and |ττ2 | − |ξ2 | = −ττ2 − |ξ2 | = −τ + τ1 − |ξ2 | = −τ + κ + λ. From (3.139) we deduce

∞
λ − |ξ|θ F ∗ F ∗ dS Sξ1 dλdκ, |III(τ, ξ)| ≤ θ s τ − κ − λθ 1 2 κ |ξ − ξ | 1 0 |ξ1 |−|ξ−ξ1 |=λ (3.147) and the above argument used in Case A implies again (3.144). Using the inequalities (3.137), (3.138) and (3.144), we complete the proof of the theorem.


4. Concentration of the local energy 4.1. Introduction In this chapter we shall study a special property of the wave maps with inhomogeneous right-hand side Dα ∂α u = F ; (4.1) we recall that ∂ ∂α = , (xα ) = (t, x) ∈ R1+m , α = 0, 1, . . . , m ∂xα denote the partial derivatives with respect to the variables xα , while Dα are the covariant pull-back derivatives in the bundle u∗ T N. As usual, the Greek indices α, β run from 0 to m. We use summation convention over repeated indices. The corresponding initial data are u(0, x) = u0 (x) ∈ H s (Rm ; N ),

ut (0, x) = u1 (x) ∈ H s−1 (Rm ; T N ).

(4.2)

If the target is a hyperboloid (or a manifold with negative curvature and suitable symmetry) results due to Shatah and Tahvildar-Zadeh [32], Christodoulou and Tahvildar-Zadeh [8], Grillakis [13], show that for any C ∞ initial data a global unique C ∞ solution exists, at least in the class of radial or equivariant wave maps. The key point in this approach is the following property, called the nonconcentration of energy:
|∇t,x u(t, x)|2 dx → 0, (4.3) Ω(t)

as t → 0 and Ω(t) = {x ∈ R : |x| < t}. In this chapter we study the property (4.3) for more general domains of the form Ωα (t) = {x ∈ R2 : |x|α < t}, where α ∈ (0, 1] and the target is S 2 . We shall consider the inhomogeneous problem (4.1), assuming u is an equivariant wave map. Our goal is to compare the following local norms:
t and F (s, ·)H ε (Ωα (s)) ds, u(t, ·)H 1+ε (Ωα (t)) 2

0

Wave Maps

41

for ε ≥ 0. When ε = 0 and α = 1, these are exactly the norms appearing in the local energy estimate
T 1    k ∂t u(T, ·) 1−k +C F (s, ·)H 0 (Ω(s)) ds, (4.4) u(t, ·)H 1 (Ω(t)) ≤ C H (Ω(T )) t

k=0

where 0 < t < T. This energy estimate, combined with a suitable control of L∞ norm, leads to the property (4.3) (see, e.g., [32] for details ). If one tries to improve the regularity in (4.4) by a small additional ε > 0, then a natural question arises: does an estimate like u(t, .)H 1+ε (Ωα (t)) ≤C

(4.5)

1   k  ∂t u(T, .) 1−k+ε +C H (Ωα (T )) k=0




T

t

F (s, .)H ε (Ωα (s)) ds

hold for t ∈ [0, T ]? This question of course is particularly interesting in the case of dimension n = 2, when the energy space H 1 coincides with the critical space with respect to scaling and local existence; the behavior of (4.5) as ε → 0 can be regarded as a measure of the instability of H 1 from the point of view of local existence. Our goal is to show that (4.5) fails in general in n = 2, and actually an even stronger statement holds: we shall construct wave maps for which lim u(t, ·)H 1+ε (Ωα (t)) = ∞,

t→0+

while the expression
1   k  ∂t u(T, ·) 1+ε + H (Ωα (T ))

0

k=0

T

F (s, ·)H ε (Ωα (s)) ds

  is bounded for 0 < α < 1 and ε < min 12 , 1 − α . More precisely, we have the following. Theorem 4.1. Consider the inhomogeneous wave maps equation (4.1) defined on R2+1 with target N = S2 . Then one can find positive numbers ε > 0, T > 0, initial data (u0 , u1 ) ∈ H 1+ε × H ε and a source term F ∈ L1 ((0, T ); H ε (Ωα (t))) such that the Cauchy problem for (4.1) with initial data at t = T u(T, x) = u0 (x),

ut (T, x) = u1 (x)

has a solution u defined (at least) on the domain Kα (T ) = {(t, x); t ∈ (0, T ], x ∈ Ωα (t)}, with α ∈ (0, 1 − ε), satisfying u ∈ C((0, T ]; H 1+ε (Ωα (t)))

(4.6)

42

Piero D’Ancona and Vladimir Georgiev

and lim u(t)H 1+ε (Ωα (t)) = ∞.

t→0+

Remark 4.1. The condition u ∈ C((0, T ]; H 1+ε (Ωα (t))) means that the function defined as v(t) = u(t, ·)H 1+ε (Ωα (t)) belongs to C((0, T ]). This theorem shows that the naive intuitive argument based on approximation of (weak) H 1 solutions in the light cone K(T ) = {(t, x); t ∈ (0, T ], x ∈ Ω(t)} by means of sequences of smoother H 1+ε solutions in the slightly distorted “cones” Kα (T ) = {(t, x); t ∈ (0, T ], x ∈ Ωα (t)}, might have a concentration of the local energy as shown by the property lim u(t)H 1+ε (Ωα (t)) = ∞.

t→0+

The plan of the chapter is the following. In Section 4.2 we construct the wave maps used in the proof of the theorem by a rescaling in time of solutions of a suitable elliptic problem. This method does not yield an exact solution of the wave maps equation, but produces instead an error which is exactly the source term in (4.1); we then estimate the solution and the source term in the energy norms. In Section 4.3 we compute the higher-order norms H 1+ε of the solution and the C([0, T ]; H ε )-norm of the source term, and conclude the proof of the theorem. The Appendix is devoted to the proof of some technical lemmas. 4.2. Construction of the solutions From now on we will use the following notations: if f and g are two positive, real-valued functions, we write f  g if there exists a constant C > 0 such that f ≤ Cg; we shall write f ∼ g, if there exist constants A > 0 and B > 0, such that Ag ≤ f ≤ Bg. Consider the equation (2.66) for equivariant wave maps with initial data φ(1, r) = φ0 (r),

φt (1, r) = φ1 (r),

(4.7)

where φ = φ(t, r) depends only on t and r. We shall construct a solution of the following special form:   v(r) φ(t, r) = Q ; (4.8) t the function Q must satisfy a suitable ordinary differential equation, which we derive now. By the definition of ϕ we have:       v(r) v(r) (v(r))2  v(r) 2v(r)  v(r) Q Q , ∂t2 φ = + , ∂t φ = − 2 Q t t t4 t t3 t       v  (r)  v(r) (v  (r))2  v(r) v  (r)  v(r) 2 Q Q Q ∂r φ = , ∂r φ = + . t t t2 t t t

Wave Maps Plugging these quantities into (2.66) we see that ϕ satisfies the identity  2  sin 2φ ∂t − ∆ φ(t, x) + 2    2r   2 (v(r))  v(r) 2v(r)  v(r) (v  (r))2  v(r) = Q Q + 3 Q − t4 t t t t2 t       v (r)  v(r) sin 2Q v (r)  v(r) Q Q . − + − t t rt t 2r2

43

(4.9)

Our main idea is to regard all the terms involving the time derivatives as a source term, i.e., to choose     (v(r))2  v(r) 2v(r)  v(r) 2 f = ∂t u = Q + 3 Q ; (4.10) t4 t t t then the equation (4.9) (∂ ∂t2 − ∆)φ(t, x) +

sin 2φ =f 2r2

simplifies to sin 2φ = 0, 2r2 and, recalling our choice of ϕ, this leads to the following equation for Q −∆φ(t, x) +

(v  (r))2  v(r) v  (r) + rv  (r)  sin 2Q Q − Q + = 0. t2 t rv(r) 2r2

(4.11)

As v = v(r) we may choose a solution of the following ordinary differential equation: v  (r) + rv  (r) c = 2, (4.12) rv(r) r where the positive constant c is a parameter to be chosen. This is an equation of Euler-type: r2 v  (r) + rv  (r) − cv(r) = 0. A special solution to (4.12) is v = rα , provided we take c = α2 . With these choices, the equation for Q becomes α2

α 1 r2α  sin 2Q 2 1 r Q − Q + α = 0. r 2 t2 r2 t 2r2

Setting for brevity z=

(4.13)

rα , t

we can rewrite (4.13) as follows: α2 z 2 Q + α2 zQ −

sin 2Q = 0. 2

Now, making change of the variable s=

1 ln z, α

(4.14)

44

Piero D’Ancona and Vladimir Georgiev

we get sin 2Q(s) = 0. (4.15) 2 We have not yet chosen the initial data for Q. Multiplying the equation by Q we obtain 1 + cos 2Q = const (4.16) (Q )2 + 2 and this means that the quantity Q (s) −

1 + cos 2Q ≡ (Q )2 + (cos Q)2 2 is constant on the integral curves of (4.15) or, in other words, is a first integral of the equation. Now we may choose the initial data for Q such that I(s) is equal to 1: indeed, it is sufficient to take π Q(0) = , Q (0) = 1 = =⇒ I(s) = (Q )2 + (cos Q)2 = 1. 2 The last equation has the two solutions   Q tan = c0 e±s , 2 I(s) = (Q )2 +

and our choice of the initial data for Q implies c0 = 1. Discarding the solution with sign − we finally obtain  1 Q(z) = 2 arctan z α . (4.17) Our next step is to study the regularity properties of the remainder f (t, r) defined as above by     r2α  rα rα  rα f (t, r) = 4 Q +2 3Q (4.18) t t t t in the region rα < t. In fact, we shall prove the following: Lemma 4.1. For any α ∈ (0, 1) and T > 0 we have f ∈ L1 ((0, T ); H 0 rad (Ωα (t))). Proof. As before we use the notation rα . t From the equations (4.18) and (4.14) for Q we have:   1 sin 2Q  f = 2 zQ + . t 2α2 z=

So, the norm of f in H 0 rad (Ωα (t)) = L2 rad (Ωα (t)) for fixed t ∈ (0, T ) is: 2
α1  1 t sin 2Q 2  r dr, zQ (z) + f (t, ·)L2 rad (Ωα (t)) = 4 t 0 2α2

(4.19)

(4.20)

Wave Maps

45

where in terms of z we have 2 z α −1 , α 1 + z α2 1

Q (z) = so we get 2 f (t, ·)L2 rad (Ωα (t))

≤ ≤

≤




C t4

1

tα





2

zα

2

+ (sin 2Q)

2

(1 + z α )2

0

⎛




1




2 α

tα

r dr ⎞

1

tα

C⎝ z r dr + (sin 2Q)2 r dr⎠ 2 2 t4 α (1 + z ) 0 0 ⎛ 1 ⎞
α 2 C C ⎝ t r3 α⎠ = 2 dr + t 2 , t4 tα t4− α 0

where we have used the inequality

1 2

1+z α

≤ 1 and C = C(α, T ) > 0 is a constant

independent of t. Finally, we have the following inequality: C (4.21) f (t, ·)L2 rad (Ωα (t)) ≤ 2− 1 . t α Now it is obvious that the function t−2+1/α is in L1 (0, T ) if α < 1, and this completes the proof.
The next lemma shows that the solution ϕ(t, ·) belongs to the energy space H 1 : Lemma 4.2. The solution φ(t, r) of (2.66), defined according to (4.8), belongs to H 1 rad (Ωα (t)) for every fixed t > 0. Proof. For any fixed t > 0 we have
t α1
2 2 = |∂ ∂r φ| r dr ≤ C φ(t, ·)H˙ 1 rad

0

0


≤

1 tα

C

1

tα





2

zα (1 + z

r

2 α

)2

1 dr r

2 dr = const. tα Note that the reverse inequality is also true. Indeed, we have 

t α1  2 zα 1 2 2 dr φ(t, .)H˙ 1 |∂ ∂r φ| r dr = C 2 (Ωα (t)) = 2 rad(loc) r (1 + z α ) Ωα (t) 0

(4.22)

0


≥C

1

tα

r 2

0

tα

dr = const ,

where we used inequalities 1 1 ≤ ∀z ∈ [0, 1]. 2 ≤ 1, 2 1 + zα

(4.23)

46

Piero D’Ancona and Vladimir Georgiev

The solution of the equation has the form  1 1 φ = 2 arctan z α  z α =: w, and this gives the bound 2

2

u(t, ·)L2 rad (Ωα (t))  w(t, ·)L2 rad (Ωα (t)) . Then we get
φ(t, ·)L2 rad (Ωα (t))

≤ C

2 w(t, ·)L2 rad (Ωα (t))

1

tα

= 0

r2 t

2 α

1

r dr = Ct α .


This concludes the proof of the lemma. 4.3. Higher regularity of the solution

We now estimate higher-order norms of the solution constructed in the previous section and of the corresponding source term. In order to carry an explicit computation of the fractional Sobolev norms, the following well-known alternative definition is useful: Definition 4.1. (see, e.g., [Triebel 2.5.1]) Let s > 0. The fractional H s (R2 ) norm can be defined as follows: 
1/2
2   −(1+2s)  [s]+1 f H s (R2 ) = f L2 (R2 ) + |h| dh . ∆ h f (x) R2

L2 (R2 )

(4.24) The Sobolev space H s consists of all functions f such that the above norm is finite. Here ∆k h f (x) is the difference of order k defined as follows: the difference of order 0 and 1 of the function f are simply ∆0 h f (x) = f (x) and ∆1 h f (x) = f (x + h) − f (x); then the difference of order k may be defined inductively as ∆k h f (x) = ∆1 h (∆k−1 h f (x)). In order to extend the definition of fractional Sobolev spaces to subdomains Ω of R2 , it is sufficient to consider, for each function f on Ω, the set of all its extensions f˜ to R2 (i.e., f˜
Ω = f ): Definition 4.2. Let s > 0. We say that the function f ∈ H s (Ω) if the following norm is finite: f H s (Ω) = inf f˜H s (R2 ) , f˜

(4.25)

where the inf is taken over all extensions f˜ ∈ H s (R2 ) of f . We now specialize the above definitions to our (radial) situation. We consider first the case s ∈ (0, 1):

Wave Maps

47

s Definition 4.3. Let 0 < s < 1. We say that f = f (r) belongs to the space Hrad (R2 ) if the following norm is finite: 
1/2
1
∞ 1 −(1+2s) ∆ h f (r) 2 r dr dh f H s rad (R2 ) = f L2 rad (R2 ) + |h| . 0

0

(4.26) Now we consider the case s ∈ (1, 2): Definition 4.4. Let s = 1 + ε ∈ (1, 2). We say that f = f (r) belongs to the space 1+ε 1 Hrad (R2 ), if f ∈ Hrad (R2 ) and the following norm is finite: 
1/2
1
∞ 1 −(1+2ε) ∆ h ∂r f (r) 2 r dr dh f H 1+ε rad (R2 ) = f H 1 rad (R2 ) + |h| . 0

0

(4.27) The definition of the fractional Sobolev spaces on a ball Ω of radius R centered at 0 is identical to the above one (4.25). Moreover, for any radially symmetric function f (x) = f (|x|), we have the following obvious inequality 
1/2
R/2 2 [s]+1 −(1+2s) f H s (|x| 1. Then

Wave Maps

55

Proof. We first estimate some norms with respect to space variables at a fixed time t > 0. The L2 -norm of the term F = f ∂φ u(φ) can be computed immediately as follows: f ∂φ u(φ)L2 = f L2 .

(4.61)

µ

Next we use the fact that for µ > 1 the space H is an algebra, and this gives the estimate f ∂φ u(φ)H µ  f H µ ∂φ u(φ)H µ . 2

(4.62)

2

µ

Thus, if we consider f as an operator from L into L and from H into H µ , an interpolation argument implies that f is bounded on H s for each 0 < s < µ, with a norm bounded by f H s . Hence we have f ∂φ u(φ)H s  f H s ∂φ u(φ)H s . s

(4.63) 

But from the previous lemma we may control the H norm of u (φ) with the H s norm of φ. In conclusion, the result follows by integrating the above estimate and applying H¨ ¨older’s inequality with respect to time.
An analogous result holds on the restricted cones: Lemma 4.5. Let φ ∈ C([0, T ]; H s (Ωα (t))) and f ∈ L1 ((0, T ); H s−1 (Ωα (t))) for some s > 1. Then F = f ∂φ u(φ) ∈ L1 ((0, T ); H s−1 (Ωα (t))). Proof. The argument is identical to the above one (use Proposition 4.1).




5. Non-uniqueness result in the subcritical case 5.1. Introduction The general wave map problem in local coordinates has the form (see equation (2.40)) (5.1)
u + Γbc (u)∂α ub ∂ α uc = 0. The natural problem for (5.1) is the Cauchy problem with initial data at t = 0 u(0, x) = u0 (x),

ut (0, x) = u1 (x).

(5.2)

When N is the unit sphere in R with the metric induced by the Euclidean metric, the equations become very symmetrical using the full set of functions (u1 , . . . , ud ), indeed we obtain (see equation (2.50)) d


u + (|ut |2 − |∇x u|2 )u = 0

(5.3)

subject to the constraint |u| = 1. It is easy to see that if a smooth function u(t, x) solves (5.3) and moreover satisfies the constraint at t = 0, i.e., |u(0, x)| = 1, u(0, x) · ut (0, x) = 0, then we have |u| = 1 for all t. It is also possible to consider system (5.3) without geometric constraints, from a purely analytical point of view.

56

Piero D’Ancona and Vladimir Georgiev

The minimal regularity required of u in order to give a meaning to the nonlinear term (in distributional sense) is Dt,x u ∈ L2loc , provided |u| = 1. This can be further relaxed if we remark that
u is parallel to u by the equation, hence we have also u ∧
u = 0, |u| = 1 or equivalently ∂t (ut ∧ u) =

n 

∂xj (uxj ∧ u),

(5.4)

j=1

since u · ut = u · uxj = 0 by the condition |u| = 1. For smooth functions with norm 1, (5.4) and (5.3) are equivalent; moreover, equation (5.4) has a distributional sense for Du ∈ L1loc only, provided |u| = 1. Thus in the following we shall say that u is a weak solution of equation (5.3) if Du ∈ L1loc , |u| = 1 a.e. and u satisfies (5.4) in the sense of distributions. One of the basic open questions of the theory is the well-posedness of the Cauchy problem for (5.3) in two dimensions, i.e. with u : R × R2 → S2 ⊆ R3 . In this case the local existence of smooth solutions follows by classical arguments, while global existence meets essential difficulties. The critical space for equation (5.4) in two dimensions is H 1 , which is also the energy space; thus an important question is the well-posedness of (5.4) in H 1 . The aim of this chapter is to investigate the behavior of (5.3) for solutions of low regularity, i.e., below the energy space H 1 × L2 , and indeed to show that in general the problem is not well-posed in this situation. Our main result is the following: Theorem 5.1 (Nonuniqueness). (See Theorems 5.2 and 5.3 below.) It is possible to construct two weak solutions u, v : R × R2 → S2 to the equation (5.4), continuous in time with values in H 1−ε (Ω) for all ε > 0 and all bounded Ω, and also L∞ with 1 values in the Besov space B2,∞ , such that u ≡ v for t < 0 and u ≡ v for t > 0. We can construct u, v such that their Besov norm is arbitrarily small. 1 1 , recalling Tataru’s local well-posedness result in B2,1 , we may say Since H 1 = B2,2 1 that the question of local well-posedness in H is confined in the gap between the 1 1 and B2,1 . In this regard it is necessary to mention Tao’s two Besov spaces B2,∞ result [40] showing that the well-posedness holds provided the data are slightly smoother (H 1+ε × H ε ) and small in the H˙ 1 norm. See also the paper [27], where a model (scalar) equation with the same type of nonlinearity is considered. The plan of this chapter is the following: in Section 5.2 we recall the definition of equivariant and self-similar solutions, which are necessary for the following constructions. The self-similar ansatz leads to an ODE which is studied in detail in Appendix A; in particular all the solutions are computed explicitly. Section 5.3 is devoted to the non-uniqueness result Appendix B collects some technical lemmas.

Wave Maps

57

5.2. Equivariant and self-similar solutions When the manifolds have rotational symmetry, two interesting classes of special solutions arise, the equivariant and the self-similar solutions. The general construction is standard and can be found in [5], [30], [32]. We recall briefly the equivariant ansatz. Assume that N is a smooth dimensional rotationally symmetric manifold defined as N = {(φ, χ); φ ∈ [0, φ∗ ), χ ∈ S−1 } (φ∗ may be +∞), with metric dφ2 + g(φ)2 dχ2 ,

(5.5)

where dχ is the standard metric on S . In the coordinates (φ; χ1 , . . . , χ−1 ), denoting by hij the coefficients of the metric dχ2 , the only nonzero Christoffel symbols for (5.5) are 2

−1

Γφχi χj = −g  (φ)g(φ)hij ,

Γχχij φ =

g  (φ) δij , g(φ)

i Γχχij χs = γjs ,

(5.6)

i where γjs are the Christoffel symbols for the metric hij . The equivariant wave maps are the maps u satisfying the ansatz

u(t, x) = (φ, χ),

φ = φ(t, r),

χ = χ(ω),

(5.7)

where (r, ω) are the spherical coordinates on Rn . Under this assumption, the equations for χ decouple and in fact we obtain that χ : Sn−1 → S−1 must be a harmonic map. For suitable choices of the dimensions n,  (and of χ) further simplifications occur. In the special case  = n = 2 under consideration here, it is easy to see that χ must be a rotation of degree k = 1, 2, 3, . . . of S1 into itself. With this choice the equation for φ decouples and we obtain 1 k2 φtt − φrr − φr + 2 g  (φ)g(φ) = 0. (5.8) r r This is the equivariant wave map equation. When N is the sphere S2 , the above framework corresponds to the standard choice of coordinates (φ, χ) with φ ∈ [0, π] and χ ∈ S1 . Then the metric can be written dφ2 + sin2 φ dχ2 , and equation (5.8) becomes 1 k2 φtt − φrr − φr + 2 sin(2φ) = 0. (5.9) r 2r Actually it is more convenient to embed N = S2 in R3 ; with the usual coordinate system on N ⎛ ⎞ sin φ sin χ ⎝sin φ cos χ⎠ , cos φ

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where φ ∈ [0, π], χ ∈ [0, 2π], φ = 0 corresponding to the north pole and φ = π to the south pole of the unit sphere. Then we can express the solution u(t, x) to (5.3) as the vector (u1 , u2 , u3 ) with u1 = sin φ(t, |x|) · |x|−k Re (x1 + ix2 )k u2 = sin φ(t, |x|) · |x|−k Im (x1 + ix2 )k u3 = cos φ(t, |x|); since cos χ = cos(kω) = |x|−k Re (x1 + ix2 )k , sin χ = sin(kω) = |x|−k Im (x1 + ix2 )k . In order to introduce the self-similar solutions, we recast the equation in the hyperbolic coordinates  r (5.10) ρ = t2 − r 2 , τ= ; t notice that the inverse transformations are given by τρ ρ , r= √ . (5.11) t= √ 2 1−τ 1 − τ2 Then we obtain 1 2 ∆H (5.12) ∂t2 − ∂r2 − ∂r = ∂ρ2 + ∂ρ − 2 , r ρ ρ where ∆H is the Laplace operator on the hyperboloid ρ = 1, that is, (1 − τ 2 )(2τ 2 − 1) ∂τ . τ In the new coordinates ρ, τ the equation (5.8) becomes ∆H = (1 − τ 2 )2 ∂τ2 −

(1 − τ 2 ) 2 1 sin 2φ = 0. ∂ρ2 φ + ∂ρ φ − 2 ∆H φ + k 2 ρ ρ 2τ 2 ρ2

(5.13)

(5.14)

We can now define the self-similar solutions as the solutions which are independent of ρ, i.e., r φ(t, r) = ψ =: ψ(τ ). t Under this assumption the first two terms in (5.14) drop, and we obtain immediately the following equation for ψ = ψ(τ ): τ 2 (τ 2 − 1)ψ  + τ (2τ 2 − 1)ψ  + k 2 sin ψ cos ψ = 0.

(5.15)

Notice that if one could find global smooth solutions to this equation, an immediate consequence would be a blow-up result for the wave map equation (5.3); but this is not possible, as shown in [30] (while in higher dimensions this idea is correct and was exploited in [32]). Nevertheless it is possible to utilize the singular (i.e., not in H 1 ) solutions thus obtained, as we shall do in the following section. It is not difficult to see that all solutions to (5.15) are analytic and defined for τ = 0, 1; in Appendix A we give a complete study of the equation, and we represent all its solutions using Jacobi’s elliptic functions. In the next sections, in particular, we shall use the following special solutions:

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(i) The function defined as  ψ(τ ) =

arcsin τ π/2

for 0 ≤ τ ≤ 1 for τ > 1

(5.16)

is a solution to the equation (for τ = 1) in the case k = 1. Notice that the only constant solutions are the integer multiples of π/2, and that the value of the constant chosen here for τ > 1 ensures (H¨ o¨lder) continuity. The fact that (5.16) is a solution can be verified directly, or can be obtained by setting λ = π/2, k = 1 in the general expression (5.72). (ii) A more general class of solutions in the case k = 1 is given by the expressions (δ ∈ (0, 1))  ⎧ ⎪ ⎪2 arctan tan λ · τ for 0 ≤ τ ≤ 1, ⎪ ⎨ 2 1 ± 1 − τ2   (5.17) ψ(τ ) = ⎪ 1 1 ⎪  am δ arctan for τ > 1. ⎪ ⎩ τ2 − 1 δ Here the two-parameter function am (τ |m) is the Jacobi amplitude in the case m > 1; a precise definition is given in Appendix A, here we shall only need to know that it is an analytic periodic function of τ , with the property am (0|m) = 0, hence (5.17) tends to 0 as τ → ∞. Again, we can ensure H¨¨older continuity at τ = 1 by imposing a suitable condition on the constants λ, δ. In the following section we shall also need to compute sin ψ and cos ψ for ψ given by (5.17); writing γ = tan we have

sin ψ(τ ) =

and

λ 2

⎧ 4γτ ⎪   ⎪ ⎪ ⎨ (1 ± 1 − τ 2 ) + γ 2 (1 ∓ 1 − τ 2 ) ⎪ ⎪ ⎪ ⎩

  δ · sn arctan √τ 12 −1 δ

⎧   2 ) − γ 2 (1 ∓ 2 ⎪ (1 ± 1 − τ ⎪ ⎪  1 − τ ) ⎪ 2 ⎨ (1 ± 1 − τ 2 ) + γ (1 ∓ 1 − τ 2 ) cos ψ(τ ) = ⎪  ⎪  ⎪ ⎪ ⎩dn arctan √ 12 δ , τ −1

for 0 ≤ τ ≤ 1, (5.18) for τ > 1,

for 0 ≤ τ ≤ 1, (5.19) for τ > 1

(the signs ± are the same as in (5.17), the ∓ are opposite). For the definition and properties of the elliptic functions sn (τ |δ), dn (τ |δ) see Appendix A. These formulas are proved in Appendix A, see Remark 5.4 and (5.75), (5.76), (5.77).

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5.3. Low regularity self-similar solutions Our purpose here is to construct weak solutions to the wave map equation (5.3) from R × R2 to S2 below critical regularity. As mentioned in the Introduction, it is convenient to transform the equation in the form of a conservation law ∂t (∂ ∂t u ∧ u) =

2 

∂j (∂ ∂j u ∧ u)

(5.20)

j=1

in order to handle weak solutions of very low regularity; indeed if u is a locally bounded function such that ∂u ∈ L1loc (R × R2 ), all the terms in (5.20) have a well-defined meaning in the sense of distributions. Starting from a solution ψ(τ ) of (5.15), we can construct a self-similar solution to the wave map equation by setting ⎛ ⎞ u1 (t, x) u(t, x) = ⎝ u2 (t, x) ⎠ , (5.21) u9 (t, x) with u1 (t, x) = |x|−k Re (x1 + ix2 )k sin (ψ (|x|/t)) , u2 (t, x) = |x|−k Im (x1 + ix2 )k sin (ψ (|x|/t)) , u3 (t, x) = cos (ψ (|x|/t)) . We shall restrict ourselves to the case k = 1 and the special solutions examined in the preceding section. Our first result is based on the solution (5.17); this means simply ⎛ ⎞ x1 1⎝ ⎠ x2 (5.22) u(t, x) = t 2 2 t − |x| inside the cone, and

⎞ x1 /|x| u(t, x) = ⎝x2 /|x|⎠ 0 ⎛

(5.23)

outside the cone. Our aim is to prove Theorem 5.2. The function u(t, x) defined in (5.22), (5.23) is a solution in the distributional sense of the wave map equation (5.20). Moreover, given any bounded open set Ω ⊆ R2 , we have u ∈ C(R, H s (Ω)),

∀s < 1

and

1 u ∈ L∞ (R, B2,∞ (Ω)).

(5.24)

A second solution is the function v(t, x), independent of t, defined as (5.23) every1 (Ω)) for any Ω, and v ≡ u for t ≤ 0, thus the weak where; we have v ∈ C(R, B2,∞ solution to (5.20) is not unique in the spaces (5.24).

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Remark 5.1. Actually the solution u is continuous with values in Besov spaces for t ≥ 0 and t ≤ 0, and has a jump at t = 0 only. We also notice that, thanks to the 1 scaling properties of Besov spaces, the B˙ 2,∞ norm of u(t, x) on the ball B(0, t) is constant as t ↓ 0, i.e., it concentrates at 0. Proof. We remark that the first derivatives of u in the sense of distributions coincide with the derivatives a.e., which are locally integrable functions on R × R2 . This follows at once by Lemma 5.3 in the Appendix (with obvious choices); indeed, the singularities of u are concentrated along the positive cone t = |x|, where u is Holder ¨ continuous, and along the line x = 0 for negative t, where it is bounded. Thus we can compute the derivatives ut , uxj directly by differentiating the above formulas, obtaining locally integrable functions. A similar remark holds for v. Outside the positive light cone, i.e., for t ≤ |x| (including negative t) we have u ≡ v, and an explicit computation gives ∂t u ∧ u = ∂t v ∧ v = 0, ⎞ ⎛ 0 ∂x1 u ∧ u = ∂x1 v ∧ v = ⎝ 0 ⎠ , x2 /|x|2 ⎞ ⎛ 0 ⎠. 0 ∂x2 u ∧ u = ∂x2 v ∧ v = ⎝ 2 −x1 /|x|

(5.25) (5.26)

(5.27)

These formulas hold on the whole space R × R2 for the second solution v. To check that v is a solution of (5.20) we may proceed directly; indeed, we have to check that   x1 x2 ∂x1 − 2 + ∂x2 =0 |x| |x|2 in the sense of distributions, and this follows from the identities xj ∂xj |x| = |x| (notice that these functions are locally integrable). To check that u is a solution, it is simpler to express the equation in spherical coordinates: 1 1 ∂t (ut ∧ u) − ∂r (ur ∧ u) − ur ∧ u − 2 ∂ω (∂ ∂ω u ∧ u) = 0. (5.28) r r Outside the light cone we have u = (cos ω, sin ω, 0), while inside it holds u=

 1 (r cos ω, r sin ω, t2 − r2 ). t

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We can write the wedge products in a global form valid everywhere: writing for brevity ⎛ ⎞ − sin ω Z(ω) = ⎝ cos ω ⎠ 0 and  √ 1 if r ≤ t, t2 −r 2 (5.29) w(t, r) = 0 otherwise, we find after some computations r ut ∧ u = w(t, r) · Z, t ur ∧ u = −w(t, r) · Z,

(5.30) (5.31)

while

r2 t − r2 · Z. (5.32) t2 Thus, to check that u is a solution we must show that √ 1 rw t2 − r 2 ∂t χK = 0 (5.33) + ∂r w + w − t r rt2 in distribution sense, where χK is the characteristic function of the future light cone K. Given any test function φ, we must show that  √
 rw 1 t2 − r 2 wφ − ∂t φ − w∂ ∂r φ − φ rdrdt = 0. r t rt2 K ∂ω (∂ ∂ω u ∧ u) = ∆ω u ∧ u =

This amounts to say that, writing K ε = {(t, r) : t ≥ r + ε}, the limit of the same integral over K ε tends to zero when ε → 0. We integrate by parts, the integrals over K ε cancel since the solution is smooth in K ε and the terms on the boundary t = r + ε give (keeping into account the fact that the components νt and νr of the normal unit vector are opposite)
∞
 r ε rwφdS = rw(r + ε, r)φ(r + ε, r)dr → 0. 1− t r+ε 0 ∂K ε It remains now to show that the solutions belong to the stated spaces. For the solution v this follows directly from Lemma 5.1 in Appendix B, and the remark 1 that the H s -norms for s < 1 are controlled by the Besov norm of B2,∞ . Consider now the solution u; for t ≤ 0 it coincides with v, hence the same argument is applicable and we obtain the Besov-valued continuity on (−∞, 0]. We now consider the case t > 0. We know by Lemma 5.1 in Appendix B that x1 /|x| 1 is locally in B = B2,∞ , and this implies that also the function x1 x1 − |x|

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is locally in B since x1 is smooth. Then also the function  x1 x1 − |x| for |x| ≤ 1, g(x) = 0 for |x| ≥ 1, is in B locally, since g(x) is Lipschitz continuous at |x| = 1. But g(x) is compactly 1 (R2 ) globally. Recalling that the homogeneous norm supported, hence g ∈ B2,∞ 1 2 ˙ B2,∞ (R ) is invariant for the scaling x → x/t, we obtain that the function g(x/t) 1 (R2 ). This implies immediately that is continuous on [0, +∞) with values in B2,∞ x1 + g(x/t) (5.34) u1 = |x| 1 is continuous on [0, +∞) with values in B2,∞ (Ω) for any bounded open Ω; u2 is identical. As to u3 , we only have to remark that

u3 (t, x) = λ(x/t),

(5.35)

where λ(x) is the function defined in Lemma 5.2 in Appendix B, which belongs to 1 B2,∞ (R2 ), and argue as before. 1 Continuity with values in H s for t = 0 follows, since the norm of B2,∞ is stronger. Continuity also at t = 0 follows from (5.34) and (5.35) since the scaling properties of H s for s < 1 imply that g(·/t)H s (R2 ) → 0 as t → 0.




The situation does not improve if we consider small solutions. Indeed, we can construct two different weak solutions depending on a parameter δ which are small and coincide for t < 0. The first solution is the self-similar solution obtained using the functions (5.17) with the plus sign; recalling (5.18), (5.19), we have inside the light cone, i.e., for |x| < |t|, 2γxj  uj (t, x) = , (5.36) 2 (1 + γ )t + (1 − γ 2 ) t2 − |x|2 for j = 1, 2, and

 (1 − γ 2 )t + (1 + γ 2 ) t2 − |x|2  u3 (t, x) = ; (1 + γ 2 )t + (1 − γ 2 ) t2 − |x|2

(5.37)

here γ = tan λ2 is a small parameter. On the other hand, outside the light cone, i.e., for |x| > |t|, we have   t xj (5.38) δ sn arctan  uj = δ |x| |x|2 − t2 for j = 1, 2, and

 u3 = dn

t

arctan  |x|2 − t

δ 2

 .

(5.39)

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For a small fixed δ we choose γ such that 2γ = δ · sn (π/2|δ) (5.40) 1 + γ2 to ensure continuity; recall that for δ < 1 it holds d sn (s|δ) ≤ 1, |sn (s|δ)| ≤ 1, ds while  d 2 1 − δ ≤ dn (s|δ) ≤ 1, ds dn (s|δ) ≤ δ. So the function u(t, x) thus constructed takes its values in a δ-neighborhood of the North pole (0, 0, 1). The second solution v(t, x) coincides with u(t, x) for negative t and outside the forward light cone; inside it, i.e., for t > |x|, we define it using again (5.17) but we choose now the minus sign; in a more explicit form we obtain (see again (5.18), (5.19)) 2γxj  vj (t, x) = , (5.41) 2 (1 + γ )t − (1 − γ 2 ) t2 − |x|2 for j = 1, 2, and  (1 − γ 2 )t − (1 + γ 2 ) t2 − |x|2  (5.42) v3 (t, x) = (1 + γ 2 )t − (1 − γ 2 ) t2 − |x|2 (compare with (5.36), (5.37)). Remark 5.2. Notice that the solution u(t, x) takes its values in a δ-neighborhood of the North pole (0, 0, 1); on the other hand, the second solution v(t, x) covers the entire sphere, and for t > 0, x = 0 we have v(t, 0) = (0, 0, −1), the South pole. Then we have: Theorem 5.3. Consider the function u(t, x) defined by (5.36), (5.37) for |x| ≤ |t| and by (5.38), (5.39) for |x| > |t|. Moreover, consider the function v(t, x) defined as u(t, x) for t < |x| and by (5.41), (5.42) inside the future light cone t > |x|. Then both u(t, x) and v(t, x) are weak solutions of the wave map equation (5.20), and have the same regularity properties (5.24). Moreover, for any bounded Ω ⊆ R2 and t < 0 fixed we have 1 1 u(t, ·) − N B2,∞ (Ω) ≡ v(t, ·) − N B2,∞ (Ω) = O(γ) as γ → 0,

(5.43)

where N denotes the North pole N = (0, 0, 1). Proof. The regularity of u, v is proved as before, and we obtain as above the continuity with values in a Besov space for t > 0 and for t < 0, with a jump in t = 0 (u is actually continuous with values in a Besov space also at t = 0). Also the estimate (5.43) follows by a simple argument. To prove that u, v solve (5.20) we begin as above by applying Lemma 5.3 in the Appendix. The singularity of u and v is concentrated on the light cones

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K = {|t| = |x|}. Choosing a ball Ω near a point of the cone different from the origin, and setting f = u, h(t, x) = u(|x|, x), we see that assumption (5.92) of this lemma is satisfied, using the H¨¨older continuity of u near the point; thus the distributional derivative of u on (R × R2 ) \ (0, 0) coincides with the derivative a.e. Now we can apply again this lemma choosing K = the origin, and in this case assumption (5.92) follows from the fact that u is bounded near the origin. In conclusion, the first distributional derivative of u on R × R2 coincides with its derivative a.e. The argument for v is identical. Thus it is sufficient to prove that for any test function φ the following identity holds:
 1 1 φt ut ∧ u − φr ur ∧ u + φ ur ∧ u + 2 φ uωω ∧ u r dr dω dt = 0 r r and an analogous one for v(t, x). Now u is a function of the form ⎛ ⎞ cos ω sin ψ(r/t) u(t, x) = u(t, r cos ω, r sin ω) = ⎝ sin ω sin ψ(r/t) ⎠ , cos ψ(r/t) and this gives, writing

the identities ut ∧ u = Z(ω)

r  r , ψ t2 t

⎞ ⎛ − sin ω Z(ω) = ⎝ cos ω ⎠ 0 1 r ur ∧ u = −Z(ω) ψ  , t t

uωω ∧ u = Z(ω) sin ψ cos ψ.

ω). We are It is not restrictive to consider test functions of the form φ(t, r)φ(t, thus reduced to prove that  
 r  r  1 1  r 1 φ ψ · φ φ + 2 φ sin(2ψ) r dr dt = 0; (5.44) ψ + + t r t2 t t t r 2r recall that ψ(s) is smooth for s = ±1 and ψ  (s) means the derivative a.e. Introduce now the sets Aε = {|t| ≥ |x| + ε}, whose boundary is made of the two cones ∂ ± Aε = {±t = |x| + ε}, and Bε = {|x| ≤ |t| + ε}, whose boundary is made of the two sets ∂ + Bε = {t = |x| − ε, t ≥ 0},

∂ − Bε = {−t = |x| − ε, t ≤ 0}.

Identity (5.44) will follow if we prove that the integral restricted to Aε ∪ Bε converges to 0 as ε → 0. On Aε ∪ Bε the functions are smooth, hence we can integrate by parts and the integrals on the interior cancel (since ψ solves the self-similar

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ODE away from the singularity). Only the boundary terms remain, i.e., we must prove that  
∞
 r r r −εr  νt + νr ψ  φ dS = C ψ φ(r+ε, r)dr → 0, (5.45) (r + ε)2 r+ε 0 ∂ + Aε t t with a similar relation on ∂ − Aε , and  

∞  r r t+ε  t+ε  νt + νr ψ φ dS = εψ φ(t, t + ε)dt → 0 t2 t 0 ∂ + Bε t t

(5.46)

with a similar relation on ∂ − Bε . To prove (5.45), (5.46) it is sufficient to recall that ψ satisfies in all cases the condition τ 2 (1 − τ 2 )ψ  (τ )2 − sin ψ 2 = const , whence the estimate, valid for any τ = ±1, C . |ψ  (τ )| ≤  τ |1 − τ 2 | This implies

  3/2  r ψ ≤ C (r +√ε) r+ε r ε

which gives (5.45), and

   t+ε t2 ψ ≤C √ t (t + ε)3/2 ε

which gives (5.46). The proof for v is identical.




5.4. Appendix A: The self-similar ODE This section is devoted to a complete study of the equation τ 2 (τ 2 − 1)ψ  + τ (2τ 2 − 1)ψ  + k 2 sin ψ cos ψ = 0

(5.47)

which governs the profile of self-similar solutions φ(t, r) = ψ(r/t) to equation (5.9). Here k ≥ 1 is any integer. We can express all solutions to (5.47) in an explicit form using Jacobi elliptic functions. These functions are usually introduced as doubly periodic meromorphic functions with suitable additional properties; but the standard definition is only given for restricted values of the parameters, hence from our standpoint it is both too general and too restrictive. For convenience of the reader, we construct them from scratch in a very short but complete way. Remark 5.3 (Jacobi elliptic functions on R). Consider the system of ODEs for the functions f, g, h : R → R f  = gh,

(5.48)



g = −hf, 

h = −m f g, 2

(5.49) (5.50)

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where m is a fixed real number, subject to the initial conditions f (0) = 0,

g(0) = 1,

h(0) = 1.

(5.51)

It is clear that the system admits a unique C 1 solution (f, g, h) which is in fact real analytic and can be prolonged on the whole R using the first integrals of the system (5.52) f 2 + g 2 = 1, m2 f 2 + h2 = 1, m2 g 2 − h2 = 1 which follow at once from the equations. The standard notation for these functions is f = sn (t|m), g = cn (t|m), h = dn (t|m). Notice that in the literature the parameter m is usually restricted to the range [0, 1[, which is not sufficient for our purposes. It is clear by the definition that sn (t|m) = sn (t| − m),

cn (t|m) = cn (t| − m),

dn (t|m) = dn (t| − m).

Squaring the equations and using the conservation laws (5.52), we obtain immediately 2 (5.53) f  = (1 − f 2 )(1 − m2 f 2 ), g  = (1 − g 2 )(1 − m2 + m2 g 2 ), 2

h

2

= −(1 − h2 )(1 − m2 − h2 ).

(5.54) (5.55)

From (5.53)–(5.55) it is immediate to obtain several properties of the elliptic functions. Consider first the case |m| < 1. Then sn (t|m) and cn (t|m) are periodic with period 4K(m), where
π/2 ds  K(m) = (5.56) 0 1 − m2 sin2 s is called the complete integral of the first species. The couple (sn (t|m), cn (t|m)) has a behavior similar to the couple (sin, cos), i.e., they oscillate between ±1, and indeed we have in the special case m = 0 sn (t|0) = sin t,

cn (t|0) = cos t.

Moreover, the zeros of sn (t|m) are t = 2jK(m) and those of cn (t|m) are (2j + 1)K(m), j ∈ Z. On the other hand, the √ function dn (t|m) has period 2K(m) and oscillates between the values 1 and 1 − m2 , thus it is strictly positive (and is identically 1 when m = 0). A fundamental property connecting elliptic functions with elliptic integrals is the following: for fixed |m| < 1 it holds
α ds √ = β, then sin α = sn (β|m), cos α = cn (β|m). (5.57) if 2 sins s 1 − m 0 This follows from (5.53), (5.54) through the change of variables f → sin f , g → cos g.

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When |m| > 1 the behavior changes. The function sn (t|m) oscillates between ±1/m, and indeed we have the formula 1 (5.58) sn (t|m) = sn (mt|m−1 ) m from which we see that the period is 4K(m−1 )m−1 and the zeros 2jK(m−1 )m−1 . The functions cn (t|m) and dn (t|m) exchange with each other according to the formulas cn (t|m) = dn (mt|m−1 ), dn (t|m) = cn (mt|m−1 ), (5.59) √ so that, for |m| > 1, cn (t|m) is strictly positive, oscillates between 1 and 1 − m−2 and has period 2K(m−1 )m−1 , while dn (t|m) oscillates between ±1 and has period 4K(m−1 )m−1 . When |m| = 1 we have simply et − e−t 2 1 = t , cn (t|1) = dn (t|1) = . (5.60) t −t e +e cosh t e + e−t We finally introduce the fourth Jacobi function called the amplitude and connected to the above through the relations sn (t|1) = tanh t =

sn (t|m) = sin(am (t|m)),

cn (t|m) = cos(am (t|m)).

(5.61)

When |m| < 1 we can compute for |t| < K(m) (i.e., between the first zeros of cn) am (t|m) = arctan

sn (t|m) . cn (t|m)

(5.62)

Actually am (t|m) extends as an analytic function for all t ∈ R; indeed, by (5.57) we have immediately
α ds √ = β, then α = am (β|m) (|m| < 1), (5.63) if 2 sins s 1 − m 0 which means that am (t|m) is the inverse function of the integral to the right, regarded as a function of α. This equivalent definition of am (t|m) is meaningful for any t, provided |m| < 1. When |m| > 1 we can use for all t definition (5.62), since cn (t|m) is strictly positive in this case; thus am (t|m) is 4K(m−1 )/m−1 periodic and oscillates between the values ± arctan[(m2 − 1)−1/2 ]. Notice that also in this case am (t|m) inverts the elliptic integral as in (5.63), but only on a finite interval:
α ds √ if = β, then α = am (β|m) (|m| > 1, |α| < arcsin(|m|−1 )). 1 − m2 sins s 0 (5.64) The zeros of am (t|m) are the same of sn (t|m), that is to say 2jK(m−1 )m−1 . When |m| = 1 we have am (t|1) = 2 arctan(et ) − We are now ready to prove the

et − 1 π = 2 arctan t . 2 e +1

(5.65)

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Proposition 5.1. Consider the equation τ 2 (τ 2 − 1)ψ  + τ (2τ 2 − 1)ψ  + k 2 sin ψ cos ψ = 0.

(5.66)

(i) If ψ is a solution, also jπ ± ψ are solutions, j ∈ Z. (Thus it is sufficient to study the solutions in the range [0, π/2].) The only constant solutions are given by ψ(τ ) = jπ/2. (ii) If ψ is a C 1 solution near a point τ0 ∈]0, 1[, then ψ can be extended to an analytic function on the whole interval ]0, 1[. Moreover, the limits  µ = lim ψ  (τ ) 1 − τ 2 λ = lim ψ(τ ), τ ↑1

τ ↑1

exist and characterize uniquely the solution ψ. Indeed, for λ ∈ (0, π/2) and any µ, or for λ = 0, π/2 and µ = 0 (the excluded cases correspond to the constant solutions), we can represent ψ(τ ) as follows:      π ψ(τ ) = − am (1 + q0 )1/2 q1 + sgn µ · k · arctanh 1 − τ 2 (1 + q0 )−1/2 , 2 (5.67) where
π/2 ds µ2 2  q1 = (5.68) q0 = 2 − sin λ, k λ q0 + sin2 s and sgn µ must be replaced by −1 when µ = 0. (iii) If ψ is a C 1 solution near a point τ0 > 1, then ψ can be extended to an analytic function on the whole interval ]1, ∞[. Moreover, the limits  µ = lim ψ  (τ ) τ 2 − 1 λ = lim ψ(τ ), τ ↓1

τ ↓1

exist and characterize uniquely the solution ψ. Indeed, for λ ∈]0, π/2[ and any µ, or for λ = 0, π/2 and µ = 0 (the excluded cases correspond to the constant solutions), we can represent ψ(τ ) as follows:      1/2 −1/2 q1 + sgn µ · k · arctan τ 2 − 1 q0 , (5.69) ψ(τ ) = am q0 where q0 =

µ2 + sin2 λ, k2


q1 = 0

λ

ds  q0 − sin2 s

(5.70)

and sgn µ must be replaced by −1 when µ = 0. Remark 5.4. Before sketching the proof, we single out a few solutions with special properties that are used in the paper. (a) For τ ∈ (0, 1), most of the solutions given by (5.67) have a nasty behavior near 0, indeed arctanh is unbounded near 1 and am (t|m) either has a linear growth

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(|m| < 1) or oscillates (|m| > 1). The only good solutions are obtained when q0 = 0, i.e., with the choice
π/2 λ ds = − log tan ; λ ∈ (0, π/2], µ = ±k sin λ ⇒ q0 = 0, q1 = sin s 2 λ (5.71) recalling (5.65) and the identities √  π 1 1 + 1 − τ2 2 , − arctan x = arctan exp(arctanh 1 − τ ) = τ 2 x we obtain !  k " τ λ √ ψ(τ ) = 2 arctan tan · , (5.72) 2 1 ± 1 − τ2 where ± = sgn µ. It is useful to compute also 2γτ k √ √ , (1 ± 1 − τ 2 )k + γ 2 (1 ∓ 1 − τ 2 )k

λ γ = tan , 2

(5.73)

√ √ (1 ± 1 − τ 2 )k − γ 2 (1 ∓ 1 − τ 2 )k √ √ cos ψ(τ ) = , (1 ± 1 − τ 2 )k + γ 2 (1 ∓ 1 − τ 2 )k

λ γ = tan , 2

(5.74)

sin ψ(τ ) = and

where ± = sgn µ (and ∓ = − sgn µ). (b) The solutions for τ > 1 have a nice behavior, and in particular they are monotone; we shall be interested in small solutions with the property ψ → 0 as τ → ∞. Actually we have: for any fixed k ≥ 1 and small δ > 0 we can find (unique) λ, µ such that q0 = δ 2 and the solution given by (5.69) tends to 0 as τ → +∞. The solution is monotone decreasing for odd k and monotone increasing for even k. In particular, for k = 1 we can choose q1 = π/2 and the solution is given by   1 1 , (5.75) ψ(τ ) = am δ arctan √ τ2 − 1 δ so that, recalling formulas (5.59),   1 sin ψ(τ ) = δ · sn arctan √ (5.76) δ , 2 τ −1   1 (5.77) cos ψ(τ ) = dn arctan √ δ . 2 τ −1 Indeed, since q0 < 1, the amplitude function in (5.69) has a periodic behavior, 1/2 1/2 with zeros in the points 2jK(q0 )q0 . Recall that
π/2
arcsin √q0 ds ds 1/2   = , K(q0 ) = 2 0 1 − q0 sin s 0 q0 − sin2 s

Wave Maps while


q1 = 0

Notice that q1 ≤

λ

71

ds  . q0 − sin2 s

1/2 K(q0 )

for small λ, µ since # √ λ ≤ arcsin q0 = arcsin sin2 λ + µ2 /k 2 ;

more precisely, if we keep q0 fixed and change the values of λ, µ, we see that q1 1/2 takes all the values from 0 (when λ = 0) to K(q0 ) (when µ = 0). Moreover, K(m) is a strictly increasing function for 0 < m < 1, with K(m) ↓ π/2 as m ↓ 0 and K(m) ↑ ∞ as m ↑ 1. Now the solution (5.69) tends to 0 as τ → ∞ provided −1/2 ) approaches one of the zeros, i.e., provided we can the argument s of am (s|q0 find j ∈ Z such that π 1/2 (5.78) q1 + sgn µ · k = 2jK(q0 ). 2 If k > 0 is odd, we can write k = −2j + 1 for a negative integer j and choosing µ < 0 condition (5.78) becomes π  π 1/2 − K(q0 ) . q1 = + |2j| 2 2 The right-hand side is slightly less than π/2 for small q0 , and keeping the value of q0 fixed we can find λ, µ such that the condition is satisfied (since q1 ranges from 0 1/2 to K(q0 ) > π/2). Thus our claim is proved for odd k. A similar argument holds for even k (we choose now 2j = k and µ > 0). Proof. The claims of part (i) are self-evident. We now prove (ii). It is clear that ψ defined near τ0 ∈]0, 1[ is analytic; if we multiply the equation by ψ  we obtain the identity $ 2 % τ (1 − τ 2 )ψ  (τ )2 − k 2 sin2 ψ = 0, i.e., τ 2 (1 − τ 2 )ψ  (τ )2 − k 2 sin2 ψ = const. A first consequence is that we can extend the solution ψ of (5.66) on the whole  interval ](0, √ 1) since ψ must be bounded on any compact subinterval. Moreover, 2 setting g( 1 − τ ) = ψ(τ ), we see that the function g(s) satisfies the differential equation   τ 4 g  ( 1 − τ 2 )2 − k 2 sin2 g( 1 − τ 2 ) = const , i.e., (1 − s2 )2 g  (s)2 − k 2 sin2 g(s) = const. Thus we see that analytic function near√s = 0, and this implies that √ g(s) is an √ both ψ(τ ) = g( 1 − τ 2 ) and 1 − τ 2 ψ  (τ ) = −τ g  ( 1 − τ 2 ) have a limit as τ ↑ 1, as claimed. Defining λ, µ, q0 , q1 as in the statement, we can write τ 2 (1 − τ 2 )ψ  (τ )2 − k 2 sin2 ψ = q0 k 2 ,

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or equivalently,

ψ  (τ )2 k2 , = 2 2 τ (1 − τ 2 ) q0 + sin ψ

and hence

ψ  (τ ) k =± √ , (q0 + sin2 ψ)1/2 τ 1 − τ2 where ± = sgn µ. To solve this we set χ = −ψ + π/2 and obtain χ (τ ) k =∓ √ . (q0 + 1 − sin2 χ)1/2 τ 1 − τ2 We can now integrate between τ and 1; for τ close enough to 1, recalling (5.63), we obtain immediately that (5.67) holds in a left neighborhood of 1 and hence on ]0, 1[ by analyticity. The proof of part (iii) is analogous.
5.5. Appendix B: Some technical lemmas We recall the standard definitions j 1 uB2,∞ uL2 (2j−1 ≤|ξ|≤2j+1 ) , (R2 ) = uL2 + sup 2  j∈Z

and, for any open set Ω, 1 2 1 1 uB2,∞ (Ω) = inf{u1 B2,∞ (R2 ) : u1 ∈ B2,∞ (R ), u1 | Ω = u}. 1 (Ω) can be defined as the space of restrictions to Ω of functions Of course B2,∞ 1 (R2 ), which in turn are the temperate distributions for which the above from B2,∞ 1 norm uB2,∞ (R2 ) is defined and finite. For details see [45], Section 4.2.1.

Lemma 5.1. The function

x |x| 1 1 , i.e., it belongs to B2,∞ (Ω) for any bounded belongs locally to the Besov space B2,∞ 2 open set Ω ⊆ R . θ(x) =

1 (Ω), it is sufficient to show that ψ(x)θ(x) is in Proof. By the definition of B2,∞ 1 2 B2,∞ (R ) for any cut-off function ψ ∈ C0∞ (Rn ). First of all we recall that the Fourier transform of the function θ from L∞ (R2 ) can be expressed as

 = P.V. ξ = lim χε ξ , θ(ξ) ε↓0 |ξ|3 |ξ|3

χε (ξ) = 1 for |ξ| ≥ ε, 0 elsewhere,

where P.V. (meaning principal value) is exactly defined as the limit in the distributional sense at the right-hand side (see, e.g., [34], p. 164 ff.). Thus we are led to estimate the quantities 
   η    − η) dη ψ(ξ    |ξ|≥ε |η|3  2 j−1 j+1 L (2

≤|ξ|≤2

)

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73

uniformly with respect to ε. We split the integral as follows:


 − η) η dη = ψ(ξ + =: Iε (ξ) + II(ξ). |η|3 |η |≥ε 1≥|η|≥ε |η |≥1 The first part can be handled by the standard trick

η  − η) − ψ(ξ)]   − η) η dη = [ψ(ξ dη, ψ(ξ Iε = 3 |η| |η|3 1≥|η|≥ε 1≥|η|≥ε since η has average 0 on the sphere. We recall now that ψ is rapidly decreasing and hence satisfies   η2 = 1 + |η|2 , (5.79) + |ψ(η)| ≤ CN η−N , |∇η ψ(η)| for any N ; thus by Taylor’s formula, if |η| ≤ 1,   − η) − ψ(ξ)|  |ψ(ξ ≤ sup ∇η ψ(ξ − θη) · |η|, ≤ CN ξ−N |η|, 0≤θ≤1

where CN is independent of ε. This gives
−N |IIε (ξ)| ≤ CN ξ |η|−1 dη ≤ CN ξ−N , |η |≤1

whence, choosing N = 3, IIε L2 (2j−1 ≤|ξ|≤2j+1 ) ≤ C0 2−j

(5.80)

with C0 independent of ε. We now estimate II(ξ), which corresponds to the integration on {|η| ≥ 1}. When |ξ| ≤ 1 it is sufficient to remark that for any N it holds  − η)| ≤ Cη−N |ψ(ξ and this gives immediately IIL2 (|ξ|≤1) ≤ C.

(5.81)

It remains to consider the L2 norm of II(ξ) when |ξ| ∼ 2j , j ≥ 0. We can split II as

 − η) η dη +  − η) η dη =: II1 + III2 . ψ(ξ ψ(ξ II(ξ) = 3 |η| |η|3 |η |≤|ξ|/2 |η |≥|ξ|/2 When |η| ≤ |ξ|/2 we have  − η)| ≤ Cξ − η−N ≤ Cξ−N |ψ(ξ which implies

II1 L2 (2j−1 ≤|ξ|≤2j+1 ) ≤ C2−j ; on the other hand, when |η| ≥ |ξ|/2, we have directly
 − η)|dη ≤ C|ξ|−2 , |III2 (ξ)| ≤ |ξ|−2 |ψ(ξ and this gives

III2 L2 (2j−1 ≤|ξ|≤2j+1 ) ≤ C2−j .

(5.82)

(5.83)

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Piero D’Ancona and Vladimir Georgiev

In conclusion, by (5.80), (5.82), (5.83) we obtain 
   η    ψ(ξ − η) 3 dη    |ξ|≥ε  2 j−1 |η| L (2

≤ C2−j ,

≤|ξ|≤2j+1 )




whence the result follows.

For our second lemma we need an alternative expression of the Besov norm (see, e.g., [34], Section 2.5.1): given δ > 0, which may be chosen arbitrarily, −1 1 uB2,∞ ∆2h uL2 (R2 ) . (R2 )  uL2 + sup |h| |h|≤δ

Here

∆2h

is the second difference operator ∆2h f (x) = f (x + 2h) − 2f (x + h) + f (x),

while the first difference operator is simply ∆1h f (x) = f (x + h) − f (x). Lemma 5.2. Consider the function λ = λ(x) on R2 which is defined by  λ(x) = 1 − |x|2 for |x| ≤ 1, 0 elsewhere. Then, given any f (s, x) ∈ C 2 (R+ × R2 ) with bounded derivatives, the composition 1 1 f (λ(x), x) belongs locally to the Besov space B2,∞ , i.e., it belongs to B2,∞ (Ω) for 2 1 any bounded open set Ω ⊆ R . In particular, λ(x) itself belongs to B2,∞ (R2 ). Proof. Choose any cut-off function ψ(x), and define  g(x) = ψ(x)f ( 1 − |x|2 , x). We need a suitable estimate for ∆2h g(x). Notice that, given any C 2 function Φ(y) on RN and any locally bounded function γ : Rn → RN , the following formulas hold: ∆1h Φ(γ(x)) = a, ∆1h γ(x) where a = a(γ(x + h), γ(x)), for a suitable C 1 vector-valued function a(y, z), and ∆2h Φ(γ(x)) = a, ∆2h γ(x) + A1 ∆1h γ(x + h), ∆1h γ(x) + A2 ∆1h γ(x), ∆1h γ(x), where a = a(γ(x + h), γ(x)), Aj = Aj (γ(x + 2h), γ(x + h), γ(x)), for a(y, z), Aj (y, z, p) continuous functions of their arguments (vector- and matrix-valued respectively). If we apply these formulas to g(x), since x, h run on a compact set we get |∆1h g(x)| ≤ C|∆1h λ(x)| (5.84) and |∆2h g(x)| ≤ C|∆2h λ(x)| + C|∆1h λ(x)|2 + C|∆1h λ(x + h)|2 (5.85) with C independent of x, h. We are thus reduced to estimate the differences of λ(x). For a fixed |h| ≤ 1/4, we split R2 in the three domains |x| > 1 + 3|h|, 1 + 3|h| > |x| > 1 − 3|h| and |x| < 1 − 3|h|. In the first one we have simply ∆1h λ(x) = ∆1h λ(x + h) = ∆2h λ(x) = 0.

(5.86)

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75

In the region 1 + 3|h| > |x| > 1 − 3|h| we have directly |∆1h λ(x)| ≤ 6|h|1/2 ,

|∆1h λ(x + h)| ≤ 7|h|1/2 ,

|∆2h λ(x)| ≤ 16|h|1/2 ,

and since the measure of the region is 12π|h| we obtain for some constant independent of h  1  |∆h λ(x)| + |∆1h λ(x + h)| 2 ≤ C|h|, (5.87) L (1+3|h|>|x|>1−3|h|) ∆2h λL2 (1+3|h|>|x|>1−3|h|) ≤ C|h|.

(5.88)

In the last  region |x| < 1 − 3|h| some more computation is needed; notice that here λ(x) = 1 − |x|2 . We have ∆1h

 |h|2 + 2x · h  1 − |x|2 = −  1 − |x|2 + 1 − |x + h|2

and this implies |∆1h the inequality |∆1h

 1 − |x|2 | ≤ 2|h|1/2 ;

 1 − |x + h|2 | ≤ 2|h|1/2

is analogous. Thus we have      1   ∆h 1 − |x|2 + ∆1h 1 − |x + h|2 

L2 (|x| 0, we can construct a smooth solution v : R × R2 → Sn to the wave map system so that E(0, u − v) ≤ δ while the Lipschitz condition (6.1) is not satisfied at t = 1. We remark that the solutions constructed in the counterexample are radially symmetric, hence the symmetry assumption does not improve the regularity of the solution map. By a different method we can prove a more precise result concerning the uniform continuity of the solution map. Indeed, assume a solution map Φ : H 1 × L2 → C([−T, T ], H˙ 1) ∩ C 1 ([−T, T ], L2) possibly non unique, is defined on some neighborhood U of 0 in H 1 × L2 . Then we can prove that Φ is not uniformly continuous in these spaces. More precisely we have:

Wave Maps

79

Theorem 6.2. Let U be a neighborhood of zero in H 1 (R2 ) × L2(R2 ), and let T > 0. Then for any δ > 0 we can construct two smooth solutions u, v : R × R2 → Sn to the wave map system, belonging to the fixed neighborhood U , such that ∂ ∂t (u − v)(0, ·)L2 (R2 ) + ∇x (u − v)(0, ·)L2 (R2 ) ≤ δ while at t = T ∂ ∂t (u − v)(T, ·)L2 (R2 ) ≥ 1,

∇x (u − v)(T, ·)L2 (R2 ) ≥ 1.

In the last section of this chapter we shall show that this result is not confined to the setting considered here but it is a quite general phenomenon due to the properties of the nonlinearity in the wave map equation; indeed, the uniform continuity in critical spaces is violated for arbitrary dimension and (non flat) targets. 6.2. Well-posedness of the Cauchy problem for semilinear wave equations The linear wave equation ∂t2 u − ∆u = 0

(6.2)

with initial data u(0, x) = u0 (x),

ut (0, x) = u1 (x)

(6.3)

satisfies the energy estimate ∇x u(t, ·)L2 + ∂ ∂t u(t, ·)L2 ≤ C (∇x u0 (·)L2 + u1 (·)L2 ) , provided the initial data u0 , u1 belong to the Hilbert space H = H˙ 1 (Rn ) × L2 (Rn ).

(6.4)

(6.5)

In this case, for any T > 0 the solution map Φ0 is a well-defined linear bounded (i.e., continuous) operator Φ0 : H → C(I; H),

I = [−T, T ],

which takes the data (u0 , u1 ) into a solution u(t, x) = Φ0 (u0 , u1 ) of (6.2) in distribution sense on [−T, T ] × Rn , satisfying the initial conditions (6.3) in a strong sense. The above setting can be slightly extended if we consider a Banach space X = X(I) ⊆ C(I; H) such that Φ0 is a continuous linear operator Φ0 : (u0 , u1 ) ∈ H → X. Now consider the nonlinear Cauchy problem utt − ∆u = F (u),

t ∈ [−T, T ],

x ∈ Rn ,

(6.6)

with initial data (6.3). Here F is a continuous map F : u ∈ X → F (u) ∈ Y 

(6.7)

for some subset Y of the space of distributions D ((−T, T ) × R ). Classical wellposedness is usually formulated in terms of the continuity of the mapping datasolution, (u0 , u1 ) → u(t). More precisely, fixed T > 0 and a suitable Banach space n

80

Piero D’Ancona and Vladimir Georgiev

X = X(T ) ⊂ C(I, H) we shall say that the Cauchy problem (6.6) is well posed in H if there exists a positive r > 0 and a continuous operator Φ : {(u0 , u1 ) ∈ H; (u0 , u1 )H ≤ r} → X = X(T ), so that u(t) = Φ(u0 , u1 )(t) is a solution in distribution sense of (6.6) and satisfies the initial condition (6.3). For an early example of this type of study see [39] where the well-posedness of the Cauchy problem for the wave maps in (t, x) ∈ R × R is considered, and a similar question is investigated by Kenig, Ponce and Vega in a series of papers (see in particular [22]) where, for several classes of equations, an even weaker regularity condition on Φ is assumed, namely uniform continuity. (This type of approach will be considered in the following f Section 7). In the situations when a standard contraction argument is sufficient to prove well-posedness and to construct the solution map Φ (see [33]), one can easily show that Φ is locally Lipschitz continuous. More precisely, one can find a positive r > 0 and a constant C such that u0 , u ˜1 )H u − u ˜X ≤ C(u0 , u1 ) − (˜

(6.8)

u0 , u˜1 ) ∈ H, satisfying for all (u0 , u1 ), (˜ u0 , u ˜1 )H ≤ r. (u0 , u1 )H + (˜

(6.9)

Our aim is to show that (6.8), (6.9) cannot hold in the present case. A preliminary step will be a reduction of the problem to a simpler form using stereographic projection. 6.3. The wave map system in stereographic projection In the following it will be sufficient to consider the case when the target is the twodimensional sphere S2 , since the general case will follow from the trivial embedding S2 ⊂ Sn . Thus we consider the Cauchy problem (utt − ∆u) + Q(∂u)u = 0,

(6.10)

with initial data u(0, x) = u0 (x) ∈ H˙ 1 (R2 ),

ut (0, x) = u1 (x) ∈ L2 (R2 ),

(6.11)

where the quadratic form Q is given by Q(∂u) = |∂ ∂t u|2 − |∇x u|2 .

(6.12)

In order to reduce the vector-valued wave system (6.10) to a scalar one, we compose the map u : (t, x) ∈ R × R2 −→ u = u(t, x) ∈ S2 ⊂ R3 with the stereographic projection u = (u1 , u2 , u3 ) ∈ S2 −→ z ∈ C ∪ ∞, given by z=

u1 + iu2 ; 1 + u3

(6.13)

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81

the South pole S = (0, 0, −1) is mapped in ∞. The inverse map is u1 =

2 Re z 2 Im z 1 − |z|2 , u = , u = . 2 3 1 + |z|2 1 + |z|2 1 + |z|2

(6.14)

The metric induced by the projection is (1 + |z|2 )2 |dz|2 . The lines through the origin are geodesics on C. Hence, we can take a geodesic γ(s) given by γ(s) = (1 + Ai)s = s + h(s)i, h(s) = As, in C, where A is a real constant. The composition of a geodesic with any realvalued solution of the free wave equation generates a wave map u = uγ (see [30]). To see the form of the resulting equation, we shall write it in terms of the scalar function u1 X(t, x) = Re z(t, x) = 1 + u3 and a generic curve s + h(s)i; from (6.14) we get u1 =

2X , 1 + X 2 + h2 (X)

u2 =

2h(X) , 1 + X 2 + h2 (X)

u3 =

1 − X 2 − h2 (X) . (6.15) 1 + X 2 + h2 (X)

Substitution of this ansatz into the wave map equation gives the scalar equation M (X)2X − L(X)Q(∂X) = 0,

(6.16)

where L(X) = 4h(X)h (X)(−3X 2 +h2 (X)+1) − (1 − (h (X))2 )(2X 3 −6Xh2(X)−2X), (6.17) 4 2 2 3  2  M (X) = −X +(1+h (X)) −2X h(X)h (X)−2X(1+h (X))h(X)h (X). (6.18) Indeed, to prove (6.16), we start with the relations   1 − X 2 + h2 (X) − 2Xh(X)h (X) 2X ∂xj = 2∂ ∂ X , x j 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))2   −2Xh(X) + (1 + X 2 − h2 (X))h (X) 2h(X) = 2∂ ∂xj X ∂xj , 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))2 and   X + h(X))h (X) 1 − X 2 − h2 (X) = −4∂ ∂ X ∂xj , xj 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))2 which imply 1 + (h (X))2 . Q(∂u) = 4Q(∂X) (1 + X 2 + h2 (X))2

(6.19)

(6.20)

(6.21)

(6.22)

Computing the second-order derivative (note that h = 0) gives ∂x2j xj

2(∂ ∂x2j xj X)M1 (X) − 4(∂ ∂xj X)2 L1 (X) 2X = , 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))3

(6.23)

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where L1 (X) = X(1 + X 2 + h2 (X))(1 + (h (X))2 ) + 2(X + h(X)h (X))(1 − X 2 + h2 (X) − 2Xh(X)h(X)), M1 (X) = (1 − X 2 + h2 (X) − 2Xh(X)h(X))(1 + X 2 + h2 (X)). From these identities we get   2X 2(2X)M1 (X) − 4Q(∂X)L1(X) 2 , = 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))3

(6.24) (6.25)

(6.26)

and combining (6.26) and (6.22), we finally obtain (6.16). In the special case h(X) = AX, where A is a real constant, we obtain L(X) = 4A2 X(−3X 2 + A2 X 2 + 1) − 2X(1 − A2 )(X 2 − 3A2 X 2 − 1) (6.27) = 2X(1 + A2 )(1 − X 2 (1 + A2 )), M (X) = −X 4 + (1 + A2 X 2 )2 − 2A2 X 4 − 2A2 X 2 (1 + A2 X 2 ) = (1 − X 2 (1 + A2 ))(1 + X 2 (1 + A2 )). The equation (6.16) suggests us to take X so that the equation 2X + f (X)Q(∂X) = 0

(6.28)

is satisfied. Here

2X(1 + A2 ) . (6.29) 1 + X 2 (1 + A2 ) It is clear that (6.28) implies (6.16). This scalar nonlinear wave equation can be transformed into a linear wave equation (see [27]) by the aid of the transform
X
s Y = G(X) ≡ eF (s) ds, F (s) = f (σ)dσ. f (X) = −

0

0

√ So using (6.29), we find F (s) = − ln(1 + B s ), where B = 1 + A2 and Y = B −1 arctan(BX). In conclusion, given any solution of the linear wave equation 2 2

2Y = 0

(6.30)

the function (6.31) X = B −1 tan(BY ) is a solution of the scalar nonlinear wave equation (6.28) and from (6.15) we see that the function u = uA (t, x) defined by  sin(2BY ) A sin(2BY ) , u2 = , u3 = cos(2BY ), B = 1 + A2 , (6.32) u1 = B B is a wave map. Now we can choose the special solutions of the free wave equation (6.30) that we shall use in the following: we take
dξ Y (t, x) = Re (6.33) sin(t|ξ|)eix·ξ ϕ(ξ) , |ξ| R2

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83

which corresponds to the choice of initial data Y (0, x) = 0,

ˆ Yt (0, x) = Re ϕ.

This implies in particular ∂ ∂t Y (t, ·)L2 + ∇x Y (t, ·)L2 ≤ CϕL2 for any t ≥ 0. Thus for the resulting wave map uA , by (6.32) we have ∂ ∂t uA (0, ·)L2 + ∇x uA (0, ·)L2 ≤ CϕL2 ,

(6.34)

and noticing that ∂t uA (0) = ∂t Y (0, x)(2, 2A, 0),

∇x uA (0) = ∇x Y (0, x)(2, 2A, 0).

we have also C −1 |A1 − A2 |ϕL2 ≤ ∂ ∂t (uA1 (0, ·) − uA2 (0, ·))L2 + ∇x (uA1 (0, ·) − uA2 (0, ·))L2 ≤ C|A1 − A2 |ϕL2

(6.35)

for some constant C independent of ϕ, A, A1 , A2 . 6.4. Conclusion of the proof of Theorem 6.1 Assume by contradiction that the solution map is (locally) Lipschitz continuous. Then for any real numbers A, A˜ such that ˜ 0 < A < A, we can consider the wave maps u and uA˜ constructed in (6.32), and estimate (6.35) implies that ˜ ∂ ∂t (u(t, ·) − uA˜ (t, ·))L2 + ∇x (u(t, ·) − uA˜ (t, ·))L2 ≤ C|A − A|ϕ L2 . ˜ and taking the limit A˜ → A, we get Dividing by |A − A| ∂ ∂t ∂A u(t, ·)L2 + ∇x ∂A u(t, ·)L2 ≤ CϕL2 . The A-derivatives can be computed explicitly; from (6.32) we obtain ∂A u1 = ∂A u2 =

2A B2 Y 2A2 B2 Y

cos(2BY ) − cos(2BY )

A B 3 sin(2BY ), + B13 sin(2BY ),

∂A u3 = − 2A B Y sin(2BY ). Then taking the time derivative, we find the following pointwise estimate: |∂ ∂t ∂A uA (t, x)| ≥ C0 (A)|Y (t, x)||∂ ∂t Y (t, x)| − C1 (A)|∂ ∂t Y (t, x)|, where C0 (A) > 0 provided A > 0. For space derivatives we have an analogous estimate: |∇x ∂A uA (t, x)| ≥ C0 (A)|Y (t, x)||∇x Y (t, x)| − C1 (A)|∇x Y (t, x)|.

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Therefore, if we fix an A > 0, the assumption that the solution map is Lipschitz continuous implies the inequality ∂t Y (t, ·)L2 + C0 (A)Y (t, ·)∇x Y (t, ·)L2 − C1 (A)∂ ∂t Y (t, ·)L2 C0 (A)Y (t, ·)∂ − C1 (A)∇x Y (t, ·)L2 ≤ ∂ ∂t ∂A u(t, ·)L2 + ∇x ∂A u(t, ·)L2 ≤ CϕL2 . On the other hand, the classical energy identity gives ∂ ∂t Y (t, ·)L2 + ∇x Y (t, ·)L2 ≤ CϕL2 and we arrive at Y (t, ·)∂ ∂t Y (t, ·)L2 + Y (t, ·)∇x Y (t, ·)L2 ≤ CϕL2 .

(6.36)

Recall that this estimate is valid locally, i.e., only for ϕL2 ≤ r according to (6.9); but by a standard rescaling argument we obtain that the estimate (6.36) with ϕL2 ≤ r implies the scale invariant estimate Y (t, ·)∂ ∂t Y (t, ·)L2 + Y (t, ·)∇x Y (t, ·)L2 ≤ Cϕ2L2

(6.37)

also for large values of ϕL2 . In conclusion we have proved that for any solution Y (t, x) of the linear wave equation of the form (6.33), an estimate like (6.37) holds, with a constant C independent of the norm of the arbitrary initial datum ϕ. Clearly this implies that (6.37) is valid also for all solutions of the form
dξ Y (t, x) = (6.38) sin(t|ξ|)eixξ ϕ(ξ) |ξ| R3 for arbitrary ϕ ∈ L2 . The proof will be concluded if we show that such an estimate is impossible. Indeed, (6.37) implies
Ψ(x)Y (t, x)∂ ≤ CΨL2 ϕ2 2 ∂ Y (t, x)dx (6.39) t L for any Ψ ∈ L2 . Using the Plancherel identity and (6.38) we see that this inequality yields

dη  Ψ(ξ − η) cos(t|ξ|) sin(t|η|)ϕ(ξ)ϕ(η)dξ ≤ CΨL2 ϕ2L2 . (6.40) |η| Now for any even integer M > 2 we define the function (compare with [27]) 1 ϕM (ξ) = H(AM ) , (6.41) |ξ| ln5/8 |ξ| where (6.42) AM = {ξ ∈ R2 : 2 ≤ |ξ| ≤ M, dist(|ξ|, 8Z + 1) < 1/2} while H(A) denotes the characteristic function of the set A. The condition dist(|ξ|, 8Z + 1) < 1/2 ensures that the following inequalities are satisfied: ϕM (ξ) sin(t0 |ξ|) ≥ CϕM (ξ) ≥ 0, ϕM (ξ) cos(t0 |ξ|) ≥ CϕM (ξ) ≥ 0

(6.43)

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with C > 0 and t0 = π/4 (by the non-negativity of sin). The function Ψ will be chosen as follows:  Ψ M (ξ) = H(2 ≤ |ξ| ≤ M )

1 |ξ| ln9/16 |ξ|

.

(6.44)

Notice that for any M > 3 we have the estimates ϕM L2 (R2 ) ≤ C, ΨM L2 (R2 ) ≤ C

(6.45)

with some constant C independent of M > 3. Further, we take N ∈ 8Z + 1 and M ∈ 16Z + 1 so that M . 3 0 we can find δ > 0 such that the condition ˜)|t=0 L2 ≡ φL2 < δ Dt,x (u − u implies, for any fixed T > 0, Dt,x (u − u ˜)|t=T L2 < ε. In view of inequality (6.51), this means that for any ε > 0 we can find δ > 0 such that 2 2 2 z(T, 0) φL2 < δ =⇒ µ sin < δ + ε; (6.53) = 3 2 recall that z(t, x) is a generic solution of the free wave equation, with data z(0, x) = 0, zt (0, x) = φ(x), subject to the only constraint φL2 < r0 − µ = r0 /2. Thus the proof is reduced to a property of the linear wave equation, and it is not difficult to show that (6.53) is absurd. Here we shall give a proof based on harmonic analysis methods; a different proof based on the explicit representation of the fundamental solution can be found in Section 7. More precisely, we shall construct a sequence of data φj (x) which converges to 0 in the norm L2 , such that the corresponding solution z(t, x) satisfies z(T, 0) = C > 0, and this is a contradiction with (6.53) and concludes the proof of the theorem.
Proposition 6.1. Fix T > 0. There exists a sequence φj (x) of real-valued, smooth functions (actually in the Schwartz class) such that φj → 0 in L2 and the solution to the Cauchy problem
zj = 0,

zj (0, x) = 0,

∂t zj (0, x) = φj (x)

satisfies zj (T, 0) = 1 for all j. Proof. To prove this statement we observe first of all that for any function g(ξ) with the properties g(−ξ) = g(ξ), the function

g(ξ) is real-valued


z(t, x) = R2

sin(t|ξ|) cos(x · ξ)g(ξ)

dξ |ξ|

is a real-valued solution of the Cauchy problem
z = 0,

z(0, x) = 0,

zt (0, x) = gˆ(x)

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89

(since gˆ(x) is also real-valued). Thus to achieve the proof it will be sufficient to construct a sequence of functions φN (ξ) with the properties φN (ξ) = φN (−ξ) ≥ 0,

φ(ξ) is real-valued,

φN → 0 in L2 ,

and such that, at a fixed point x0 which we may choose, e.g., equal to x0 = (1, 0),
dξ zN (T, x0 ) = ≥ 1. sin(t|ξ|) cos(ξ1 )φN (ξ) |ξ| 2 R To this end we define

Dj,m = ξ :

the sets Dj,m in R2 , m ∈ Z, j ≥ 1, & ' ( |ξ| π − (2j + 1) ≤ π ∩ ξ : |ξ1 − 2mπ| ≤ π ; T 2 4 4

the first condition ensures that ξ ∈ Dj,m

sin(T |ξ|) ≥ C0 > 0,

= =⇒

while the second one implies that ξ ∈ Dj,m

cos(ξ1 ) ≥ C0 > 0

= =⇒

for some constant C0 independent of j, m. Moreover, denoting by ) Dj = Dj,m m∈Z

we shall choose the functions φN of the following form: φN (ξ) =

bN 

cj χDj ,

cj ≥ 0,

j=aN

where χDj are the characteristic functions of the sets Dj . Notice that this implies immediately that φN are real-valued, non-negative, and φN (−ξ) = φN (ξ) since the sets Dj are symmetric for reflections. If we notice that, by the first condition, ξ ∈ Dj,m

|ξ| ≥ T πj,

=⇒ =

we can write zN (T, x0 ) ≥ C02 T π

bN 

cj j −1 µ(Dj ),

j=aN

where µ is the Lebesgue measure; on the other hand we have φN 2L2 =

bN 

c2j µ(Dj ).

j=aN

It is an elementary exercise to obtain the estimates T 2 π 3 (j + 1/4) ≥ µ(Dj ) ≥

π2 j, 16

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Piero D’Ancona and Vladimir Georgiev

and hence we obtain zN (T, x0 ) ≥ C02 T

bN π3  cj , 16 j=a

φN 2L2 ≤ 2T 2π 3

bN 

jc2j .

j=aN

N

We need a sequence cj ≥ 0 and indices aN ≤ bN such that the first sum is bounded from below by some positive constant independent of N , while the second sum tends to 0 as* N → ∞; this can be * obtained immediately if we choose a sequence ∞ ∞ cj such that 1 cj diverges while 1 jc2j converges, such as cj =

1 . j log j

The proof of the proposition is concluded.




7. Ill-posedness in the critical case (fundamental solution approach) 7.1. Introduction This final section is devoted to a more detailed study of the properties of the solution map in the critical spaces Φ : H˙ n/2 × H˙ n/2−1 → C([0, T ]; H˙ n/2 ) ∩ C 1 ([0, T ]; H˙ n/2−1 ) in the case of a general target and arbitrary base space dimension. Notice that it is not known if such a map is defined at all, or is unique if it exists; however, our counterexamples are always based on explicit smooth solutions, for which uniqueness holds, hence any possible solution map must contain our counterexamples in its image. The behavior of the wave map system in the critical case s = n/2 is largely an open problem (while in the subcritical case s < n/2 one has in general illposedness in a sharp sense, i.e., non-uniqueness, see, e.g., [39] and [10]). A possible line of attack was suggested by Bourgain (see [2], [3] and also Tzvetkov [46]) who proved that the map data
→ solution to some nonlinear evolution equations is not C 2 in the subcritical Sobolev spaces. This holds for the cubic NLS, for KdV and mKdV with different critical indices. The result was sharpened by Kenig, Ponce and Vega [22] who proved that the solution map actually is not (locally) uniformly continuous in the subcritical spaces. We also mention [4] and [24] where the case of the supercritical nonlinear wave equation and of the Benjamin-Ono equation are considered. Our aim here is to prove a similar result for the wave map system in the critical case s = n/2. Our assumption on N will be quite general; essentially we only require that N is not flat. More precisely, we assume that there exists a geodesic curve γ :] − s0 , s0 [→ N with γ(0) = 0, γ  (0) = 0.

(7.1)

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91

From such generality it will be clear that the ill-posedness in the sense of uniform continuity is a general property of nonlinear systems like (1.1) more than a geometric property. Our result is the following: Theorem 7.1. Let N be a smooth Riemannian manifold, isometrically embedded in Rm , such that there exists a geodesic curve γ :] − s0 , s0 [→ N with γ  (0) = 0.

γ(0) = 0 ∈ N,

(7.2)

Assume a solution map Φ : (u0 , u1 ) → u for system (1.1) with data (1.2) is defined on some neighbourhood U of 0 in X × Y = H n/2 (Rn ; N ) × H n/2−1 (Rn ; N ). Then, for any T > 0, Φ is not uniformly continuous between the spaces Φ : U ⊆ H n/2 × H n/2−1 → C([0, T ]; H˙ n/2 (Rn , Rm )) or Φ : U ⊆ H n/2 × H n/2−1 → C 1 ([0, T ]; H˙ n/2−1 (Rn , Rm )). As usual, we say that Φ : U ⊆ X × Y → L is uniformly continuous on U if: (1)

(1)

(2)

(2)

for any ε > 0 there exists δ > 0 such that, for any (u0 , u1 ) and (u0 , u1 ) in U it holds (1)

(1)

(2)

(2)

(u0 , u1 ) − (u0 , u1 )X×Y ≤ δ (1)

(1)

(2)

⇒

u(1) − u(2) L ≤ ε,

(7.3)

(2)

where u(1) = Φ(u0 , u1 ), u(2) = Φ(u0 , u1 ). Thus the above result excludes in particular that Φ is (locally) Lipschitz or H¨ o¨lder continuous. Remark 7.1. It is not difficult to prove by similar arguments that the solution map is not uniformly continuous also in the subcritical case, i.e., from H s × H s−1 , 1 ≤ s < n/2 with values in C([0, T ], H s ) or C 1 ([0, T ], H s−1 ). However, it is already known, at least in the case of a rotationally symmetric target, that a much stronger ill-posedness result holds, namely the local non-uniqueness can be proved. This was obtained for n = 3 in [32], for n ≥ 4 in [5] and for n = 2 in [10]. Since the arguments in these results have a local nature, it is reasonable to argue that non-uniqueness may hold also in the general nonsymmetric case. Remark 7.2. The proof of the theorem is based on an explicit construction of sequences of data such that the corresponding solutions violate (7.3); such solutions are of geodesic type, i.e., of the form γ◦v(t, x), where v(t, x) is a real-valued solution of the homogeneous wave equation. We recall that if γ(s) = (γ1 (s), . . . , γm (s)) is an arbitrary curve in Rm with values in N , and v(t, x) an arbitrary real-valued function, for the composition u(t, x) = γ(v(t, x)) we can write  
u + Γbc (u)∂α ub ∂ α uc ≡ γ ·
v + γ + Γbc (γ)γb γc · ∂α v∂ α v and this is identically zero as soon as
v = 0 and γ(s) is a geodesic curve.

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Remark 7.3. The ill-posedness for the wave map problem in the case n = 1, s = 1/2 is proved in [39]. It is interesting to mention also the paper [27], where a scalar wave equation of the form
u + f (u)∂α u∂ α u = 0 is studied in the critical spaces H n/2 × H n/2−1 . Of course in the scalar case it is possible to prove a much stronger ill-posedness result (actually, a blow-up result). Remark 7.4. It is important to notice that in the proof of Theorem 7.1 the fact that Γjk are Christoffel symbols of some Riemannian manifold is not essential. In other words, the result holds for any system of the form (1.1), provided the curves locally defined by the system of equations γ + Γbc (γ)γb γc = 0 satisfy an assumption like (7.2) near some point. This means that the ill-posedness in the sense of uniform continuity is a general property of systems of the wave map type. 7.2. Proof of Theorem 7.1 It is not restrictive to assume that γ is parameterized by arc length; moreover, in the following we shall take T = 1 for simplicity of notations but the proof is unchanged in general. Assumption (7.2) implies that for some component γj of γ  one has |γ γj (s)| ≥ c1 (N ) for |s| ≤ c0 (N ) (7.4) for suitable constants c0 , c1 depending only on the manifold N . Let v, w be two C ∞ real-valued solutions of the homogeneous wave equation
v =
w = 0 with data v(0, x) = w(0, x) = v0 (x),

vt (0, x) = v1 (x),

wt (0, x) = w1 (x).

Notice that v(0, x) ≡ w(0, x) and only the second datum is different. Moreover, we shall always work with data of compact support, so that v(t, ·), w(t, ·) will have support in a fixed ball (say B(0, 10)) for all t ∈ [−1, 1]. Then the functions u(1) = γ ◦ v, u(2) = γ ◦ w are solutions of the wave map equation (see Remark 7.2), provided v, w take their values in the domain of γ(s); more precisely we shall assume that |v| ≤ c0 (N ), |w| ≤ c0 (N ), (7.5) and these conditions will be verified in the explicit construction of v and w. The corresponding Cauchy data are given by u(1) (0, x) = u(2) (0, x) = γ(v0 )(x),

ut (0, x) = γ  (v0 )v1 (x), (1)

ut (0, x) = γ  (v0 )w1 (x). (2)

(7.6)

Assume now that the solution map is defined and uniformly continuous on some neighbourhood U of 0 in H n/2 (Rn ; N ) × H n/2−1 (Rn ; N ) with values in the

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93

space C 1 ([0, 1]; H˙ n/2−1 ) (the case of C([0, 1]; H˙ n/2 ) is completely analogous). If we apply this to the data (7.6), we obtain that: for any ε > 0 there exists δ > 0 such that ∂t u(2) (0, ·)H n/2−1 ≤ δ ⇒ sup ∂ ∂t u(1) (t, ·)−∂ ∂t u(2) (t, ·)H˙ n/2−1 ≤ ε ∂ ∂t u(1) (0, ·)−∂ t∈[0,1]

(7.7) (1) (2) for all data u(1) (0, x) = u(2) (0, x) and ut (0, x), ut (0, x) in U . We can express this condition in terms of the data for v, w. Indeed, we have ∂t u(1) (0, x) − ∂t u(2) (0, x) = γ  (v0 )(v1 − w1 )(x), where γ  (v0 ) is a smooth function, equal to a constant outside some compact set in Rn . Applying Lemma 7.3 from the Appendix for s = n/2 − 1, we have γ(f )gH n/2−1 ≤ cn γ(f )L∞ ∩H n/2 · gH n/2−1 , where we are using the notation uX∩Y = uX + uY . Since γ(0) = 0 we can apply the standard Moser type estimate γ(f )H n/2 ≤ ρ0 (f L∞ ) · f H n/2

(7.8)

for a suitable continuous increasing function ρ0 (s) (see e.g. [43], Vol.III, Chapter 13, Proposition 10.2), we obtain an inequality like γ(f )gH n/2−1 ≤ ρ1 (f L∞ ∩H n/2 ) · gH n/2−1

(7.9)

for some continuous increasing ρ1 (s), which is valid provided the range of the realvalued function f is contained in a compact subset of the domain of the smooth function γ(s). Then we have ∂ ∂t u(1) (0) − ∂t u(2) (0)H n/2−1 ≤ ρ1 (v0 L∞ ∩H n/2 )v1 − w1 H n/2−1 ,

(7.10)

hence property (7.7) implies the following: for all ε > 0 there exists δ > 0 such that v1 − w1 H n/2−1 ≤ δ

⇒ sup ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 ≤ ε

(7.11)

t∈[0,1]

for all data (v0 , v1 ) and (w0 , w1 ) belonging to a suitable neighbourhood V of 0 in H n/2 (Rn ) × H n/2−1 (Rn ) and such that v0 = w0 and |v0 | ≤ c0 , where c0 = c0 (N ) is defined in (7.4). We now estimate from below the second term in (7.11) ∂t v − γ  (w)∂ ∂t wH˙ n/2−1 . ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 = γ  (v)∂ We have ∂t v−γ  (w)∂ ∂t wH˙ n/2−1 ≥ (γ  (v)−γ  (w))∂ ∂t vH˙ n/2−1 −γ  (w)∂ ∂t (v−w)H˙ n/2−1 . γ  (v)∂ (7.12) We apply (7.9) to the last term and obtain γ  (v)∂ ∂t (v − w)H˙ n/2−1 ≤ ρ1 (vL∞ ∩H n/2 )∂ ∂t (v − w)H n/2−1 ;

94

Piero D’Ancona and Vladimir Georgiev

by the energy identity for
(v − w) = 0, v0 = w0 , we know that ∂ ∂t (v − w)H n/2−1 ≤ v1 − w1 H n/2−1 and in conclusion γ  (v)∂ ∂t (v − w)H˙ n/2−1 ≤ ρ1 (vL∞ ∩H n/2 )v1 − w1 H n/2−1 .

(7.13)

To estimate from below the first term in the right-hand side of (7.12) we use Taylor’s formula and get γ  (b) − γ  (a) = γ  (a)(b − a) + F (a, b)(b − a)2 ,

γ  (a) = γ  (0) + G(a) · a,

where F (a, b), G(a) are smooth functions of their arguments whose explicit expression is not relevant. Then γ  (v) − γ  (w) = γ  (0) · (v − w) + R(v, w) · (v − w), where we have written for short R(u, v) = G(v) · v + F (v, w)(v − w). 

Recalling that |γ (0)| ≥ c1 (see (7.4)), we have γ  (v)(v − w)∂ ∂t vH˙ n/2−1 ≥ c1 (v − w)∂ ∂t vH˙ n/2−1 − R(v, w)(v − w)∂ ∂t vH˙ n/2−1 . Now we can apply (7.59) of Lemma 7.3 from the Appendix to obtain ∂t vH n/2−1 , R(v, w)(v − w)∂ ∂t vH˙ n/2−1 ≤ R(v, w)L∞ ∩H n/2 (v − w)∂ while using (7.8) it is standard to obtain R(v, w)L∞ ∩H n/2 ≤ ρ2 (v, wL∞ ∩H n/2 ) · v, wL∞ ∩H n/2 (v, w = v+w) with some continuous increasing function ρ2 (s) whose precise form is not relevant. In conclusion, recalling also (7.12) and (7.13), we have proved the inequality ∂t vH˙ n/2−1 ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 ≥ c1 (v − w)∂ − ρ2 (v, wL∞ ∩H n/2 ) · v, wL∞ ∩H n/2 (v − w)∂ ∂t vH n/2−1 − ρ1 (vL∞ ∩H n/2 )v1 − w1 H n/2−1 . (7.14) To proceed, we must construct explicitly the functions v and w. This is done with the help of a few lemmas. Lemma 7.1. Let n ≥ 2. There exists a sequence of real-valued functions φj ∈ C0∞ (Rn ) supported in the ball {|x| ≤ 2}, with φj → 0

in

H n/2−1 (Rn )

as

j→∞

(7.15)

such that, denoting by zj (t, x) : R × Rn → R the solution of the linear problem
z = 0,

z(0, x) = 0,

zt (0, x) = φj (x)

(7.16)

one has zj (1, 0) = 1

for any

j.

(7.17)

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95

The functions φj and hence zj (t, x) can be chosen as radial functions in x, i.e., depending only on |x|. Proof. We begin with the case n = 2. For 0 < p < q < 1, we define ψp,q (y) on R2 as follows: I{p≤|y|≤q} (y) , (7.18) ψp,q (y) = −  1 − |y|2 log(1 − |y|2 ) where IA (y) denotes the characteristic function of the set A. An elementary computation gives 1 1 − . (7.19) 2ψp,q 2L2 (R2 ) = 2 log (1 − q ) log (1 − p2 ) Notice that taking any 0 < p1 < p < q < q1 < 1, and an arbitrary smooth radial cut-off function χp,q with I{p≤|y|≤q} (y) ≤ χp,q (y) ≤ I{p
≤|y|≤q
} (y), we can modify definition (7.18) as follows: χ{p≤|y|≤q} (y) ψ p,q (y) = −  1 − |y|2 log(1 − |y|2 )

(7.20)

in order to obtain a smooth initial datum with similar norm: 1 1 1 1 − ≤ 2ψ p,q 2L2 (R2 ) ≤ − . (7.21) log (1 − q 2 ) log (1 − p2 ) log (1 − q12 ) log (1 − p21 ) On the other hand, the solution zp,q (t, x) of the problem
z = 0,

z(0, x) = 0,

is explicitly given by zp,q (t, x) =

t 2π


|x−y|≤t

zt (0, x) = ψp,q ψp,q (y)  dy, t2 − |x − y|2

and in particular at (t, x) = (1, 0) one has
log(1 − q 2 ) 1 1 1 zp,q (1, 0) = − dy = log log(1 − p2 ) . 2π p≤|y|≤q (1 − |y|2 ) log(1 − |y|2 ) 4π By the positivity of the kernel we have immediately, for the solution z p,q obtained by replacing ψp,q with ψ p,q , 2 log(1 − q 2 ) 1 ≤ z p,q (1, 0) ≤ 1 log log(1 − q1 ) . log log(1 − p2 ) 4π log(1 − p2 ) 4π 1

If we now choose for δ ∈ (0, 1) 1 − p21 = δ,

1 − p2 = δ 2 ,

and write ψδ = ψ p,q , we obtain 1 √ | log δ|−1/2 ≤ ψ δ L2 ≤ 12

1 − q2 = δ3 , +

3 | log δ|−1/2 → 0 8

1 − q12 = δ 4

as δ → 0

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while zδ = z p,q satisfies 3 1 1 log ≤ zδ (1, 0) ≤ log 4. 4π 2 4π Defining φj = zδj (1, 0)−1 ψδj for any δj ↓ 0 we obtain the thesis. The general case for even n ≥ 2 follows easily by modifying the above example, using the fact that the solution of (7.16) can be represented as
 |α|+1 aα t y α Dα φ(x + ty)(1 − |y|2 )−1/2 dy, z(t, x) = |y |≤1

0≤|α|≤(n−2)/2

which, for a radial function φ, gives 




(n−2)/2

z(1, 0) =

1

cν

ν=0

∂rν φ(r)(1 − r2 )−1/2 rν+n−1 dr.

0

Here of course we shall choose a datum φ such that its radial derivative of order n/2 − 1 is of the form ψ p,q as seen above. Let us now consider the case of odd n, starting from n = 3. In this case it is sufficient to use the well-known fact (see, e.g., Theorem 11.1 in Volume I of [23]) that, for any bounded Ω ⊂ Rn with C ∞ boundary, C0∞ (Ω)

is dense in H 1/2 (Ω)

and also in H s for s ≤ 1/2. Since we shall need a special version of this result for radial functions, we shall give here a self-contained proof adapted to our situation. Indeed, consider the space Z = {φ ∈ C0∞ (R) : φ(x) = φ(−x), φ ≡ 0 near 1 and − 1}

(7.22)

(where “near ±1” means “on some neighbourhood of these two points, depending on φ”). It is easy to see that Z is a dense subset of the space of even functions from H 1/2 (R), that is, 1/2 Heven (R) = {u ∈ H 1/2 (R) : u(x) = u(−x)}

(7.23)

1/2

by the following argument: in the Hilbert space Heven (R) we can certainly choose a u0 orthogonal to Z, and we must only prove that u0 = 0. The tempered distribution T whose Fourier transform is given by T = ξ u0 belongs to H −1/2 (R) and by the identity  2 = (u , φ) 1/2 = 0 T (φ) = (ξ u0 , φ) 0 L H for any test function in Z, we see that the support of T is contained in the set {±1}, i.e., T is a linear combination of a finite number of derivatives of δ1 , δ−1 .

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Hence T(ξ) is a function of the form T(ξ) =

N 

(c eiξ + d e−iξ )ξ 

=0

for a suitable N ≥ 0 and complex numbers c , d , and at the same time ξ−1/2 T(ξ) must belong to L2 . It is trivial to see that the only such function T is 0, and this implies u0 ≡ 0 too. 1/2 Thus we have proved that C0∞ (]1, 1[) is dense in Heven (] − 1, 1[) since this last 1/2 space coincides with the space of restrictions of functions in Heven (R) to ] − 1, 1[, with the restriction norm (the norm of u is the infimum of the norms of its possible extensions). It would be not difficult to prove the same result for higher dimensions, but actually here we only need to construct a sequence of radial functions ψj ∈ C0∞ (B1 ) which converges to 1 in H 1/2 (B1 ), where B1 is the unit ball B1 = {x ∈ Rn : |x| < 1}. To this end, it is sufficient to remark that the operator s ((−1, 1)) → H s (B1 ) A : Heven

defined as A(f )(x) = f (|x|) is bounded for all 0 ≤ s ≤ 1: this is proved directly for s = 0, 1 and follows, e.g., by interpolation for the intermediate values of s. Hence taken any sequence fj (x) in C0∞ ((−1, 1)), with f (x) = f (−x), converging to 1 in the H 1/2 ((−1, 1))-norm, we only need to define ψj (x) = fj (|x|) to obtain the desired result. Now, setting φj = 1 − ψj we obtain a sequence of radial smooth functions on B1 , converging to 0 in the norm of H 1/2 (B1 ), and identically equal to 1 on some neighbourhood of ∂B1 (depending on j). Then by Kirchhoff’s formula we obtain
1 z(1, 0) = φj (y)dS = 1 4π ∂B1 as needed. In the general case n ≥ 3 odd, we proceed in a similar way using the general representation of the solution; notice that for radial φ the following formula holds 

(n−3)/2

z(1, 0) =

bν ∂rν φ(1)

ν=0

with suitable constants bν .




After the constructions of the lemma we have no control on the L∞ -norm of the functions zj ; if we give up the requirement that the zj be radial, however, it is easy to obtain the following result:

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Corollary 7.1. Let n ≥ 2. There exists a sequence of real-valued functions φj ∈ C0∞ (Rn ) supported in the ball {|x| ≤ 5} with φj → 0

in

H n/2−1 (Rn )

as

j→∞

(7.24)

such that, denoting by zj (t, x) : R × R → R the solution of the linear problem n


z = 0,

z(0, x) = 0,

zt (0, x) = φj (x)

(7.25)

one has zj (tj , 0) = 1

for some sequence

tj ∈ (0, 1]

(7.26)

and |zj (t, x)| ≤ 1

for all (t, x, j) ∈ [0, 1] × Rn × N.

(7.27)

Proof. The functions zj constructed in the previous lemma are smooth and compactly supported, let (tj , xj ) be a point where |zj | attains its maximum value mj on the strip [0, 1] × Rn , and define z j (t, x) = m−1 j zj (t, x − xj ) (and possibly multiply by the sign of zj (tj , xj )). Notice that tj > 0 since z(0, x) ≡ 0. This concludes the proof.
Before passing to the main part of the proof, a last elementary rescaling lemma is necessary. Lemma 7.2. Let χ(x) with χH˙ n/2−1 = 0 be a smooth compactly supported (radial) function, vanishing for |x| ≥ 2, and with the property
χ(x)dx = 0. (7.28) Rn

Let R ≥ 1, M ≥ 0 be positive numbers, 0 ≤ T ≤ 1, and denote by vR,M,T (t, x) the (radial) solution of the homogeneous wave equation
v = 0,

v(T, x) = 0,

vt (T, x) = χR,M (x) := M χ(Rx)

(7.29)

with data at t = T > 0. Denote by v0 , v1 the traces v0 = vR,M,T (0, x),

v1 = ∂t vR,M,T (0, x)

(7.30)

so that (7.29) is equivalent to a Cauchy problem for the homogeneous wave equation with data v0 , v1 at t = 0. Then the following estimates hold, for a constant cn depending only on the space dimension n and on the function χ(x): M (7.31) v0 H n/2 + v1 H n/2−1 ≤ cn , R and, for all (t, x) ∈ [0, 1] × Rn , M (7.32) |vR,M,T (t, x)| ≤ cn . R Finally, for all 0 ≤ s ≤ n/2 and all t ∈ R M v(t, ·)H˙ s + ∂ · Rs−n/2 . ∂t v(t, ·)H˙ s−1 ≤ cn (7.33) R

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Proof. Rescale v(t, x) as v(t, x) = w(Rt, Rx) so that M χ(x). R By the energy estimates we have for all real s and all t ∈ R
w = 0,

w(RT, x) = 0,

wt (RT, x) =

w(t, ·)H˙ s + ∂ ∂t w(t, ·)H˙ s−1 ≤ 2

M χH˙ s−1 R

(7.34)

which scaling back to v gives M s−n/2 R χH˙ s−1 . R Notice that (7.34) gives a finite bound also for s = 0; indeed, by assumption (7.28) we have χ (0) = 0 and hence χ/  |ξ| ∈ L2 , i.e., χ ∈ H˙ −1 . Thus for all 0 ≤ s ≤ n/2 we obtain (R ≥ 1) ∂t v(t, ·)H˙ s−1 ≤ 2 v(t, ·)H˙ s + ∂

M s−n/2 R (χH˙ n/2−1 + χH˙ −1 ). R This proves (7.33); inequality (7.31) is just the special case s = n/2 computed at t = 0. To prove (7.32) we use (7.34) again for s = n/2 + 1, which gives v(t, ·)H˙ s + ∂ ∂t v(t, ·)H˙ s−1 ≤ 2

sup w(t, ·)H˙ n/2+1 ≤ cn t∈R

M χH˙ n/2 , R

while for s = 0 it gives sup w(t, ·)L2 ≤ cn t∈R

M χH˙ −1 , R

and this is bounded by (7.28) as already remarked. Thus, by Sobolev embedding, we have M wL∞ (R×Rn ) ≤ cn sup w(t, ·)H n/2+1 ≤ cn (χH˙ n/2 + χH˙ −1 ). R t∈R Since vL∞ = wL∞ , this concludes the proof; the constant cn depends only on n and the quantity χH˙ n/2 + χH˙ −1 .
We revert now to the main part of the proof. The next step is the explicit construction of sequences of functions v, w appearing in (7.14). As v we shall choose the function vR,M,T constructed in the preceding lemma, with a suitable choice of the parameters. Notice that by (7.31) we can assume that the initial data v0 , v1 belong to the neighbourhood V of 0 in H n/2 × H n/2−1 on which property (7.11) holds, as soon as M/R is small enough; e.g., if V contains a ball of radius r0 (V ) centered in 0, we may assume that 4cn χH n/2−1

M < r0 . R

(7.35)

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Notice also that, in order to define the composition γ ◦ v, we must ensure that |v| < s0 (at least on the strip [0, 1] × Rn ) since the geodesic curve is only defined on the interval (−s0 , s0 ), or even better, that |v| < c0 given by (7.4). Using (7.32), we see that therefore it is sufficient to decrease M/R, e.g., to impose the condition M < c0 /2. (7.36) R In connection with Remark 7.2, we observe that condition (7.36) is not necessary when we assume that γ(s) is defined for all s ∈ R. Then we define vj = vR,M,T with the following choices: the parameter T will be chosen as ⇒ vR,M,T (tj , x) = M χ(Rx), (7.37) T = tj cn χH n/2

where tj are given by Corollary 7.1; the parameter R = Rj will be chosen such that 1 on the ball {|x| ≤ 2Rj −1 }; (7.38) zj (tj , x) ≥ 2 this is possible in view of (7.26) and of the continuity of zj ; it is not restrictive to assume that Rj ↑ +∞. On the parameter M = Mj , besides (7.35), (7.36) further smallness conditions will be imposed in the following. We now define wj ; let µ > 0 be a small parameter, and set (w0 := v0 and) wj := vj + µzj ,

(7.39)

where vj was defined above and zj is given by Corollary 7.1. Thus the data for wj are w0 := v0 , w1 := v1 + µφj , where v0 , v1 are the traces of vj at t = 0, studied in Lemma 7.2. Again, in order to define the composition γ ◦ w, we must ensure that |w| < c0 , at least for 0 ≤ t ≤ 1. Using (7.27) and recalling (7.32), (7.36), we see that it is sufficient to impose the condition (7.40) 0 < µ < c0 /2. Notice that the data w0 , w1 belong to the given neighbourhood V as soon as j is large enough, since φj → 0 in H n/2−1 . Consider inequality (7.14); our aim is to estimate its right-hand side from below. The first term at t = tj gives (vj − wj )∂ ∂t vj H˙ n/2−1 = µzj (tj , ·)χRj ,M ˙ n/2−1 ; Mj H

(7.41)

we can apply (7.62) from the Appendix with s = n/2 − 1; since zj ≥ 1/2 on the support of χR,M , we have c  zj (tj , ·)χRj ,M ˙ n/2−1 ≥ χRj ,M ˙ n/2−1 − c zj H n/2 χRj ,M Mj H Mj H Mj H n/2−1 . 2 Now we have for R large enough χR,M H˙ n/2−1 =

M κ, R

χR,M H n/2−1 ≤ 2

M κ, R

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where by assumption κ = χH˙ n/2−1 = 0. Moreover, by the energy identity we have for all t the relations zj (t, ·)H˙ n/2 ≤ cφj H˙ n/2−1 → 0 as j → ∞ and also for all |t| ≤ 1    sin(t|ξ|)  j  ≤ φj L2 → 0 zj (t, ·)L2 =  φ  2  |ξ| L

⇒

(7.42)

zj (t, ·)H n/2 → 0. (7.43)

Hence we have proved that ∂t vj H˙ n/2−1 ≥ c(n, κ) (vj − wj )∂

Mj . Rj

(7.44)

In view of the second term in (7.14) we need also a bound from above for the quantity (v − w)∂ ∂t v; by (7.59) from the Appendix with s = n/2 − 1, we have zj (tj , ·)χRj ,M Mj H n/2−1 ≤ Czj L∞ ∩H n/2 χRj ,M Mj H n/2−1 ≤ C

Mj zj L∞ ∩H n/2 , Rj

for j large enough, and recalling that zj → 0 in H n/2 uniformly in |t| ≤ 1 as remarked above, and |zj | ≤ 1 by construction, we finally obtain ∂t vj H n/2−1 ≥ c (n, κ) (vj − wj )∂

Mj Rj

(7.45)

provided j is large enough. We notice that, by (7.32), (7.33), vj L∞ ∩H n/2 ≤ c

Mj , Rj

while, recalling that |zj | ≤ 1 and that zj H n/2 ≤ 1 for j large enough, we have wj L∞ ∩H n/2 = vj + µzj L∞ ∩H n/2 ≤ cµ + c

Mj . Rj

Together with (7.45) this gives us the following estimate for the second term in (7.14): ∂t vj H n/2−1 ρ2 (vj , wj L∞ ∩H n/2 )vj , wj L∞ ∩H n/2 (vj − wj )∂   Mj Mj ≤ ρ3 (µ + Mj /Rj ) · µ + . (7.46) Rj Rj We can impose now the last smallness condition on µ and Mj (recall that Mj /Rj is bounded):   Mj 1 (7.47) ρ3 (µ + Mj /Rj ) · µ + ≤ c(n, κ), Rj 2

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where c(n, κ) is the constant appearing in (7.44). Thus we get Mj 1 c(n, κ) . 2 Rj (7.48) The last term in (7.14) is quite easy to estimate: we have for j → ∞

ρ2 (vj , wj L∞ ∩H n/2 )vj , wj L∞ ∩H n/2 (vj − wj )∂ ∂t vj H n/2−1 ≤

ρ1 (vL∞ ∩H n/2 )v1 − w1 H n/2−1 ≤ ρ4 (µ + Mj /Rj ) · µφj H n/2−1 → 0.

(7.49)

We can finally choose Mj and µ; we define Mj = λ · Rj , and λ, µ are two positive constants so small that conditions (7.35), (7.36), (7.47) are satisfied. Summing up, by (7.44), (7.48), (7.49), we obtain Mj 1 1 ∂ ∂t u(1) (tj , ·) − ∂t u(2) (tj , ·)H˙ n/2−1 ≥ c(n, κ) (7.50) = c(n, κ) · λ 4 Rj 4 provided j is large enough. We can now conclude the proof of the theorem. Recalling (7.30), we can choose as data for v the sequences (j)

(j)

v0 := vRj ,M Mj ,tj (0, x),

v1 := ∂t vRj ,M Mj ,tj (0, x)

while the data for w are chosen as (j)

(j)

(j)

w0 := v0 ,

(j)

(j)

w1 := v1 + µφj = v1 + c0 φj /2.

By (7.35) the data for v belong to V ; as a consequence, the data for w belong to V provided j is large enough, since φj → 0 in H n/2−1 . Thus we are in the position (j) (j) to apply the uniform continuity property (7.11); since w1 − v1 = µφj /2 we have that for all ε > 0 there exists δ > 0 such that φj H n/2−1 < δ

⇒

sup ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 ≤ ε;

t∈[0,1]

hence in particular at t = tj we must have φj H n/2−1 < δ

⇒

∂ ∂t u(1) (tj , ·) − ∂t u(2) (tj , ·)H˙ n/2−1 ≤ ε

and this is in clear a contradiction with (7.50).




7.3. Appendix The aim of this Appendix is to prove two multiplicative estimates needed in the proof of Theorem 7.1. The first one has the following form: f gH s ≤ Cf H s gL∞∩H n/2 , s < n/2.

(7.51)

Notice that this estimate is asymmetric in f, g. We can obtain this estimate from the Kato-Ponce estimate (see Lemma 2.2 in [18]) f gH s ≤ Cf Lp1 J s gLp2 + CJ s f Lp3 gLp4 −1 −1 −1 which is valid for all s ≥ 0, for all p2 , p3 ∈ (1, ∞), and 1/2 = p−1 1 +p2 = p3 +p4 ; s s/2 here J = (1 − ∆) . Then (7.51) follows taking p3 = 2, p4 = ∞, 2n n , p2 = p1 = n − 2s s

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and using the Sobolev embeddings f Lp1 ≤ Cf H s ,

J s gLp2 ≤ CgH n/2 .

Also the second commutator estimate we need, i.e., J s (f g) − gJ s f L2 ≤ Cf H s gH n/2 ,

s < n/2,

n ≥ 3,

(7.52)

can be proved by a similar argument based on the the Kato-Ponce commutator estimate (see Lemma 2.2 in [18]) J s (f g) − gJ s f L2 ≤ C∇gLp1 J s−1 f Lp2 + CJ s gLp3 f Lp4 −1 −1 −1 which is valid for all s ≥ 0, for all p2 , p3 ∈]1, ∞[, and 1/2 = p−1 1 + p2 = p3 + p4 . Now (7.52) follows taking p1 = n, p3 = s/n,

2n , n−2 and using the Sobolev embeddings

p4 =

p2 =

∇gLp1 ≤ CgH n/2 ,

2n n − 2s

J s−1 f Lp2 ≤ Cf H s ,

J s gLp3 ≤ CgH n/2 ,

f Lp4 ≤ Cf H s .

For completeness, we give a self-contained proof of (7.51), (7.52) and a refined version of (7.52) involving homogeneous Sobolev norms; we hope that our method is of independent interest. To this end, we must introduce some basic tools from the theory of Sobolev and Besov spaces. 1) Difference operators. Given h ∈ Rn and a function f : Rn → C, we denote by fj (x) the j-th translate of f in the direction h: fj (x) = f (x + j · h),

j ∈ Z,

and the difference operator ∆h = ∆ defined as ∆f = f1 − f,

i.e.,

∆f (x) = f (x + h) − f (x).

We denote by ∆ the iterates of ∆. Trivial properties are f0 ≡ f , (ffi )j = fi+j , ∆i (∆j f ) = ∆i+j f , ∆(ffj ) = (∆f )j ≡ ∆ffj . Here of special interest will be the behavior of the difference operator with respect to products. We have immediately 

∆(f g) = f1 g1 − f g = f1 (g1 − g) + (f1 − f )g which can be written shortly ∆(f g) = ∆f · g + f1 · ∆g. By induction one proves easily the Leibnitz rule  k  k ∆ (f g) = ∆ fm ∆m g.  +m=k

(7.53)

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2) Sobolev spaces with fractional index. All the functions (and the spaces) considered here are defined on the whole Rn . The homogeneous Sobolev seminorms ˙ k,p with k ≥ 0 integer, 1 < p < ∞, are defined as W  Dα uLp ; uW˙ k,p = α=k

˙ k,2 . Thus the standard Sobolev norms can be written as we write H˙ k for W uW k,p = uLp + uW˙ k,p . For our purposes it is not necessary to enter into the topological details of the definition of the corresponding spaces; only the norms are sufficient, and we shall ˙ s,p , W s,p (semi)norms with noninalways apply them to smooth functions. The W teger s > 0 are more troublesome; the usual definition by interpolation is not well suited to prove multiplicative estimates. A handier equivalent characterization can be given using the fractional integrals 

1/p [s]+1 |∆h u(x)|p dxdh , Is,p (u) = |h|n+sp where [s] is the integer part of the noninteger s > 0, 1 < p < ∞, and integration is performed over R2n ; we shall write Is,2 = Is . Then we have uW k,p  uLp + Is,p (u)

(7.54)

(see, e.g., 2.3.1 and Theorem 2.5.1 in [44]). The integral Is,p (u) plays the role of the homogeneous norm; this can be seen by a simple rescaling argument. For the following application it will be sufficient to consider the L2 case, in which we have a simple definition using the Fourier transform uH˙ s = |ξ|s u L2 ,

uH s ≡ (1 + |ξ|2 )s/2 u L2  uL2 + uH˙ s .

Indeed, let Sλ be the scaling operators for λ > 0 which are defined by (Sλ u)(x) = u(λx); it is easy to check the scaling properties Sλ uL2 = λ−n/2 uL2 ,

Sλ uH˙ s = λs−n/2 uH˙ s ,

Is (Sλ u) = λ2s−n Is (u).

Thus, for fixed u ∈ H s , if we apply the two equivalent definitions for Sλ u we obtain λs−n/2 uH˙ s + λ−n/2 uL2  λs−n/2 Is (u) + λ−n/2 uL2 . Letting λ → ∞, we obtain immediately uH˙ s  Is (u).

(7.55)

3) Besov spaces. With the same type of norms it is possible to define the Besov s as follows (see Theorem 2.5.1 in [44]): for any s > 0, 1 < p < ∞, spaces Bp,q 1 < q < ∞ set 
1/q [s]+1 ∆h f qLp s f Bp,q = f Lp + dh (7.56) |h|n+sq

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and define the spaces accordingly. From this definition in particular it is evident s = W s,p for non-integer s. We shall use the fact that that Bp,p s B2,2 ≡ Hs

for all values of s (including integers). We finally recall the continuous embedding (see, e.g., Theorem 7.58 in [1] and Theorem 2.8.1 in [44]): for s, t ≥ 0 n n ⇒ W s,p ⊆ W t,q (7.57) 1 < p ≤ q < ∞, s − = t − p q and, more generally, the Besov version r ∈ [1, ∞],

1 < p ≤ q < ∞,

s−

n n =t− p q

⇒

s t Bp,r ⊆ Bq,r .

(7.58)

We are ready to prove our lemma. We use the notation uX∩Y := uX + uY for any two Banach spaces X, Y and u ∈ X ∩ Y . We state the following lemma for smooth functions, the extension to f, g belonging to the appropriate spaces is obvious. Lemma 7.3. For all real 0 ≤ s < n/2 and any smooth functions f, g, the following inequalities hold: (7.59) f gH s (Rn ) ≤ Cf H s gL∞ ∩H n/2 and, for all λ with s < λ < n/2, f gH s (Rn ) ≤ Cf H n/2+s−λ gH λ .

(7.60)

|g(x)| ≥ C1 > 0 on the support of f ;

(7.61)

f gH˙ s ≥ cC1 f H˙ s − c f H s gH n/2

(7.62)

Moreover, assume that

then we have also for some constants c, c > 0 depending only on s, n. Remark 7.5. Estimate (7.59) can be regarded as the limit case of (7.60) as λ → n/2; when λ → s we obtain (7.59) with f and g exchanged. Proof. Notice that in order to prove (7.59), (7.60) it is sufficient to prove them with the H s -norm on the left-hand side replaced by the homogeneous H˙ s -norm, since the estimates are trivially true for the term f gL2 . We need two different (but parallel) proofs in the cases s integer or non-integer, since we have two different representations of the norm in these cases. The proof for integer s is simple. Indeed, by the Sobolev embedding 2n n H s (Rn ) ⊆ L n−2s (Rn ), ∀0≤s< 2

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(see (7.57)), and by H¨ o¨lder’s inequality we have uvL2 ≤ uLn/µ vL2n/(n−2µ) ≤ CuH n/2−µ vH µ

(7.63)

for any real number n . 2 We can apply (7.63) to the product of two derivatives (here and in the following we shall use the shorthand notation D to denote any derivative of order ): 0 0 and n − 2(λ − σ) ≥ 2σ > 0. Thus by (7.57) we have the embeddings H n/2+s−λ ⊆ W ρ,2p , H λ ⊆ W σ,2q . (7.71) In conclusion we have proved for all , m ≥ 1 with  + m = [s] + 1, and any s ≤ λ ≤ n/2, the inequality 

1/2 |∆ fm ∆m g|2 dxdh ≤ cf H n/2−λ+s · gH λ . (7.72) |h|n+2s Two terms are left. The term with m = 0,  = [s] + 1 is bounded simply by writing 1/2 

|g∆[s]+1 f |2 dxdh ≤ gL∞ · Is (f ) ≤ cgL∞ f H s . (7.73) |h|n+2s On the other hand, the term with  = 0 and m = [s] + 1 is more delicate since we o¨lder’s inequality can not use the L∞ norm of f . We proceed as follows: we apply H¨ in dx to obtain 1/2 

1/2 

ffm 2Ln/s ∆[s]+1 g2L2n/(n−2s) |ffm ∆[s]+1 g|2 dxdh ≤ dh , |h|n+2s |h|n+2s and we notice that the norm ffm Ln/s = f Ln/s is independent of h and can be drawn out of the integral. What remains is exactly a Besov-norm (see (7.56)) and

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we conclude 



|ffm ∆[s]+1 g|2 dxdh |h|n+2s

1/2 ≤ f L2n/(n−2s) gB sn ,2 ≤ cf H s gH n/2

(7.74)

s

by the continuous embeddings n/2

H s ⊆ L2n/(n−2s) ,

B sns ,2 ⊆ B2,2 ≡ H n/2

(see (7.57), (7.58)). By (7.72) for λ = n/2, (7.73) and (7.74) we obtain (7.59). By the same method we can write 



|ffm ∆[s]+1 g|2 dxdh |h|n+2s

1/2





ffm 2Lp ∆[s]+1 g2Lq dh |h|n+2s

≤

1/2 ,

where, for an arbitrary λ with s < λ < n/2, p and q are chosen as 2p =

n , λ−s

2n ; n − 2(λ − s)

2q =

proceeding exactly as in the proof of (7.74) we obtain 



|ffm ∆[s]+1 g|2 dxdh |h|n+2s

1/2 ≤ f Ln/(λ−s) gB s

2n ,2 n−2(λ−s)

≤ cf H n/2+s−λ gH λ .

(7.75) By (7.72) and (7.75) we obtain immediately (7.60) for non-integer s. The proof of (7.62) for non-integer s proceeds in a similar way. Using again the Leibnitz rule (7.53) we can write 

Is (f g) ≥

|g∆[s]+1 f |2 dxdh |h|n+2s

1/2 −c





 +m=[s]+1 m≥1

|∆ fm ∆m g|2 dxdh |h|n+2s

1/2

which by (7.72) for λ = n/2 and (7.74) implies 

Is (f g) ≥

|g∆[s]+1 f |2 dxdh |h|n+2s

1/2 − cf H s gH n/2 .

Using now assumption (7.61) we have Is (f g) ≥ C1 Is (f ) − cf H s gH n/2 and recalling that Is (u)  uH˙ s , we conclude the proof.




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References [1] R. Adams, Sobolev Spaces. Academic Press, New York, 1975. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, II, I Geom. Funct. Anal. 3 (1993) 107–156, 209–262. [3] J. Bourgain, Periodic Korteweg–de Vries equation with measures as initial data, Selecta Math. 3 (1997) 115–159. [4] Ph. Brenner and P. Kumlin, On wave equations with supercritical nonlinearities, Arch. Math. 74 (2000) 129–147. [5] T. Cazenave, J. Shatah, and A. S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. H. Poincare´ Phys. Th´eor. 68, 3 (1998), 315–349. [6] Y. Choquet-Bruhat, Global existence for non-linear σ-models, Rend. Sem. Mat. Univ. Pol. Torino, Special Issue (1988), 65–86. [7] Y. Choquet-Bruhat, Global wave maps on curved space times, Mathematical and quantum aspects of relativity and cosmology (Pythagoreon, 1998), 1–29, Lecture Notes in Phys., 537, Springer, Berlin, 2000. [8] D. Christodoulou and A. S. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46, 7 (1993), 1041–1091. [9] P. D’Ancona and V. Georgiev, On the continuity of the solution operator to the wave map system, accepted in CPAM. [10] P. D’Ancona and V. Georgiev, Low regularity solutions for the wave map equation into the 2-D sphere, accepted in Math. Zeitschrift. [11] A. Freire, S. M¨ u ¨ller and M. Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. Henri Poincare´ 15 No.6 (1998) 425–759. [12] V. Georgiev and A. Ivanov, Concentration of local energy for two-dimensional wave maps, preprint 2003. [13] M. G. Grillakis, Classical solution for the equivariant wave maps in 1+2 dimensions, preprint, 1991. [14] M. G. Grillakis, The wave map problem, In Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 227–230. [15] J. Ginibre and G. Velo, The Cauchy problem for the O(N ), CP(N −1), and GC (N, p) models, Ann. Physics 142 (1982), no. 2, 393–415. [16] C. H. Gu, On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. Pure Appl. Math. 33, 6 (1980), 727–737. [17] H. Karcher and J. C. Wood , Non-existence results and growth properties of harmonic maps and forms, J. Reine Angew. Math. 353 (1984) 165–180. [18] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907. [19] S. Klainerman and M. Machedon, On the regularity properties of a model problem related to wave maps, Duke Math. J. 87 (1997), no. 3, 553–589. [20] S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22, 5-6 (1997), 901–918.

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[21] S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations (English. English summary), Commun. Contemp. Math. 4 (2002), no. 2, 223–295. [22] C. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. Journal 106 (2001) 617–632. [23] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol.I, Springer Verlag, Berlin 1972. [24] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations., SIAM J. Math. Anal. 33 (2001) 982–988. [25] S. M¨ u ¨ller and M. Struwe, Global existence of wave maps in 1 + 2 dimensions with finite energy data, Topol. Methods Nonlinear Anal. 7, 2 (1996), 245–259. [26] A. Nahmod, A. Stefanov and K. Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, Comm. Anal. Geom. 11, Number 1, 49–83, 2003. [27] K. Nakanishi and M. Ohta, On global existence of solutions to nonlinear wave equations of wave map type, Nonlinear Anal. TMA 42, (2000), 1231–1252. [28] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin 1996. [29] J. Shatah, Weak solutions and development of singularities of the su(2) σ-model, Comm. Pure Appl. Math. 41, 4 (1988), 459–469. [30] J. Shatah and M. Struwe, Geometric wave equations, New York University Courant Institute of Mathematical Sciences, New York, 1998. [31] J. Shatah and M. Struwe, The Cauchy problem for wave maps, Preprint; to appear on International Math. Research Notices. [32] J. Shatah and A. Sh. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754. [33] I. Sigal, Nonlinear semi-groups, Ann. of Math. 78 No. 2 (1963) 339–364. [34] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971, Princeton Mathematical Series, No. 32. [35] M. Struwe, Wave maps, In Nonlinear partial differential equations in geometry and physics (Knoxville, TN, 1995). Birkh¨ ¨ auser, Basel, 1997, pp. 113–153. [36] M. Struwe, Equivariant wave maps in two space dimensions, preprint, to appear in Comm. Pure and Appl. Math. [37] M. Struwe, Radially symmetric wave maps from 1 + 2–dimensional Minkowski space to the sphere, Preprint; to appear on Math. Zeitschrift. [38] M. Struwe, Radially symmetric wave maps from 1 + 2–dimensional Minkowski space to general targets, Preprint. [39] T. Tao, Ill-posedness for one-dimensional wave maps at the critical regularity, Amer. J. Math. 122 (2000), no. 3, 451–463. [40] T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys. 224, (2001), 443–544. [41] D. Tataru, Local and global results for wave maps I , Comm. Part. Diff. Eq. 23 (1998) 1781–1793.

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[42] D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37–77. [43] M. Taylor, Partial differential equations, Vol.III, I Springer Verlag, New York, 1997. [44] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland Co., Amsterdam 1978. [45] H. Triebel, Interpolation theory, function spaces, differential operators, second ed., Johann Ambrosius Barth, Heidelberg, 1995. [46] N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Ser. ` I Math. 329 (1999) 1043–1047. [47] Y. Zhou, Global weak solutions for (1 + 2)-dimensional wave maps into homogeneous spaces, Ann. Inst. H. Poincar´ ´e Anal. Non Lin´eaire 16, 4 (1999), 411–422. Piero D’Ancona Universita ` di Roma “La Sapienza” Dipartimento di Matematica Piazzale A. Moro 2 I-00185 Roma Italy e-mail: [email protected] Vladimir Georgiev Universit` ` a di Pisa Dipartimento di Matematica Via Buonarroti No. 2 56127 Pisa Italy e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 159, 113–211 c 2005 Birkhauser
¨ Verlag Basel/Switzerland

On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations Hideo Kubo and Masahito Ohta Abstract. The aim of this work is twofold. One is to develop an approach for dealing with semilinear wave equations adopted by John [38]. In Section 2, the basis of the argument will be explained in a self-contained way. The other is an application of the approach to systems of wave equations. We shall make use of it to handle the semilinear case in Sections 3,4 and 5, and to consider the quasilinear case in Section 6. In these argument we bring such systems that the single wave components obey different propagation speeds into focus. Mathematics Subject Classification (2000). Primary 35L70; Secondary 35B20. Keywords. Nonlinear hyperbolic systems, global solutions, blow-up, lifespan, self-similar solutions, asymptotic behavior.

1. Introduction This article is concerned with systems of nonlinear wave equations (∂ ∂t2 − c2i ∆)ui = Fi (u, ∂u, ∂∇u),

(t, x) ∈ [0, ∞) × Rn (1 ≤ i ≤ N ),

(1.1)

where ∂ = (∂0 , ∂1 , . . . , ∂n ) = (∂ ∂t , ∇), ∇ = (∂ ∂x1 , . . . , ∂xn ), ci is a positive constant, and u = (u1 , . . . , uN ) is an RN -valued unknown function of (t, x). We study the global behavior of solutions of the initial value problem to (1.1). The equation has a long history. Indeed, the case where N = 1, c1 = c, and F1 (u, ∂u, ∂∇u) = |u|p with p > 1, that is (∂ ∂t2 − c2 ∆)u = |u|p ,

(t, x) ∈ [0, ∞) × Rn ,

(1.2)

The first author is partially supported by: Grant-in-Aid for Science Research #14740114, JSPS. The second author is partially supported by: Grant-in-Aid for Science Research #14740099, JSPS.

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was considered by Strauss [100] in 1968. Since we know well the feature of the homogeneous wave equation (∂ ∂t2 − c2 ∆)u = 0,

(t, x) ∈ [0, ∞) × Rn ,

(1.3)

one of the basic question is if it is possible to regard (1.2) as a small perturbation of (1.3). When n = 3, John [38] proved in 1979 that √ there exists a unique classical solution to (1.2) for small initial data if p > 1 + 2,√and that the solution may develop a singularity in a finite time if 1 < p < 1 + 2 however small the initial data is. (For the other spatial dimensional case, see the introductive part of Section 2 below.) This result means that in the former case, the effect of the nonlinearity in (1.2) is not so strong that one can think of it just as a small perturbation. On the other hand, we see that (1.2) has a different structure from (1.3) in the latter case. Roughly speaking, the behavior of the blowing-up solution is governed by the ordinary differential equation v  (y) + v  (y) = |v(y)|p ,

y ≥ 0,

(1.4)

or “almost” equivalently to (2.33) below. Having these points in mind, our first problem for (1.1) is formulated as follows. Problem: Find a sharp condition about the small data global existence and blowup for the system (1.1). Here small data global existence means that the initial value problem to (1.1) admits a unique global (mild) solution for all “small” initial data. On the contrary, we say blow-up occurs if small data global existence does NOT hold. In other words, it means that one can choose a pair of initial data such that the lifespan of the corresponding solution is finite. On the one hand, the equation (1.2) is invariant with respect to the scale transform u −→ uλ , where 2

uλ (t, x) = λ p−1 u(λt, λx)

(λ > 0).

(1.5)

This will give us a possibility to construct a self-similar solution to (1.2). When n = 3, the existence and non-existence of the self-similar solution is again characterized √ by the exponent of p = 1 + 2. Our next question is concerned with the case where the √ nonlinearity of (1.2) is regarded as a small perturbation of (1.3) (i.e., p > 1 + 2 when n = 3). What we wish to know is whether the effect of the nonlinearity to the solution of (1.2) remains for large time or not. For the equation (1.2), the answer is the following. The global solution of (1.2) tends to a solution of (1.3) in the sense of the energy for large values of t. In other words, we can say that (1.2) is a nonlinear perturbation from (1.3), or the nonlinearity in (1.2) is of “short-range” type. In Section 2, we give a complete proof of all the results mentioned in the above for n = 2, 3, in the spirit of the work of John [38]. The core of the argument is the point-wise estimate which is involved by the following weight t + |x|−

n−1 2

ct − |x|−ν

(ν > 0).

(1.6)

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In analyzing the case where the exponent p is close to the critical value, the factor ct − |x|−ν plays an essential role. It reflects the fact that the wave propagates along the characteristic cone ct = |x|. Most of the argument presented in Section 2 below contains modification and simplification of earlier works, so that it can be applicable to the system (1.1). We underline that Subsection 2.3 is devoted to a new proof for establishing optimal lower bounds of the lifespan with respect to the size of the initial data. One of the advantages of this approach is that we find point-wise behaviors of the solutions by product. While, we need to pose higher regularity of the initial data than that of the solution itself. So, for example, to consider the well-posedness of the problem in Sobolev spaces, Strichartz estimates come into play. For the estimates, we refer to [24, 52, 78, 84] and the references cited therein. We also refer to Strauss [102] concerning the other aspect of the equation (1.2) and do not go further in these direction. Based on the preparation in Section 2, we next consider the system of semilinear wave equations

2 (t, x) ∈ [0, ∞) × Rn , (∂ ∂t − c21 ∆)u1 = λ1 |u1 |p1 |u2 |p2 , (1.7) 2 2 q1 q2 (∂ ∂t − c2 ∆)u2 = λ2 |u1 | |u2 | , (t, x) ∈ [0, ∞) × Rn where λ1 , λ2 ∈ R and n = 2, 3, in three different cases. In Section 3, we treat the case where p1 = q2 = 0. Then the feature of the system (1.7) becomes similar to that of the single wave equation (1.2) in the sense that we can treat it without paying attention to the propagation speeds c1 and c2 . We underline that the proof of the blow-up part given in Subsection 3.1 improves the treatment of the reduced integral equations in the previous works [65, 66]. Actually, we reduce the problem to the single integral equation (3.21) below. In Section 4, we deal with the case where p1 , p2 , q1 , q2 ≥ 1.

(1.8)

In this case, the discrepancy of the propagation speeds comes into play through the interaction between u1 and u2 in the nonlinearity. Indeed, if c1 = c2 , then c1 t − |x| is equivalent to t + |x| when (t, x) is close to the light cone c2 t = |x|, and c2 t − |x| is so when (t, x) is close to c1 t = |x| vice versa. Therefore it is possible to extract an additional decay from the nonlinearity, because the solution n−1 ui (t, x) is supposed to behave like t + |x|− 2 ci t − |x|−ν with some ν > 0. On the other hand, if c1 = c2 , then we don’t have such an extra decay, so that we can simply adopt the argument in the preceding section. In Section 5, we consider a mixed case of the preceding two sections. In particular, we study the case where n = 3 and p1 = p2 = 1, q1 = 3, q2 = 0.

(1.9)

Then we have the following remarkable result: When 0 < c1 < c2 , small data global existence holds good. On the contrary, if c1 > c2 > 0, then blow-up occurs. In order to show the existence part, we need a different type of weight from the

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standard one given in (1.6). While, in proving the blow-up part, we modify the reduction of the problem to a single integral equation. As for the viewpoint of mathematical physics, we refer to Menzala and Ebihara [81] and the references cited therein. Now we turn our attention to the general case where the nonlinearity of the system (1.1) depends also on the derivatives of the unknown functions. This case is treated in Section 6. In order to avoid the loss of derivatives, we suppose the solution to be sufficiently smooth. This in turn requires that the nonlinear term is also smooth in its arguments. Then the quadratic nonlinearity is of special interest, as long as the small amplitude solutions are concerned. When n = 3, it was shown by John [39] that blow-up occurs for the system (1.1) with some quadratic nonlinearity. Therefore we need some structure of the nonlinearity for establishing small data global existence for that case. Christodoulou [12] and Klainerman [54] independently introduced a condition on the nonlinearity called null condition. When the propagation speeds are common, it ensures small data global existence for (1.1). We underline that if the propagation speeds are distinct, then we can relax the condtion on the nonlinearity, and there are many contributions which aim at finding a wider class of nonlinearities for which (1.1) admits a global solution. In Subsection 6.1, we present a blow-up result due to Ohta [85]. To our knowledge, only few results on blow-up for (1.1) with multiple speeds are known. An interpretation of the null condition will be given in Subsection 6.2, together with the key identity (6.57) below. They are not only useful to show small data global existence for the system (1.1) with multiple speeds, but also to examine asymptotic behaviors of the global solution to that. As we shall see, the system can be regarded as a nonlinear perturbation from the system of homogeneous wave equations (∂ ∂t2 − c2i ∆)ui = 0,

(t, x) ∈ [0, ∞) × Rn

(1 ≤ i ≤ N ),

(1.10)

in most of the cases. In the appendix, we give an elementary proof for the uniqueness of classical solutions for the system (1.1). We conclude this section by collecting several notations which will be used in the sequel.  – For any x ∈ Rn , the symbol x denotes 1 + |x|2 . – For A ≥ 1 we denote A[a]+ = Aa

if

a > 0 ; A[a]+ = 1

if a < 0 ; A[0]+ = 1 + log A.

– We put a ∨ b := max{a, b} and a ∧ b := min{a, b} for a, b ∈ R. Besides, [a]+ = max{a, 0}. – We set R∗ := R \ {0} and denote by S n−1 the unit sphere in Rn . – For a set A, χA denotes the characteristic function of A.

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In what follows, the letter C stands for various constants which may change from line to line. Especially C(. . . ) denotes a constant depending essentially only on the quantities indicated in the parentheses.

2. Single wave equation This section is concerned with the initial value problem to (∂ ∂t2 − c2 ∆)u = |u|p ,

(t, x) ∈ [0, ∞) × Rn ,

(2.1)

where c > 0 and p > 1, with u(0, x) = ϕ(x),

∂t u(0, x) = ψ(x),

x ∈ Rn ,

(2.2)

where ϕ ∈ C0∞ (Rn ) and ψ ∈ C0∞ (Rn ). For this problem Strauss [101] introduced a number p0 (n) which is the positive root of the following quadratic equation:  n−1 n+1 p− Φ(n, p) ≡ p − 1 = 0. (2.3) 2 2 Note that p0 (n) is strictly decreasing with respect to n and for instance √ √ p0 (2) = (3 + 17)/2, p0 (3) = 1 + 2, p0 (4) = 2. It was conjectured by [101] that p0 (n) plays a role as the critical exponent for the problem (2.1)–(2.2). More precisely, small data global existence holds if p > p0 (n), while blow-up occurs if 1 < p < p0 (n). Though the number seems to be strange at first glance, one can understand it based on the scaling invariance of the semilinear equation. The scaling invariance means that if u(t, x) is a solution of (2.1), then Dλ,p u(t, x) also satisfies the same equation for all λ > 0, where we denoted by Dλ,p u(t, x) the dilation of u(t, x) defined by 2

Dλ,p u(t, x) = λ p−1 u(λt, λx)

(λ > 0).

(2.4)

Then the quadratic equation (2.3) follows from the self-similarity of the function w(r, ˙ t) = (t + r)−

n−1 2

|ct − r|−(

n−1 n+1 2 p− 2 )

for

r, t ∈ [0, ∞).

Namely, if p = p0 (n), then we have the dilation invariance Dλ,p0 (n) w(|x|, ˙ t) = w(|x|, ˙ t) for all λ > 0. Now we recall the historical background about the initial value problem (2.1)– (2.2). When n = 3, John [38] firstly established that W. Strauss’ conjecture is true, provided that the initial data is of compact support. This result was extended by Glassey [25, 26] to the case n = 2. Moreover, it was shown by Schaeffer [92] that blow-up occurs for the critical case p = p0 (n) when n = 2, 3. In the case where 1 < p ≤ p0 (n), the lifespan of the solution was studied by Lindblad [76] and by Zhou [116, 117], independently. For the case n ≥ 4, Sideris [95] proved that blow-up occurs if 1 < p < p0 (n). Recently Yordanov and Zhang [115] extends the result to the critical case. On the other hand, the existence part was considered by so many authors. In fact, Zhou [118] treated the case n = 4 for p > p0 (4), and Christodoulou [12], Li

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and Yun [74] and Choquet-Bruhat [11] handled the general case for sufficiently large p (see also the references cited therein). Moreover, when n ≥ 5, the case where p0 (n) < p < (n + 3)/(n − 1) was treated by Lindblad and Sogge [79] and by Georgiev [21], independently. In the sequel, it was shown by Georgiev, Lindblad and Sogge [23] that a weak global solution of the problem (2.1)–(2.2) exists for p0 (n) < p < (n + 3)/(n − 1) and n ≥ 2. The proof is based on the weighted Strichartz estimate. It was simplified by Georgiev [22] and by Tataru [109] independently, by using the Fourier transform on the hyperboloid. Coming back to the work [38], we discuss about the assumption on the initial data. When we are interested in the asymptotic behavior of the global solution, this kind of consideration is important. For the case n = 3, Asakura [10] showed that small data global existence holds without assuming the compactness of the support of the initial data. More precisely, if the initial data (ϕ, ψ) behaves like (x−κ , x−κ−1 ) as |x| → ∞ with κ > 2/(p − 1), then small data global existence holds for p > p0 (n). Kubota [71] and Tsutaya [112] independently extended this result to the case κ = 2/(p − 1). On the contrary, if we take the initial data in such a way that (2.5) ϕ(x) ≡ 0, ψ(x) ≥ εx−κ−1 for x ∈ Rn with κ < 2/(p − 1), then blow-up occurs even though p > p0 (n) and ε is arbitrary small. (Notice that 2/(p − 1) is the number related to the exponent of the scaling invariance for (2.1).) For the case n = 2, the existence part of the above result was proven by [71] and Tsutaya [110], independently. While the blow-up part was shown by Agemi and Takamura [4] and by [110], independently. When n ≥ 4, Takamura [108] proved the blow-up part, while Kubo [59], Kubo and Kubota [61] showed the existence part under the assumption that the initial data is radially symmetric. Finally, D’Ancona, Georgiev and Kubo [14] established small data global existence for initial data being in some weighted L2 space. Next we turn our attention to the asymptotic behavior of the global solution of (2.1)–(2.2). Then the condition p > p0 (n) is necessary for this purpose. It was shown by Strauss [101], Mochizuki and Motai [82, 83] that the scattering operator exists on a dense set of a neighborhood of zero in the energy space, provided p1 (n) < p ≤ (n + 3)/(n − 1), where p1 (n) is the larger root of the quadratic equation: nΦ(n, p) − (n − 1)p + (n + 1) = 0, (2.6) where Φ(n, p) is the one in (2.3). Note that p1 (n) > p0 (n), since Φ(n, p0 (n)) = 0 and −(n − 1)p0 (n) + (n + 1) = −2/p0(n) < 0. Moreover, when 2 ≤ n ≤ 4 and p > p0 (n), analogous results were obtained by Pecher [87] for n = 3, by Tsutaya [111], Kubota and Mochizuki [72] for n = 2 independently, and by Hidano [28] for n = 4. As for the case p ≥ (n + 3)/(n − 1), Lindblad and Sogge obtained similar results in [78] for n ≥ 3.

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Next we consider the self-similar solution of the initial value problem to (2.1). The existence of it was proved by Pecher [88, 89] for p > p0 (3), and by Hidano [29] for p0 (n) < p < (n + 3)/(n − 1) and n = 2, 3, independently. It was also proved by [89] that even for small data no self-similar solution exists in general if p ≤ p0 (3). On the one hand, Ribaud and Youssfi [91] treated the problem for general n(≥ 2) and for p > p1 (n) with p1 (n) is the number in (2.6). This result was extended by Kato and Ozawa [47, 48, 49] for p0 (n) < p < (n + 3)/(n − 1), provided that the initial data is radially symmetric. Moreover, without assuming the radial symmetricity, Kato, Nakamura and Ozawa [50] obtained the same result for 2 ≤ n ≤ 5. In particular, the argument in [50] gives us a new proof based on the weighted Strichartz estimate for the earlier results of [89] and [29]. As we have seen, there are so many contributions that it is difficult to cover all issues in this article. For this reason, we shall concentrate on the approach initiated by John [38]. It is based on the point-wise estimation of the fundamental solution from above and below. To realize this, we consider only the case where n = 2, 3. In the following subsections, we shall explain the details in a selfcontained and simplified way, so that they provide basic ideas to handle the hyperbolic systems later on. This section is organized as follows. In the next subsection, we establish the blow-up result for the problem (2.1)–(2.2). The point is to consider the quantity uc,p∗ defined by (2.30) and to reduce the problem to an integral inequality (2.33) with (2.32) concerning the quantity. Subsection 2.2 is devoted to the existence part. The main step is to establish the basic estimate (2.49). In Subsection 2.3 we extend the argument in the preceding subsection to the critical case (see (2.65)) and obtain the “almost” global existence result. Subsection 2.4 is concerned with the selfsimilar solution whose existence is shown by the application of (2.92). Finally, we consider asymptotic behaviors in Subsection 2.5. In the super-critical case where p > p0 (n), the effect of the nonlinearity is not so strong that the global solution tends to some free solution as t → +∞ in the sense of the energy. Moreover, the densely defined scattering operator related to (2.1) exists in a neighborhood of the origin of the energy space. These results essentially follow from (2.100) and (2.132). 2.1. Blow-up In this subsection we sketch the proof of the blow-up result. Suppose that u(t, x) is a classical solution of the problem (2.1)–(2.2). Then it satisfies the following integral equation: u = Kc [ϕ, ψ] + Lc [|u|p ] in [0, ∞) × Rn ,

(2.7)

where we put Kc [ϕ, ψ](t, x) = Jc [ψ](t, x) + ∂t Jc [ϕ](t, x),
t Jc [F (s, ·)](t − s, x) ds. Lc [F ](t, x) = 0

(2.8) (2.9)

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Here Jc [ψ](t, x) ≡ Jc [ψ](t, x) (n = 2, 3) is defined by
t Jc(3) [ψ](t, x) = ψ(x + ctω) dS Sω , (t, x) ∈ [0, ∞) × R3 , 4π |ω|=1 Jc(2) [ψ](t, x) =

t 2π


|ξ | 0 and g ∈ C(Rn ) satisfies g(x) ≥ 0 for all x ∈ Rn ,

g(0) > 0.

(2.13)

Theorem 2.1. Let n = 2, 3 and 1 < p ≤ p0 (n). Suppose that ε ∈ (0, 1] and g ∈ C(Rn ) satisfies (2.13). Then the solution of (2.7) with (2.12) blows up in a finite time T ∗ (ε). Moreover, there exists a positive constant C ∗ independent of ε such that

exp(C ∗ ε−p(p−1) ) if p = p0 (n), ∗ ∗ (2.14) T (ε) ≤ C ∗ ε−p(p−1)/(1−pp ) if 1 < p < p0 (n). In order to prove Theorem 2.1, we prepare a couple of estimates, Lemma 2.1 and Proposition 2.1 below. By (2.7), (2.12) and (2.13), we have u(t, x) ≥ εJ Jc [g](t, x), u(t, x) ≥ Lc [|u|p ](t, x),

(t, x) ∈ [0, ∞) × Rn , (t, x) ∈ [0, ∞) × Rn .

(2.15) (2.16)

Moreover, by (2.13), there exist δ > 0 and φδ ∈ C([0, ∞)) such that g(x) ≥ φδ (|x|) ≥ 0 for x ∈ Rn ,

φδ (ρ) > 0 for ρ ∈ [0, δ].

(2.17)

Note that we can assume that δ is sufficiently small. In the sequel we shall make use of the following identity. Lemma 2.1. Let n ≥ 2 and let g ∈ C([0, ∞)). Then we have

n−3 23−n ωn−1 ρ+r g(|x + ρω|)dS Sω = λg(λ)[h(λ, ρ, r)] 2 dλ n−2 (rρ) |ρ−r| |ω |=1

(2.18)

for ρ > 0 and x ∈ Rn with r = |x| > 0, where ωn−1 = 2π (n−1)/2 /Γ((n − 1)/2) is the area of the unit sphere in Rn−1 , and h(λ, ρ, r) is defined by h(λ, ρ, r) = {λ2 − (ρ − r)2 }{(ρ + r)2 − λ2 }. Proof. We put λ = |x + ρω|,

x · ω = r cos θ

(0 ≤ θ ≤ π).

Then we have λ2 = r2 + 2rρ cos θ + ρ2 ,

sin θ =

[h(λ, ρ, r)]1/2 , 2rρ

(2.19)

Coupled Systems of Semilinear Wave Equations and





|ω |=1

π

g(λ)ωn−1 [sin θ]n−2 dθ

g(|x + ρω|)dS Sω =

= ωn−1




121

0 ρ+r

g(λ)[sin θ]n−2 |ρ−r|

λ dλ. rρ sin θ


Thus we obtain (2.18).

Next we prepare basic estimates for Jc [G](t, x) and Lc [F ](t, x) from below. Proposition 2.1. Let n = 2, 3 and let G ∈ C(Rn ), g ∈ C([0, ∞)). If G(x) ≥ g(|x|) ≥ 0 for all x ∈ Rn , then we have
r+ct n−1 1 Jc [G](t, x) ≥ λ 2 g(λ) dλ (2.20) n−1 2cr 2 |r −ct| for all (t, x) ∈ [0, ∞) × Rn , where r = |x|. Moreover, let F ∈ C([0, T ) × Rn ), f ∈ C([0, ∞) × [0, T )) with T > 0 and suppose that F (t, x) ≥ f (|x|, t) ≥ 0 for all (t, x) ∈ [0, T ) × Rn . Then we have

n−1 1 Lc [F ](t, x) ≥ λ 2 f (λ, s) dλ ds (2.21) n−1 2cr 2 Dc (r,t) for all (t, x) ∈ [0, T ) × Rn , where we put Dc (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ s ≤ t,

(2.22)

|r − c(t − s)| ≤ λ ≤ r + c(t − s)}. Proof. First we prove (2.20). By G(x) ≥ g(|x|) for x ∈ Rn , (2.10) and (2.11) imply Jc [G](t, x) ≥ Jc [g(| · |)](t, x) for (t, x) ∈ [0, ∞) × Rn . Therefore it is easy to see from (2.10) and Lemma 2.1 that (2.20) holds for n = 3. When n = 2, by (2.11) and Lemma 2.1 we have

ct ρ dρ 1  Jc(2) [g(| · |)](t, x) = g(|x + ρω|) dS Sω 2πc 0 (ct)2 − ρ2 |ω|=1
ρ+r
ct ρ dρ λg(λ) 2   = dλ. 2 2 cπ 0 (ct) − ρ |ρ−r| h(λ, ρ, r) Since h(λ, ρ, r) = h(ρ, λ, r) by (2.19), we can write
r+ct
ct ρ dρ 2 (2)   (2.23) Jc [g(| · |)](t, x) = λg(λ) dλ 2 cπ |r−ct| (ct) − ρ2 h(ρ, λ, r) |r −λ|
[ct−r]+
r+λ 2 ρ dρ   + . λg(λ) dλ 2 − ρ2 h(ρ, λ, r) cπ 0 (ct) |r −λ| Recalling g(λ) ≥ 0 for λ > 0 and noting that for ρ ≥ |r − λ| h(ρ, λ, r) = {(r + λ)2 − ρ2 }{ρ2 − (λ − r)2 } ≤ 4rλ{ρ2 − (λ − r)2 }

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we get Jc(2) [g(| · |)](t, x)

≥

1 √ cπ r

=

1 √ 2c r





r+ct


√ λg(λ) dλ

|r −ct| r+ct √

ct |r −λ|

ρ dρ   (ct)2 − ρ2 ρ2 − (r − λ)2

λg(λ) dλ.

|r −ct|

Here we have used the following identity: For 0 ≤ a < b, we have
b ρ dρ π   = . 2 2 2 2 2 b −ρ ρ −a a

(2.24)

Thus we obtain (2.20) for n = 2. Moreover, (2.21) follows from (2.9) and (2.20). This completes the proof.
Now we shall give the proof of Theorem 2.1. In what follows, we put n−1 p∗ = mp − m − 1, m = . (2.25) 2 Step 1. We see from (2.17) and Proposition 2.1 that
r+ct 1 λm φδ (λ) dλ. Jc [g](t, x) ≥ 2crm |r−ct| Therefore, if |ct − r| ≤ δ/2 and ct + r ≥ δ, then by (2.15) we have

where we put C0 = (2c)−1




u(t, x) ≥ C0 εr−m ,

(2.26)

δ

λm φδ (λ) dλ( > 0). δ/2

Step 2. We shall show that there is a positive constant C1 = C1 (g, δ, c, n, p) such that C1 εp r1−m (2.27) u(t, x) ≥ (ct + r)(ct − r)p∗ holds for c(t − δ) ≥ r = |x|. Note that if c(t − δ) ≥ r, then we have cs + λ ≥ cδ for (λ, s) ∈ Dc (r, t). By (2.16), (2.26) and Proposition 2.1, for c(t − δ) ≥ r we have



Cεp Cεp u(t, x) ≥ m λm−mp dλ ds ≥ m (cs + λ)−m(p−1) dλ ds, r r E E where we put E = {(λ, s) ∈ [0, ∞)2 : |cs − λ| ≤ δ/2, ct − r ≤ cs + λ ≤ ct + r}. Changing the variables by cs − λ , (2.28) ξ = cs + λ, η = c we have
ct+r

Cεp δ/(2c) dξ dξ Cεp ct+r u(t, x) ≥ m dη = m . m(p−1) m(p−1) r r ξ ξ −δ/(2c) ct−r ct−r

Coupled Systems of Semilinear Wave Equations

123

Thus, using (2.29) below, we arrive at (2.27). Lemma 2.2. For any κ ∈ R,
t+r t−r

dρ Cr ≥ ρκ+1 (t + r)(t − r)κ

(2.29)

holds for 0 < r < t, where C = 2/ max{κ, 1}. Proof. Since 1 − sκ ≥ min{κ, 1}(1 − s) for 0 ≤ s ≤ 1, when κ ≥ 1, we have

 κ &
t+r t−r dρ 1 2r = . 1− ≥ κ+1 κ κ(t − r) t+r κ(t + r)(t − r)κ t−r ρ While, when κ < 1, we have
t+r
t+r dρ dρ 2r 1−κ ≥ (t − r) = . κ+1 2 (t + r)(t − r)κ t−r ρ t−r ρ


This completes the proof.

Step 3. In view of (2.27), for c, y > 0 and κ ∈ R, we introduce the following quantity: ˜ y)}, (2.30) uc,κ (y) = inf{|x|−(1−m) (ct + |x|)(ct − |x|)κ |u(t, x)| : (t, x) ∈ Σ(c, ˜ y) = {(t, x) ∈ [0, ∞) × Rn : (|x|, t) ∈ Σ(c, y)}, Σ(c, Σ(c, y) = {(r, t) ∈ [0, ∞)2 : r ≤ c(t − y)}.

(2.31)

Since we may assume 0 < δ ≤ 1, (2.27) yields uc,p∗ (y) ≥ C1 εp

for y ≥ 1.

(2.32)

Next we shall show that there exists a constant C2 > 0 such that b
y η [uc,p∗ (η)]p 1− uc,p∗ (y) ≥ C2 dη for y ≥ 1, y η pp∗ 1

(2.33)

˜ y), we where b = m + (1 − m)p. Let y ≥ 1. By (2.16) and (2.21), for (t, x) ∈ Σ(c, have u(t, x) ≥ Lc [|u|p ](t, x)   

cs − λ λb 1 ∗ ≥ u c,p p pp∗ 2crm c Dc (r,t)∩Σ(c,1) (cs + λ) (cs − λ)

p

dλ ds.

Changing the variables by (2.28), we have 
(ct−r)/c 

ct+r C (ξ − cη)b [uc,p∗ (η)]p u(t, x) ≥ dξ dη rm 1 ξ p η pp∗ ct−r
ct+r
C dξ (ct−r)/c (ct − r − cη)b [uc,p∗ (η)]p ≥ dη, rm ct−r ξ p 1 η pp∗

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and by (2.29), we have


(ct−r)/c (ct − r − cη)b [uc,p∗ (η)]p Cr1−m u(t, x) ≥ dη (ct + r)(ct − r)p−1 1 η pp∗ b
(ct−r)/c  [uc,p∗ (η)]p Cr1−m cη = dη. 1 − ∗ p (ct + r)(ct − r) ct − r η pp∗ 1 Since the function b
y η [uc,p∗ (η)]p y
→ 1− dη y η pp∗ 1 ˜ y), we have is non-decreasing, for any (t, x) ∈ Σ(c, b
y [uc,p∗ (η)]p η −(1−m) p∗ r (ct + r)(ct − r) u(t, x) ≥ C dη, 1− y η pp∗ 1 which implies (2.33).

Step 4. Now we are in a position to employ Lemma 2.3 below. Then we see that uc,p∗ (y) blows up in a finite time y = T∗ (ε), provided pp∗ ≤ 1. The last condition is equivalent to 1 < p ≤ p0 (n) according to (2.3). Therefore the solution of (2.7) with (2.12) blows up in a finite time T ∗ (ε) ≤ T∗ (ε), if 1 < p ≤ p0 (n) and (2.13) hold. Moreover, we have the upper bound (2.14) of the lifespan T ∗ (ε). Lemma 2.3. Let C1 , C2 > 0, α, β ≥ 0, b > 0, κ ≤ 1, ε ∈ (0, 1], and p > 1. Suppose that f (y) satisfies b
y f (η)p η α β dη, y ≥ 1. 1− f (y) ≥ C1 ε , f (y) ≥ C2 ε y ηκ 1 Then, f (y) blows up in a finite time T∗ (ε). Moreover, there exists a constant C ∗ = C ∗ (C1 , C2 , b, p, κ) > 0 such that

exp(C ∗ ε−{(p−1)α+β} ) if κ = 1, T∗ (ε) ≤ C ∗ ε−{(p−1)α+β}/(1−κ) if κ < 1. Proof. First, we consider the case κ = 1. We put F (z) = (C1 εα )−1 f (exp(ε−µ z)), Since the function z
→ (1 − e F (z) ≥ 1,

µ = (p − 1)α + β.

−z b

) is increasing on [0, ∞) and 0 < ε ≤ 1, we have
z b 1 − e−(z−ζ) F (ζ)p dζ, z ≥ 0. F (z) ≥ C1p−1 C2 (2.34) 0

Since F (z) blows up in a finite time (see Lemma 3.3 in Subsection 3.1), we obtain the desired estimate for the case κ = 1. Next, we consider the case κ < 1. We put (p − 1)α + β G(z) = (C1 εα )−1 f (ε−ν ez ), ν = . 1−κ Then we see that G(z) satisfies (2.34). Thus we obtain the desired estimate for the case κ < 1. This completes the proof.


Coupled Systems of Semilinear Wave Equations

125

Remark 2.4. The upper bound (2.14) for T ∗ (ε), including the critical case p = p0 (n), was first obtained by Zhou [116] for n = 3 and [117] for n = 2. Then Takamura [107] gave a unified proof for both n = 2 and 3 whose argument is applicable to the system (3.1) below if the propagation speeds c1 and c2 are the same (see [65]). However it seems to be difficult to apply the argument from [107] directly to the system (3.1) for the case c1 = c2 . On the other hand, the proof in the present subsection is applicable to the case c1 = c2 as well. One of the essential differences to the argument from [107] is to consider the quantity (2.30). In [107], instead of (2.30), the quantity inf{|x|m (t − |x|)κ |u(t, x)| : (t, x) ∈ Σ (y)}, Σ (y) = {(t, x) ∈ [0, ∞) × Rn : t/2 ≤ |x| ≤ t − y} is employed in the case c = 1. Notice that |x|m is equivalent to |x|−(1−m) (t + |x|) for (t, x) ∈ Σ (y), since Σ (y) is away from the t-axis. However the quantity does not give any information about the behavior of the solution close to the t-axis. ˜ y) On the contrary, we studied the behavior of the solution for all (t, x) ∈ Σ(1, including the region that is close to the t-axis. Eventually this will give us some advantage especially for the case n = 2, because we do not have sharp Huygens’ principle in that case. Moreover, the restriction t/2 ≤ |x| in Σ (y) makes it difficult to treat systems with different propagation speeds. 2.2. Small data global existence Following John [38], we look for a solution of the integral equation (2.7) in the weighted L∞ -space X(c, κ) defined by X(c, κ) = {u ∈ C([0, ∞) × Rn ) : uX(c,κ) < ∞}, vX(c,κ) =

sup (t,x)∈[0,∞)×Rn

t + |x|

n−1 2

ct − |x|κ |v(t, x)|,

where c > 0, κ ≥ 0. Besides, we introduce a class of initial data Y (ν) defined by Y (ν) = {(ϕ, ψ) ∈ C 1 (Rn ) × C(Rn ) : (ϕ, ψ)Y (ν) < ∞}, ν+ n−1 2

(ϕ, ψ)Y (ν) = sup {x x∈Rn

ν+ n+1 2

|ϕ(x)| + x

(2.35)

(|∇ϕ(x)| + |ψ(x)|)}.

Then we have the following theorem. Theorem 2.2. Let n = 2, 3 and let p > p0 (n). When n = 3, we take ν = p − 2. When n = 2, we take ν so that ν = (p − 3)/2 if p < 4, and 1/4 < ν < 1/2 if p ≥ 4. Suppose (ϕ, ψ) ∈ Y (ν). Then there are constants ε0 = ε0 (p, ν, c, n) > 0 and C0 = C0 (ν, c, n) > 0 such that if (ϕ, ψ)Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there is a unique solution of (2.7) in C0 ε}. {v ∈ C([0, ∞) × Rn ) : vX(c,ν) ≤ 2C First we prepare basic estimates for Jc [G](t, x) and Lc [F ](t, x) from above.

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H. Kubo and M. Ohta

Proposition 2.2. Let n = 2, 3 and let G ∈ C(Rn ), g ∈ C([0, ∞)). If |G(x)| ≤ g(|x|) for all x ∈ Rn , then we have
r+ct 1 (3) λg(λ) dλ (2.36) |J Jc [G](t, x)| ≤ 2cr |r−ct| for all (t, x) ∈ [0, ∞) × R3 , and |J Jc(2) [G](t, x)|

≤

1 c




r+ct

|r −ct|

+

1 c




0

λg(λ)  dλ (r + λ)2 − (ct)2 [ct−r]+

(2.37)

λg(λ)  dλ (ct)2 − (r + λ)2

for all (t, x) ∈ [0, ∞) × R . Moreover, let F ∈ C([0, T ) × Rn ), f ∈ C([0, ∞) × [0, T )) with T > 0 and suppose that |F (t, x)| ≤ f (|x|, t) for all (t, x) ∈ [0, T ) × Rn . Then we have

1 λf (λ, s) dλ ds (2.38) |Lc [F ](t, x)| ≤ 2cr Dc (r,t) 2

for all (t, x) ∈ [0, T ) × R3 , where Dc (r, t) is defined by (2.22), and

1 λf (λ, s)  |Lc [F ](t, x)| ≤ dλ ds 2 − c2 (t − s)2 c (r + λ) Dc (r,t)

1 λf (λ, s)  + dλ ds 2 c f c (t − s)2 − (r + λ)2 Dc (r,t)

(2.39)

for all (t, x) ∈ [0, T ) × R2 , where we put ,c (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ λ ≤ c(t − s) − r}. D

(2.40)

Proof. First we prove (2.36) and (2.37). By |G(x)| ≤ g(|x|) for x ∈ Rn , (2.10) and (2.11) imply |J Jc [G](t, x)| ≤ Jc [g(| · |)](t, x) for (t, x) ∈ [0, ∞) × Rn . Therefore it is easy to see from (2.10) and Lemma 2.1 that (2.36) holds. Besides, (2.37) follows from (2.23) and (2.24). Moreover, (2.38) and (2.39) follow from (2.9), (2.36) and (2.37), respectively. This completes the proof.
We shall carry out the proof of Theorem 2.2 by dividing the argument into three steps. Step 1. We shall show that for (ϕ, ψ) ∈ Y (ν) with ν > 0 there is a positive constant C0 = C0 (ν, c, n) such that Kc [ϕ, ψ]X(c,ν) ≤ C0 (ϕ, ψ)Y (ν) ,

(2.41)

provided 0 < ν < 1/2 when n = 2. By (2.8) it suffices to show r + t

n−1 2

r − ctν |J Jc [ψ](t, x)| ≤ C·ν+

n+1 2

ψL∞ (Rn )

(2.42)

Coupled Systems of Semilinear Wave Equations

127

and r + t ≤

n−1 2

r − ctν |∂ ∂t Jc [ϕ](t, x)|

ν+ n−1 2

C(·

ν+ n+1 2

ϕL∞ (Rn ) + ·

(2.43) ∇ϕL∞ (Rn ) )

for (t, x) ∈ [0, ∞) × Rn , provided 0 < ν < 1/2 when n = 2. To this end, we prepare the following elementary inequalities whose proof will be given at the end of this step. Lemma 2.5. For κ > 0, there exists a constant C = C(κ) > 0 such that
r+t ρ−1−κ dρ ≤ C min{t, r} r + t−1 r − t−κ , |r −t|
r+t |r −t|




ρ− 2 −κ (ρ + r − t)− 2 dρ ≤ Cr − t−κ ,

[t−r]+

1

1

(2.45)

ρ− 2 −κ (t − r − ρ)− 2 dρ ≤ Cr − t− 2 r − t[ 2 −κ]+ 1

1

(2.44)

1

1

(2.46)

0

for any (r, t) ∈ [0, ∞)2 . Here we used the notations [a]+ = max{a, 0} for a ∈ R, and A[a]+ = Aa if a > 0, A[a]+ = 1 if a < 0, and A[0]+ = 1 + log A, where A ≥ 1. Let us continue the proof of (2.41). First we prove (2.42). When n = 3, we have from (2.36)
·ν+2 ψL∞ (R3 ) r+ct −ν−1 (3) |J Jc [ψ](t, x)| ≤ λ dλ. 2cr |r −ct| This estimate yields (2.42) with n = 3, by virtue of (2.44). When n = 2, we have from (2.37) 
3 3 λλ−ν− 2 1 r+ct (2) ν+ 2  dλ |J Jc [ψ](t, x)| ≤ · ψL∞ (R2 ) c |r−ct| (λ + r + ct)(λ + r − ct)
3 1 [ct−r]+ λλ−ν− 2  + dλ . c 0 (λ + r + ct)(ct − r − λ) Since λ(λ + r + ct)−1 ≤ Cλr + t−1 for λ ≥ 0, we get 
r + t

1 2

|J Jc(2) [ψ](t, x)|

≤

ν+ 32

C·


+ 0

r+ct

ψL∞ (R2 )

[ct−r]+

|r −ct|

λ−ν− 2 √ dλ λ + r − ct 1

λ−ν− 2 √ dλ . ct − r − λ 1

Therefore (2.42) for n = 2 follows from (2.45) and (2.46), since 0 < ν < 1/2. Next we prove (2.43). It follows from (2.10) and (2.11) that |∂ ∂t Jc [ϕ](t, x)| ≤ t−1 Jc [|ϕ|](t, x) + cJ Jc [|∇ϕ|](t, x).

128

H. Kubo and M. Ohta

In view of (2.42), it suffices to show that r + t

n−1 2

r − ctν Jc [|ϕ|](t, x) ≤ Ct·ν+

n−1 2

ϕL∞ (Rn ) ,

(2.47)

provided 0 < ν < 1/2 when n = 2. When n = 3, we have from (2.36)
r + ct r+ct −ν−1 (3) ν+1 Jc [|ϕ|](t, x) ≤ · ϕL∞ (R3 ) × λ dλ. 2cr |r−ct| By (2.44) we get (2.47). When n = 2, we have from (2.37) 1

cJ Jc(2) [|ϕ|](t, x) ≤ ·ν+ 2 ϕL∞ (R2 ) (I1 + I2 ), where we have set


I1 =

r+ct

|r −ct|




λλ−ν− 2  dλ, (λ + r + ct)(λ + r − ct)

[ct−r]+

I2 = 0

1

λλ−ν− 2  dλ. (λ + r + ct)(ct − r − λ) 1

We evaluate Ij by dividing the argument into three cases. First suppose r ≥ ct ≥ 0. Then I2 = 0 and
r+ct
r+ct 1 1 I1 ≤ λ−ν− 2 dλ ≤ Cr + t 2 λ−ν−1 dλ. |r −ct|

|r −ct|

By (2.44) we get 1

r + t 2 r − ctν (I1 + I2 ) ≤ Ct.

(2.48)

Next suppose 0 ≤ r ≤ ct and r + ct ≥ 1. Then we have



r+ct 1 1 − 12 λ−ν− 2 (λ + r − ct)− 2 dλ I1 + I2 ≤ Ctr + t
+

ct−r

ct−r

& 1 1 λ−ν− 2 (ct − r − λ)− 2 dλ ,

0

which yields (2.48) by (2.45) and (2.46), because of 0 < ν < 1/2. Finally, suppose 0 ≤ r ≤ ct and r + ct ≤ 1. Then we have


&
r+ct
ct−r √ − 12 − 12 (λ + r − ct) dλ + (ct − r − λ) dλ ≤ Ct. I1 + I2 ≤ C t ct−r

0

Therefore we find (2.48), so that (2.47) is valid for n = 2. Hence (2.43) holds. Thus we conclude the proof of (2.41) from (2.42) and (2.43), assuming that Lemma 2.5 is valid. Here we give a proof of the lemma. √ Proof of Lemma 2.5. First, we note that ρ ≤ 1 + ρ ≤ 2ρ for ρ ≥ 0. To prove (2.44), without loss of generality, we can assume that r ≥ t. Then, we have 

 κ &
r+t r+t −1 1+r−t dρ 1 −κ (1 + ρ) = = 1− . 1+κ κ κ(1 + r − t)κ 1+r+t r −t (1 + ρ) r−t

Coupled Systems of Semilinear Wave Equations

129

Since 1 − sκ ≤ max{1, κ}(1 − s) for 0 ≤ s ≤ 1, we have   κ  1+r−t 1+r−t 2 max{1, κ}t . ≤ max{1, κ} 1 − 1− = 1+r+t 1+r+t 1+r+t Since min{r, t} = t, we obtain (2.44). Next we prove (2.45). We put a = |r − t|. Then we have
r+t
∞ 1 1 1 1 (1 + ρ)− 2 −κ (ρ + r − t)− 2 dρ ≤ (1 + ρ)− 2 −κ (ρ − a)− 2 dρ |r −t|

− 12 −κ




a

2a+1

≤ (1 + a)

− 12

(ρ − a)




∞

dρ +

(ρ − a)−1−κ dρ ≤ C(1 + a)−κ ,

2a+1

a

which implies (2.45). Finally, we prove (2.46). We can assume t > r. We put a = [t − r]+ = t − r, and
a 1 1 I(a) = (1 + ρ)− 2 −κ (a − ρ)− 2 dρ. 0

Since I(a) is bounded for 0 ≤ a ≤ 1, we have only to consider the case a > 1. Then, we have
  a − 12
a/2 1 1 a − 12 −κ a (1 + ρ)− 2 −κ dρ + 1 + (a − ρ)− 2 dρ I(a) ≤ 2 2 0 a/2 ≤

Ca− 2 (1 + a)[ 2 −κ]+ + C(1 + a)− 2 −κ a 2 , 1

1

1

1

which implies (2.46). This completes the proof.




Step 2. We need the following weighted decay estimate for Lc [F ](t, x) to handle the nonlinearity. Proposition 2.3. Let n = 2, 3 and let a, c, µ and κ > 0. Suppose F ∈ C([0, T )×Rn ) and κ = 1. Then there exists a constant C1 = C1 (a, c, µ, κ) > 0 such that sup (t,x)∈[0,T )×Rn

≤ C1

t + |x|m ct − |x|κ0 |Lc [F ](t, x)|

sup (t,x)∈[0,T )×Rn

(2.49)

t + |x|µ at − |x|κ |F (t, x)|,

where we put m = (n − 1)/2 and

µ−m−1 κ0 = µ+κ−m−2

if κ > 1, µ > m + 1 , if κ < 1, µ + κ > m + 2

provided in the case n = 2 that µ < 2 when κ > 1, and that µ + κ < 3 when κ < 1. Fc )(x, ct) with Fc (x, t) = c−2 F (x, t/c), it suffices to Proof. Since Lc (F )(x, t) = L1 (F show (2.49) for c = 1. When n = 3, we have from (2.38) |L1 [F ](t, x)| ≤ I ×

sup (t,x)∈[0,T )×Rn

t + |x|µ at − |x|κ |F (t, x)|,

(2.50)

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H. Kubo and M. Ohta

where we put



1 I= λ + s−µ+1 λ − as−κ dλ ds. 2r D1 (r,t) Changing the variables by β = λ − s,

α = λ + s, we have I

= ≤ ≤

1 4r




t+r

|t−r|




(2.51)

1 + a −κ 1−a α+ β dβ 2 2 r −t
α
t+r α−µ+1 dα σ−κ dσ −µ+1

α

α

dα



1 2(1 + a)r |t−r| −aα
C t+r α−µ+1 α[1−κ]+ dα, r |t−r|

since r − t > −α. Therefore, if κ > 1 and µ > 2, then we have I ≤ Ct + r−1 r − t−µ+2 , by (2.44). While, if κ < 1 and µ + κ > 3, then we have I ≤ Ct + r−1 r − t−µ−κ+3 . Thus we get I ≤ Ct + r−1 r − t−κ0 , hence (2.49) for n = 3 follows from (2.50). When n = 2, we have from (2.39) |L1 [F ](t, x)| ≤ (I1 + I2 ) where we put I1





=

(t,x)∈[0,T )×Rn

t + |x|µ at − |x|κ |F (t, x)|,

D1 (r,t)

λ + s−µ+1 λ − as−κ  dλ ds, (λ − s + t + r)(λ + s + r − t)

f1 (r,t) D

λ + s−µ+1 λ − as−κ  dλ ds. (λ − s + t + r)(t − r − λ − s)

=



I2

sup

Changing the variables by (2.51), we have
α 1−a
−κ  2 α + 1+a 1 t+r α−µ+1 2 β √ √ dα dβ, I1 = 2 |t−r| α + r − t β+t+r r −t

α 1−a −κ  2 α + 1+a 1 [t−r]+ α−µ+1 2 β √ √ I2 = dβ. dα 2 0 β+t+r t−r−α −α

(2.52)

(2.53) (2.54)

We remark that µ > 3/2 by the assumption and that we may assume µ < 2 without loss of generality. We divide the argument into three cases. First suppose 2r ≥ t ≥ 0 and t + r ≥ 1. Since β + t + r ≥ 2r if either β > −α and 0 < α < t − r or β > r − t, we

Coupled Systems of Semilinear Wave Equations

131

have β + t + r ≥ Ct + r in this case. Therefore we can handle the β–integrals as before and get 
t+r 1 α−µ+1 α[1−κ]+ 2 √ dα (2.55) t + r (I1 + I2 ) ≤ C α+r−t |t−r|
[t−r]+ α−µ+1 α[1−κ]+ √ + dα . t−r−α 0 By (2.45) and (2.46) we can conclude I1 + I2 ≤ Ct + r− 2 r − t−κ0 . 1

(2.56)

Next suppose 2r ≥ t ≥ 0 and t + r ≤ 1. Then both β–integrals in I1 and I2 are bounded. Therefore, I1 + I2 is estimated by the right-hand side of (2.55). Hence (2.56) also holds in this case. Finally suppose t ≥ 2r ≥ 0. We begin with proving
α 1−a −κ  2 α + 1+a 1 2 β √ dβ ≤ Ct + r− 2 t + r[1−κ]+ (2.57) β + t + r −α a−1 for 0 ≤ α ≤ t+r. Denoting the left-hand side by I and setting d = 1+a 2 (t+r)+ 2 α, we get    12
α  12
d/a 2 2 σ−κ σ−κ √ √ dσ ≤ dσ, I= 1+a 1+a σ+d σ+d −aα −d

since d ≥ aα for 0 ≤ α ≤ t + r. By the fact that d ≥ min{1, a}(t + r) for 0 ≤ α ≤ t + r, the application of (2.58) below gives (2.57). Now it follows from (2.53), (2.57) and (2.45) that I1 ≤ Ct + r− 2 t + r[1−κ]+ r − t−µ+ 2 , 1

3

since µ > 3/2. Moreover, (2.54), (2.57) and (2.46) yield I2 ≤ Ct + r− 2 t + r[1−κ]+ r − t−µ+ 2 , 1

3

since 3/2 < µ < 2. For t ≥ 2r ≥ 0, we see that these estimates imply (2.56). Therefore we obtain (2.49) for n = 2 via (2.52). This completes the proof.
The following lemma has been used in the proof of (2.57). Lemma 2.6. Let κ, a, d > 0. Then there exists a positive constant C = C(κ, a) such that
d/a 1 1 σ−κ (d + σ)− 2 dσ ≤ Cd− 2 d[1−κ]+ . (2.58) −d

Proof. We denote by I the left-hand side of (2.58). First suppose 0 < a < 1. Then, splitting the integral at σ = d and putting
d
d/a 1 −κ − 12 σ (d + σ) dσ, P2 = σ−κ (d + σ)− 2 dσ, P1 = −d

d

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we have I = P1 + P2 . It is easy to see that
d/a 1 1 (d + σ)− 2 dσ ≤ Cd−κ+ 2 . P2 ≤ d−κ d

To treat P1 , we consider two different cases. Let 0 < d < 1. Then we easily see 1 that P1 is estimated by Cd−κ+ 2 , since
d 1 P1 = (d + σ)− 2 dσ ≤ C. −d

On the other hand, when d ≥ 1, we further split the integral as
−d/2
d 1 1 Q1 = σ−κ (d + σ)− 2 dσ, Q2 = σ−κ (d + σ)− 2 dσ. −d

−d/2

It follows that Q2 ≤

 − 12
d d 1 σ−κ dσ ≤ Cd− 2 d[1−κ]+ . 2 −d/2

While, we have −κ

Q1 ≤ Cd




d

−d

(d + σ)− 2 dσ ≤ Cd−κ+ 2 . 1

1

Thus we have shown (2.58) for the case 0 < a < 1. Finally suppose a ≥ 1. Then I ≤ P1 . Since the argument concerning P1 in the above still works for the present case, we obtain the desired estimate. This completes the proof.
Step 3. For ε > 0 we put C0 ε}, Xε := {v ∈ C([0, ∞) × Rn ) : vX(c,ν) ≤ 2C

(2.59)

where ν is the number from the theorem. Define a map from Xε to C([0, ∞) × Rn ) by Φ[v] := Kc [ϕ, ψ] + Lc [|v|p ]. (2.60) Let C0 and C1 are the constants in (2.41) and (2.49), respectively. We shall show C0 ε0 )p−1 ≤ 1, then for 0 < ε ≤ ε0 we have that if ε0 satisfies p2p+1 C1 (C 2C C0 ε for v ∈ Xε , (2.61) 1 Φ[v1 ] − Φ[v2 ]X(c,ν) ≤ v1 − v2 X(c,ν) for v1 , v2 ∈ Xε . (2.62) 2 Once we establish (2.61) and (2.62), we find that Φ[v] is a contraction on Xε , hence (2.7) admits a unique solution in Xε for 0 < ε ≤ ε0 . Let a = c, µ = mp and κ = pν. Then µ > m + 1 is equivalent to p∗ > 0 by (2.25). Since p > p0 (n) implies pp∗ > 1 by (2.3), we see from (2.49) that Φ[v]X(c,ν)

≤

Lc [|v|p ]X(c,p∗ ) ≤ C1 [vX(c,ν)]p .

Coupled Systems of Semilinear Wave Equations

133

Recalling that ν < 1/2 ≤ p∗ for n = 2 and p ≥ 4, we can deduce Lc [|v|p ]X(c,ν) ≤ C1 [vX(c,ν) ]p . This estimate, together with (2.41) and the choice of ε0 , implies (2.61). Since ||s|p − |t|p | ≤ p(|s|p−1 + |t|p−1 )|s − t| for p > 1 and s, t ∈ R, we have the following estimate by (2.49) with the same choice of the parameters as in the above: p−1 Lc [|v1 |p ] − Lc [|v2 |p ]X(c,p∗ ) ≤ pC1 (v1 p−1 X(c,ν) + v2 X(c,ν) )v1 − v2 X(c,ν) ,

which yields (2.62). This completes the proof of Theorem 2.2. Remark 2.7. In Theorem 2.2 we can relax the condition on the initial data. We put n−1 2 − m, m = . (2.63) p∗ = p−1 2 Let p satisfy p < (n + 3)/(n − 1), so that p∗ > 0. If (ϕ, ψ) ∈ Y (p∗ ), then there exists a global solution u ∈ X(c, p∗ ) of (2.7), provided (ϕ, ψ) is sufficiently small in Y (p∗ ). Indeed, because of the fact that pp∗ < 1 for p > p0 (n), mp+pp∗ −m−2 = p∗ and that mp + pp∗ < 3 for p > 3 and n = 2, we can apply Proposition 2.3 as κ < 1. Therefore we can prove such an existence theorem analogously to the proof of Theorem 2.2. While, we see from (2.25), (2.63) and (2.3) that p∗ > p∗ for p > p0 (n), hence ∗ Y (p ) ⊂ Y (p∗ ). Therefore the above result improves Theorem 2.2 concerning the assumption on the initial data when p0 (n) < p < (n + 3)/(n − 1). Remark 2.8. If the initial data (ϕ, ψ) is more regular, then the solution u(t, x) of (2.7) also gain the corresponding regularity. For instance, let (ϕ, ψ) ∈ C 3 (Rn ) × C 2 (Rn ) and (∂ α ϕ, ∂ α ψ) (|α| ≤ 2) are sufficiently small in Y (ν) with ν is the number from Theorem 2.2. Then u ∈ C 2 ([0, ∞) × Rn ), hence it becomes a classical solution of (2.1)–(2.2). 2.3. Almost global existence In this subsection we consider (2.7) for the case of p = p0 (n). As we have seen in the Subsection 2.1, the lifespan of the solution of (2.7) is estimated from above in terms of the size of the initial data. Here we shall investigate the lower bound of the lifespan. The main result in this subsection reads as follows. Theorem 2.3. Let n = 2, 3 and let p = p0 (n). Suppose (ϕ, ψ) ∈ Y (ν) with ν > 1/p. Then there is a constant ε0 = ε0 (p, c, n) > 0 such that if (ϕ, ψ)Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution u ∈ C([0, T ∗ (ε)) × Rn ) of (2.7). Moreover, there is a constant C = C(p, c, n) > 0 such that T ∗ (ε) ≥ exp(Cε−p(p−1) ).

(2.64)

Remark 2.9. In view of (2.14), the above estimate is sharp with respect to ε. In order to obtain such an optimal order in ε, we adapt the argument used in [45]. The idea is to consider the difference between the solution of (2.7) and that of the

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homogeneous wave equation. This consideration simplifies both arguments in [3] and [67]. To prove the theorem, we need the following extension of Proposition 2.3 to the case κ = 1, since pp∗ = pp∗ = 1 for p = p0 (n). Proposition 2.4. Let n = 2, 3 and let a, c, µ > 0. Suppose F ∈ C([0, T ) × Rn ) and µ > m + 1. Then there exists a positive constant C1 = C1 (a, c, n, µ) such that sup (t,x)∈[0,T )×Rn

t + |x|m ct − |x|κ0 |Lc [F ](t, x)|

≤ C1 log(2 + T )

sup (t,x)∈[0,T )×Rn

(2.65)

t + |x|µ at − |x||F (t, x)|,

where κ0 = µ − m − 1, provided µ < 2 when n = 2. Proof. Let b = max(a, c). First we shall prove that if 2bt ≤ r := |x|, then we have t + |x|m ct − |x|κ0 |Lc [F ](t, x)| ≤C

(2.66)

t + |x| at − |x||F (t, x)|. µ

sup (t,x)∈[0,T )×Rn

We shall use Proposition 2.2. Since c(t − s) − r ≤ 0 for 2bt ≤ r and s ≥ 0, we see ,c (r, t) = ∅. Moreover, if (λ, s) ∈ Dc (r, t), then that D λ ≥ r − c(t − s) ≥ r − 2b(t − s) ≥ 2bs ≥ 2as. This means that as − λ is equivalent to s + λ when (λ, s) ∈ Dc (r, t). In the following, we assume c = 1. When n = 3, we have from (2.38) |L1 [F ](t, x)| ≤ CI ×

sup (t,x)∈[0,T )×Rn

where we put 1 I= 2r





t + |x|µ at − |x||F (t, x)|,

(2.67)

λ + s−µ dλ ds.

D1 (r,t)

Changing the variables by (2.51), we have
α
t+r 1 I = α−µ dα dβ 4r |t−r| r −t
t+r 1 ≤ α−µ+1 dα, 2r |t−r| since r − t > −α. Recalling κ0 = µ − 2 > 0, we have I ≤ Ct + r−1 r − t−κ0 . Hence (2.66) with n = 3 follows from (2.67). When n = 2, we have from (2.39) |L1 [F ](t, x)| ≤ CI

sup (t,x)∈[0,T )×Rn

t + |x|µ at − |x||F (t, x)|,

(2.68)

Coupled Systems of Semilinear Wave Equations where we put





I= D1 (r,t)

135

λ + s−µ  dλ ds, (λ − s + t + r)(λ + s + r − t)

Changing the variables by (2.51), we have

α 1 1 t+r α−µ √ √ dβ. I= dα 2 |t−r| α + r − t β+t+r r −t

(2.69)

We divide the argument into two cases. First suppose 2r ≥ t ≥ 0 and r ≥ 1. Since β +t+r ≥ 2r if β > r −t, we have β + t + r ≥ Ct+r in this case. Therefore we get
t+r 1 α−µ+1 √ t + r 2 I ≤ C dα α+r−t |t−r| ≤ Cr − t−κ0 , by (2.45), since κ0 = µ − (3/2) > 0. Next suppose t ≥ 2r ≥ 0 or 0 ≤ r ≤ 1. It is easy to see that the β–integral is 1 estimated by Ct + r 2 for α < t + r. Therefore we get
t+r 1 α−µ 2 √ dα I ≤ Ct + r α+r−t |t−r| ≤ Ct + r 2 r − t−(µ− 2 ) , 1

1

by (2.45). Since t + r is equivalent to r − t in this case, we have I ≤ Ct + r− 2 r − t−κ0 . 1

Thus we have shown (2.66) with n = 2. Next we shall prove that if 2bt ≥ r, then we have (2.70) t + |x|m ct − |x|κ0 |Lc [F ](t, x)| µ t + |x| at − |x||F (t, x)|. ≤ C log(1 + r − ct) sup (t,x)∈[0,T )×Rn

In this case, we cannot avoid the logarithmic tail in the β–integral. We can show that (2.70) holds as well, by proceeding as in the proof of Proposition 2.3 and using the lemma below instead of Lemma 2.5. In conclusion we obtain (2.65) from (2.66) and (2.70). We finish the proof, assuming the following lemma is valid.
Lemma 2.10. For κ > 0 and  > 0, there exists a constant C = C(κ, ) > 0 such that
r+t C min{t, r} [log(1 + r − t)] ρ−1−κ [log(1 + ρ)] dρ ≤ , (2.71) t + rr − tκ |r −t|
r+t 1 1 C[log(1 + r − t)] ρ− 2 −κ (ρ + r − t)− 2 [log(1 + ρ)] dρ ≤ (2.72) r − tκ |r −t|

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for (r, t) ∈ [0, ∞)2 . Besides, assuming κ < 1/2, we have
t−r 1 1 C[log(1 + r − t)] ρ− 2 −κ (t − r − ρ)− 2 [log(1 + ρ)] dρ ≤ r − tκ 0

(2.73)

for (r, t) ∈ [0, ∞)2 with t > r. Proof. It is easy to see from (2.46) that (2.73) is valid. Since (2.72) can be proved similarly to (2.71), we shall only deal with (2.71). Putting f (ρ) =

[log(2 + ρ)] , (2 + ρ)κ/2

we find that the left-hand side of (2.71) is estimated by some constant times
t+r f (ρ) I := dρ. 1+κ/2 ρ |t−r| Note that f (ρ) is increasing on (0, ρ0 ) and decreasing on (ρ0 , ∞) with ρ0 = max{e2/κ − 2, 0}. When |t − r| ≥ ρ0 , by (2.44) we have
t+r dρ C min{t, r}f (|t − r|) ≤ . I ≤ f (|t − r|) 1+κ/2 ρ t + rt − rκ/2 |t−r| This means that (2.71) holds. While, if |t − r| ≤ ρ0 , then we have 0 < f (0) ≤ f (|t − r|) ≤ f (ρ0 ) ≤ Cf (0), in particular, f (ρ0 ) ≤ Cf (|t − r|). Therefore we get
t+r C min{t, r}f (|t − r|) dρ ≤ . I ≤ f (ρ0 ) 1+κ/2 ρ t + rt − rκ/2 |t−r|


Thus we have shown (2.71). This completes the proof.

End of the proof of Theorem 2.3: As it is mentioned in Remark 2.9, we consider the difference between the solutions of (2.7) and the homogeneous wave equation. Namely, we handle the following integral equation: w = Lc [|u0 + w|p ] in [0, T ) × Rn ,

(2.74)

where u0 (t, x) = Kc [ϕ, ψ](t, x). Suppose we obtain the solution w(t, x) of (2.74). Then, putting u = w + u0 , we get the solution of (2.7) and its lifespan is the same as that of w. Thus we have reduced the problem to the analysis for (2.74). For ε > 0 and T > 0 we set Z := {w ∈ C([0, T ) × Rn ) : w ≡ wZ(c,p∗ ) ≤ 2p C1 C0p εp },

(2.75)

vZ(c,κ) =

(2.76)

sup (t,x)∈[0,T )×Rn

t + |x| ct − |x| |v(t, x)|. m

κ

Define a map from Z to C([0, T ) × Rn ) by Ψ[w] := Lc [|u0 + w|p ].

(2.77)

Let us fix ε0 as p2p+1 C1 (C C0 ε0 )p−1 ≤ 1. For ε with 0 < ε ≤ ε0 we take T such that 2

p2p

+1

C1p (C C0 ε0 )p(p−1) log(2 + T ) ≤ 1.

(2.78)

Coupled Systems of Semilinear Wave Equations

137

To apply the contraction mapping principle, we shall show that Ψ[w] ≤

2p C1 (C C0 ε0 )p for w ∈ Z, 1 w1 − w2  for w1 , w2 ∈ Z Ψ[w1 ] − Ψ[w2 ] ≤ 2 for 0 < ε ≤ ε0 , provided T satisfies (2.78). Since |s + t|p ≤ 2p−1 (|s|p + |t|p ) for p > 1 and s, t ∈ R, we have

(2.79) (2.80)

Ψ[w] ≤ 2p−1 (Lc [|u0 |p ] + Lc[|w|p ]). We see from (2.41) u0 Z(c,ν) ≤ C0 ε. Since ν > 1/p, (2.49) with a = c, µ = mp and κ = pν yields C0 ε)p . Lc [|u0 |p ]Z(c,p∗ ) ≤ C1 [u0 Z(c,ν) ]p ≤ C1 (C Using (2.65) as a = c, µ = mp and κ = pp∗ = 1, we get Lc[|w|p ]Z(c,p∗ ) ≤ C1 log(2 + T )[wZ(c,p∗ ) ]p ≤ C1 (C C0 ε)p for w ∈ Z, by virtue of (2.78). Therefore (2.79) is valid. Next we consider (2.80). It follows that Ψ[w1 ] − Ψ[w2 ] ≤

p2p−2 (2Lc[|u0 |p−1 |w1 − w2 |] + +Lc[(|w1 |p−1 + |w2 |p−1 )|w1 − w2 |]).

By (2.49) with a = c, µ = mp and κ = (p − 1)ν + p∗ we have p2p−1 Lc [|u0 |p−1 |w1 − w2 |]

≤ p2p−1 C1 [u0 Z(c,ν) ]p−1 w1 − w2  1 ≤ w1 − w2  4 by the choice of ε. (Recall ν > 1/p = p∗ for p = p0 (n).) While, by (2.65) with a = c, µ = mp and κ = 1 we get p2p−2 Lc [(|w1 |p−1 + |w2 |p−1 )|w1 − w2 |] ≤ p2p−2 C1 log(2 + T )[w1 p−1 + w2 p−1 ]w1 − w2  1 ≤ w1 − w2  4 for w1 , w2 ∈ Z, by (2.78). Hence we get (2.80). Moreover, (2.78) implies (2.64). This completes the proof of Theorem 2.3. 2.4. Self-similar solution The aim of this subsection is to study the existence and non-existence of the selfsimilar solution to (2.1) or the associated integral equation (2.7), by following Pecher [89]. We call u(t, x) a self-similar solution, if it satisfies (2.1) and Dλ,p u(t, x) ≡ u(t, x)

(2.81)

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for (t, x) ∈ [0, ∞) × Rn and λ > 0, where Dλ,p u is given in (2.4). Taking t = 0 and λ = 1/|x| in (2.81), we have u(0, x) = |x|− p−1 u(0, ω), 2

∂t u(0, x) = |x|− p−1 −1 ∂t u(0, ω) 2

(2.82)

with ω = x/|x|. This means that the initial data of a self-similar solution must be a homogeneous function. Since such an initial data does not belong to Y (p∗ ), we can not apply the existence result stated in Remark 2.7, despite of the fact that 2/(p − 1) = p∗ + m. For this reason, we introduce the following homogeneous version of Y (ν) defined by Y˙ (ν) = {(ϕ, ψ) ∈ C 1 (Rn∗ ) × C(Rn∗ ) : (ϕ, ψ)Y˙ (ν) < ∞}, ν+ n−1 2

(ϕ, ψ)Y˙ (ν) = sup {|x| x∈Rn ∗

ν+ n+1 2

|ϕ(x)| + |x|

(2.83)

(|∇ϕ(x)| + |ψ(x)|)}.

Besides, since the singularity at the origin propagates through the light cone ct = |x|, we modify the norm  · X(c,ν) as vX(c,κ) = sup{|t+|x||m |ct−|x||κ |v(t, x)| : (t, x) ∈ [0, ∞)×Rn , ct = |x|}. (2.84) ˙ Now we are in position to state the main results about self-similar solutions. Theorem 2.4. Assume that (ϕ, ψ) ∈ Y˙ (p∗ ) and n = 2, 3. (i) Let p0 (n) < p < (n + 3)/(n − 1). Then there are constants ε0 = ε0 (p, c, n) > 0 and C0 = C0 (p, c, n) > 0 such that if (ϕ, ψ)Y (p∗ ) ≤ ε for 0 < ε ≤ ε0 , then there is a unique solution of (2.7) in X˙ := {v ∈ C(([0, ∞) × Rn ) \ {ct = |x|}) : vX(c,p ≤ 2C C0 ε0 }. ˙ ∗) (ii) Let 1 < p ≤ p0 (n). Then the conclusion (i) does NOT hold. Corollary 2.1. Let n = 2, 3 and let p0 (n) < p < (n + 3)/(n − 1). Let the initial data take the form ϕ(x) = f (ω)|x|− p−1 , 2

ψ(x) = g(ω)|x|− p−1 −1 2

(2.85)

with ω = x/|x|, f ∈ C 1 (S 2 ) and g ∈ C(S 2 ). If (ϕ, ψ) ∈ Y˙ (p∗ ) is sufficiently small ˙ in Y˙ (p∗ ), then there exists a unique self-similar solution of (2.7) in X. First we show the part (ii ) of Theorem 2.4. Suppose that the conclusion (i) is still valid for 1 < p ≤ p0 (n). Let us take the initial data in such a way as ϕ(x) = 0,

ψ(x) = ε|x|− p−1 −1 2

where ε > 0. Then it follows from (2.7) that u(x, t) ≥ Jc [ψ](t, x).

for x ∈ Rn∗ ,

Coupled Systems of Semilinear Wave Equations

139

For all δ > 0 we have ψ(x) ≥ ε|x|−m (|x| + δ)−p∗ −1 (recall (2.63)). Hence Proposition 2.1 implies
r+ct ε (λ + δ)−p∗ −1 dλ (2.86) u(t, x) ≥ 2crm |r−ct|  ε  (|r − ct| + δ)−p∗ − (r + ct + δ)−p∗ , ≥ 2cp∗ rm where r = |x|, since p∗ > 0 for p < (n + 3)/(n − 1). We fix (t, x) ∈ (0, ∞) × Rn such that ct/2 < |x| < ct, and set E = {(λ, s) ∈ [0, ∞)2 : (ct − r)/2c < s < 2(ct − r)/3c,

c(t − s) − r < λ < cs}.

Notice that if (λ, s) ∈ E, then we have 2λ > cs, hence λ+cs ≥ 3(cs−λ). Therefore, for (s, y) ∈ (0, ∞) × Rn such that (|y|, s) ∈ E we get u(s, y) ≥ Cε|y|−m (||y| − cs| + δ)−p∗

(2.87)

from (2.86). Now it follows from Proposition 2.1 that

1 u(x, t) ≥ λm |u(y, s)|p dλ ds 2crm D (r,t)

c εp λ−m(p−1) (|λ − cs| + δ)−pp∗ dλ ds ≥ 2crm E
2(ct−r)/3c
cs Cεp ≥ ds (cs − λ + δ)−pp∗ dλ. rm |ct − r|m(p−1) (ct−r)/2c c(t−s)−r Since pp∗ ≥ 1 for 1 < p ≤ p0 (n), the λ-integral goes to ∞ as δ → 0. This is a contradiction. Next we show the part (i ) of Theorem 2.4 by modifying the proof of Theorem 2.2. First of all, we introduce the following analogues to Lemmas 2.5 and 2.6. We omit the proof of them, since it is analogous to that of Lemmas 2.5 and 2.6. Lemma 2.11. For κ > 0, there exists a constant C = C(κ) > 0 such that
r+t C min{t, r} ρ−1−κ dρ ≤ , (t + r)|r − t|κ |r −t|
r+t 1 1 ρ− 2 −κ (ρ + r − t)− 2 dρ ≤ C|r − t|−κ

(2.88) (2.89)

|r −t|

for (r, t) ∈ (0, ∞)2 . Besides, assuming κ < 1/2, we have
[t−r]+ 1 1 ρ− 2 −κ (t − r − ρ)− 2 dρ ≤ C|r − t|−κ 0

for (r, t) ∈ (0, ∞)2 .

(2.90)

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Lemma 2.12. Let κ, a, d > 0. If κ < 1, then there exists a positive constant C = C(κ, a) such that
d/a 1 1 |σ|−κ (d + σ)− 2 dσ ≤ C|d|−κ+ 2 . (2.91) −d

Next we introduce the following analogue to Proposition 2.3. It can be shown in the same line as Proposition 2.3, if we use Lemmas 2.11 and 2.12 instead of Lemmas 2.5 and 2.6 respectively. Proposition 2.5. Let n = 2, 3 and let a, c, µ and κ > 0. Suppose F ∈ C(([0, T ) × Rn ) \ {ct = |x|}), κ < 1 and µ + κ > m + 2. Then there exists a constant C = C(a, c, µ, κ) > 0 such that sup |t + |x||m |ct − |x||κ0 |Lc [F ](t, x)|

(2.92)

ct=  |x|

≤ C sup |t + |x||µ |at − |x||κ |F (t, x)|, ct=  |x|

where we put κ0 = µ + κ − m − 2, provided µ + κ < 3 when n = 2. End of the proof of the part (i ) of Theorem 2.4: Proceeding as in the proof of (2.41) and using Lemma 2.11, we can show that for (ϕ, ψ) ∈ Y˙ (ν) with ν > 0 there is a positive constant C0 = C0 (ν, c, n) such that Kc [ϕ, ψ]X(c,ν) ≤ C0 (ϕ, ψ)Y˙ (ν) , ˙

(2.93)

provided 0 < ν < 1/2 when n = 2. In order to show Lc [|u|p ]X(c,p ≤ C[uX(c,p ]p , ˙ ˙ ∗) ∗) we apply Proposition 2.5 by taking a = c, µ = mp and κ = pp∗ . Since µ+κ−m−2 = p∗ , κ < 1 for p > p0 (n), and µ + κ < 3 for p > 3 and n = 2, we get the desired estimate. We omit the further detail, since it is similar to Step 3 in Subsection 2.2. 2.5. Asymptotic behavior We have studied the existence of a classical global solution of (2.1) (see Remark 2.8). The aim of this subsection is to examine asymptotic behaviors as t → +∞ of such a solution u ∈ X(c, p∗ ) of (2.1). To be more precise, we shall show that u(t, x) is asymptotically free. Namely, there is a unique classical solution v(t, x) of the homogeneous wave equation (∂ ∂t2 − c2 ∆)v = 0,

(t, x) ∈ [0, ∞) × Rn ,

(2.94)

such that u(t) − v(t)E(c) → 0 where


u(t)E(c) =

&1/2 (c |∇x u(t, x)| + |∂ ∂t u(t, x)| )dx . 2

Rn

as t → +∞,

2

2

(2.95)

(2.96)

Coupled Systems of Semilinear Wave Equations

141

For the moment, we roughly explain the idea to get this type of result. Let us rewrite the integral equation (2.7) as
∞ Jc [F (u)(s, ·)](t − s, x) ds u = Kc [ϕ, ψ] + 0
∞ Jc [F (u)(s, ·)](t − s, x) ds in [0, ∞) × Rn , − t

with F (u)(t, x) = |u(t, x)|p . Having this in mind, if we set v(t, x) = u(t, x) − Rc [F (u)](t, x) in [0, ∞) × Rn ,
∞ Rc [F ](t, x) = − Jc [F (s, ·)](t − s, x) ds

(2.97) (2.98)

t

then we have ∂t2 − c2 ∆)u − F (u) = 0. (∂ ∂t2 − c2 ∆)v = (∂ Therefore the main step is reduced to show Rc [F (u)](t)E(c) → 0 as

t → +∞.

(2.99)

To realize this procedure, we first prepare the following basic estimate for Rc [F ](t, x). Proposition 2.6. Let n = 2, 3 and let a, c, µ > 0. Suppose F ∈ C([0, ∞) × Rn ) and κ > 1. Then there exists a constant C1 = C1 (a, c, µ, κ) > 0 such that sup (t,x)∈[0,∞)×Rn

≤ C1

t + |x|µ−1 |Rc [F ](t, x)|

sup (t,x)∈[0,∞)×Rn

(2.100)

t + |x|µ at − |x|κ |F (t, x)|,

provided µ > m + 1 when a = c, and that µ > 1 when a = c. Jc [g](t, x) by the definitions (2.10) and (2.11), we Proof. Since Jc [g](−t, x) = −J have
∞ Jc [F (s, ·)](s − t, x) ds.

Rc [F ](t, x) = t

Besides, we may assume c = 1. When n = 3, we have from (2.36) |R1 [F ](t, x)| ≤ I × where we put 1 I= 2r





sup (t,x)∈[0,∞)×Rn

t + |x|µ at − |x|κ |F (t, x)|,

(2.101)

λ + s−µ+1 λ − as−κ dλ ds,

E1 (r,t)

Ec (r, t) = {(λ, s) ∈ [0, ∞)2 : s > t, |c(s − t) − r| ≤ λ ≤ c(s − t) + r}. Changing the variables by (2.51), we have
∞
r−t 1 + a −κ 1 1−a α+ β dα. I= dβ α−µ+1  4r −(t+r) 2 2 t+r

142

H. Kubo and M. Ohta When a = 1, we have I

≤

t + r−µ+1 4r







r−t

∞

dβ −(t+r)

t+r



1 + a −κ 1−a α+ β dα 2 2

≤ Ct + r−µ+1 , since µ > 1 and κ > 1. While, if a = 1, then we have
r−t
∞ 1 −κ I ≤ β dβ α−µ+1 dα 4r −(t+r) t+r ≤

Ct + r−µ+1 ,

since κ > 1 and µ > 2. At the last step, we have used Lemma 2.13 below. Hence (2.100) for n = 3 follows from (2.101), assuming the lemma. Lemma 2.13. For κ > 1 there exists a constant C = C(κ) > 0 such that
1 t+r dβ C J := , t ∈ R, r > 0. ≤ κ r t−r β |t| + r

(2.102)

Proof. Without loss of generality, we may assume t ≥ 0. We easily have three bounds for J as follows: J ≤ 2 ; J ≤ 2t − r−κ

for t > r ; J ≤ Cr−1 .

(2.103)

When t + r ≤ 1, (2.102) follows from the first estimate in (2.103). When t + r ≥ 1 and t ≥ 2r, (2.102) follows from the second one in (2.103). Finally, if t + r ≥ 1 and t ≤ 2r, then (2.102) follows from the last one in (2.103). This completes the proof.
In order to handle the case n = 2, we prepare the following estimates. Lemma 2.14. Let κ > 0, 0 < γ < 1 and d ∈ R. If κ + γ > 1, then there exists a constant C = C(κ, γ) > 0 such that
∞ (1 + σ)−κ (d + σ)−γ dσ ≤ C(1 + |d|)−κ−γ+1 , (2.104) |d |

and




d

(1 + |σ|)−κ (d − σ)−γ dσ ≤ C(1 + |d|)−γ (1 + |d|)[1−κ]+ ,

(2.105)

or equivalently,
∞ (1 + |σ|)−κ (d + σ)−γ dσ ≤ C(1 + |d|)−γ (1 + |d|)[1−κ]+ .

(2.106)

−∞

−d

Proof. First we show (2.104). Since (d + σ)−γ ≤ (σ − |d|)−γ , by integration by parts, we have
∞
∞ κ −κ −γ (1 + σ) (d + σ) dσ ≤ (1 + σ)−κ−γ dσ, 1 − γ |d | |d |

Coupled Systems of Semilinear Wave Equations

143

which yields (2.104), because κ + γ > 1. Next we show (2.106). When d ≤ 0, (2.106) follows from (2.104). In what follows, we assume d > 0 and set
d
∞ −κ −γ P1 = (1 + |σ|) (d + σ) dσ, P2 = (1 + |σ|)−κ (d + σ)−γ dσ. −d

d

Then (2.104) gives P2 ≤ C(1 + |d|)−κ−γ+1 . While, seeing the proof of (2.58) with a = 1, we find that P1 is estimated by C(1+|d|)−γ (1+|d|)[1−κ]+ . Thus we have shown (2.106). The proof is complete.
Now we turn to the proof of (2.100) for n = 2. It follows from (2.37) that |R1 [F ](t, x)| ≤ (I1 + I2 ) where we put I1

sup (t,x)∈[0,∞)×Rn

=

=

(2.107)

√ 1 λλ + s−µ+ 2 λ − as−κ



E1 (r,t)

I2

t + |x|µ at − |x|κ |F (t, x)|,

f1 (r,t) E

 dλ ds, (λ − s + t + r)(λ + s + r − t) √ 1 λλ + s−µ+ 2 λ − as−κ  dλ ds (s − λ − t − r)(λ + s + r − t)

with ,c (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ λ ≤ c(s − t) − r}. E Changing the variables by (2.51), we have 1
∞
−κ α−µ+ 2  1−a α + 1+a 1 r−t 2 β √ 2 dβ dα, I1 ≤ 2 −(t+r) β+r+t t+r
∞ 1
−κ α−µ+ 2  1−a α + 1+a 1 −(t+r) 2 β  2 I2 ≤ dβ dα. 2 −∞ −(β + t + r) −β

(2.108) (2.109)

First we evaluate I1 . Suppose a = 1. Then (2.108) implies

∞ β−κ 1 1 r−t √ dβ α−µ+ 2 dα I1 ≤ 2 −(t+r) β + r + t t+r
∞ 3 β−κ √ dβ, ≤ Ct + r−µ+ 2 β+r+t −(t+r) since µ > 3/2. Thus the application of (2.106) with d = r + t gives I1 ≤ Ct + r−µ+1 ,

(2.110)

since κ > 1. On the one hand, when a = 1, we have
∞ 1
1 + a −κ 1 t + r−µ+ 2 r−t 1−a √ α+ β dα, I1 ≤ dβ  2 2 2 β + r + t −(t+r) t+r

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H. Kubo and M. Ohta

since µ > 1/2. By κ > 1 we also have (2.110) in this case. Next we evaluate I2 . When a = 1, (2.109) yields
∞
1 β−κ 1 −(t+r)  dβ I2 ≤ α−µ+ 2 dα 2 −∞ −(β + r + t) |β |
−(t+r) −κ−µ+ 32 β  ≤ C dβ −(β + r + t) −∞
∞ 3 β−κ−µ+ 2 √ = C dβ, β−r−t t+r since µ > 3/2. By (2.104) with d = −r − t we have I2 ≤ Ct + r−κ−µ+2 , since κ + µ − (3/2) > 1/2. By κ > 1, we arrive at I2 ≤ Ct + r−µ+1 .

(2.111)

Finally, we consider the case where a = 1. We have
∞
1 β−µ+ 2 1 + a −κ 1 −(t+r) 1−a  dβ α+ β dα I2 ≤  2 −∞ 2 2 −(β + r + t) |β |
∞ 1 β−µ+ 2 √ ≤ C dβ, β−r−t t+r since κ > 1. By (2.104) with d = −r − t, we have (2.111), since µ > 1. Thus we have proved (2.100) via (2.107), (2.110) and (2.111). This completes the proof.
Remark 2.15. If u ∈ C 2 ([0, ∞) × Rn ) satisfies ∂xα uX(c,ν)t + |x|−m ct − |x|−ν |∂ ∂xα u(t, x)| ≤ ∂ for |α| ≤ 2, then Rc [F (u)] ∈ C 2 ([0, ∞) × Rn ), where F (u) = |u|p . In fact, it follows from (2.100) that ∗

sup (t,x)∈[0,∞)×Rn

t + |x|p

+m

|Rc [F (u)](t, x)| ≤ C1 [uX(c,ν)]p .

(2.112)

Here we took a = c, µ = mp and κ = pν with ν the number in Theorem 2.2. (Recall that µ > m + 1 is equivalent to p∗ > 0 by (2.25) and that p > p0 (n) implies pp∗ > 1.) Since ∂xα (F (u))](t, x) ∂xα Rc [F (u)](t, x) = Rc [∂ for |α| ≤ 2, and ∂xα (F (u)(t, x)) is a continuous function in (t, x), we see from ∗ (2.112) that ∂xα Rc [F (u)](t, x) is estimated by Ct + |x|−p −m [|||u|||X(c,ν) ]p , where we put |||u|||X(c,ν) = max ∂ ∂xα uX(c,ν) . |α|≤2

Coupled Systems of Semilinear Wave Equations

145

As for the derivative in t, we have for n = 3
∞
1 ∂t Rc [F ](t, x) = − ds F (u)(s, x + c(s − t)ω) dS Sω 4π t |ω |=1

∞ 3  c − (t − s) ds ωj ∂xj (F (u)(s, x + c(s − t)ω)) dS Sω 4π t |ω |=1 j=1 and ∂t2 Rc [F ](t, x) = F (t, x)
∞
3  c ds ωj ∂xj (F (u)(s, x + c(s − t)ω)) dS Sω − 2π t |ω |=1 j=1 −

c2 4π




∞




(t − s) ds

t

3 

|ω |=1 j,k=1

ωj ωk ∂xj ∂xk (F (u)(s, x + c(s − t)ω)) dS Sω .

Therefore each term under the integral signs is continuous in (t, x) and these integrals converge, by the assumption on u(t, x). For example, ∂t Rc [F ](t, x) is estimated by some constant times
t+1
ds |F (u)(s, x + c(s − t)ω)| dS Sω t




|ω |=1

∞

(s − t)ds

+ t+1 ∞




(s − t) ds

+ t

≤


|ω |=1

|F (u)(s, x + c(s − t)ω)| dS Sω




|ω |=1

|∂ ∂x (F (u)(s, x + c(s − t)ω))| dS Sω ∗

C[uX(c,ν)]p + C[|||u|||X(c,ν) ]p t + |x|−p

−m

,

in view of (2.112). Similarly we can treat the case n = 2. Thus we find Rc [F (u)] ∈ C 2 ([0, ∞) × Rn ). Moreover, a straight forward computation shows that Rc [F ] satisfies (2.1). In order to prove (2.99), we prepare (2.120) below. To this end, we will set up the argument in more general situation. By u(t) we denote a function of t ∈ R with values in D (Rn ), the space of distributions on Rn . Besides, we denote by S(Rn ) the space of rapidly decreasing functions on Rn , and by S  (Rn ) the space of tempered distributions on Rn . We consider the initial value problem u (t) − ∆u(t) = F (t) for t ∈ R

(2.113)

u(t) = u (t) = 0 at t = s,

(2.114)

with zero initial data 

where s is an arbitrary real number and u (t) stands for the second derivative of u(t), and so on. For a function f ∈ L2 (Rn ) we denote by fˆ and fˆ∗ , respectively

146

H. Kubo and M. Ohta

the Fourier transform of f and the inverse Fourier transform of f such that fˆL2 (Rn ) = fˆ∗ L2 (Rn ) = f L2 (Rn ) . Then the following facts are in essence well known. Proposition 2.7. Assume that F (t) ∈ C(R; L2 (Rn )). (i) Let s ∈ R be fixed. For t ∈ R we define a functional u(t; s), ϕ on S(Rn ) by

s sin(c(t − τ )|ξ|) ˆ F (ξ, τ )ϕˆ∗ (ξ) dξ for ϕ ∈ S(Rn ). dτ u(t; s), ϕ = − c|ξ| t Rn (2.115) Then u(t; s), with s regarded as a parameter, is a solution of the initial value problem (2.113)–(2.114) such that u(t; s) ∈ C 2 (R; H −1 (Rn )). Moreover, the solution is unique in C 2 (R; D (Rn )). (ii) Let n ≥ 2. Assume that there are positive constants θ and C such that F (t)L2 (Rn ) ≤ C(1 + |t|)−1−θ

for t ∈ R.

For t ∈ R we define a functional u(t), ϕ on S(R ) by

∞ sin(c(t − τ )|ξ|) ˆ F (ξ, τ )ϕˆ∗ (ξ) dξ dτ u(t), ϕ = − c|ξ| n t R

(2.116)

n

for ϕ ∈ S(Rn ). (2.117)

Then u(t) ∈ C 2 (R; S  (Rn )),

(2.118)

and we have for each t ∈ R u (t) ∈ L2 (Rn ), and

∇u(t) ∈ L2 (Rn )


u(t)E(c) ≤ (n + 1)

∞

F (τ )L2 (Rn ) dτ.

(2.119)

(2.120)

t

Proof. The first part (i) is well known. (For the uniqueness see for instance [61], Lemma 5.1.) First we shall prove (2.118). Let t ∈ R be fixed. Then we claim that u(t) ∈ S  (Rn ). To see this we take a positive number δ such that δ < θ and δ < 1. Then sin(c(t − τ )|ξ|) C(τ − t)δ ≤ |ξ| |ξ|1−δ and |ξ|δ−1 ϕˆ∗ (ξ) ∈ L2 (Rn ) for n ≥ 2, hence the integrand in the right-hand side of (2.117) is integrable with respect to (ξ, τ ) over Rn × (t, ∞), according to (2.116). Therefore we have for ϕ ∈ S(Rn ) u(t), ϕ = lim u(t; k), ϕ, k→∞

where u(t; s) is given by (2.115). Since u(t; k) ∈ S  (Rn ) for all k = 1, 2, . . ., we find by the Banach-Steinhaus’ theorem that u(t) ∈ S  (Rn ). Now the desired property (2.118) follows easily from the above procedure.

Coupled Systems of Semilinear Wave Equations

147

1 ∂ Finally we shall prove (2.119) and (2.120). Let Dk = √ (k = 1, . . . , n). −1 ∂xk Then we get by virtue of (2.116)
∞ c|Dk u(t), ϕ| ≤ ϕL2 (Rn ) F (τ )L2 (Rn ) dτ. t

Hence Dk u(t) ∈ L (R ) and 2

n




cDk u(t)L2 (Rn ) ≤

∞

F (τ )L2 (Rn ) dτ.

t

Analogously we obtain (2.119) and (2.120). This completes the proof.




Now we are in position to state the main result of this subsection. Theorem 2.5. Let n = 2, 3 and let p > p0 (n). For |α| ≤ 2 suppose (∂ ∂xα ϕ, ∂xα ψ) ∈ 2 Y (ν) with ν the number chosen in Theorem 2.2. Let u ∈ C ([0, ∞) × Rn ) be a classical solution to (2.1)–(2.2) such that ∂xα u ∈ X(c, ν) for |α| ≤ 2. Then there is a unique classical solution v(t, x) of (2.94) satisfying ∗

u(t) − v(t)E(c) ≤ C[uX(c,ν)]p t−p

for

t ≥ 0.

(2.121)

Remark 2.16. We recall the choice of ν for convenience. When n = 3, we take ν = p∗ . While, when n = 2, we take ν so that ν = p∗ if p < 4, and 1/4 < ν < 1/2 if p ≥ 4. Proof. As we have seen in Remark 2.15, Rc [F (u)](t, x) is a classical solution to (2.1) if u(t, x) is as in Theorem 2.5, where F (u) = |u|p . Therefore, once we establish ∗

Rc [F (u)](t)E(c) ≤ C[uX(c,ν) ]p t−p

for t ≥ 0,

(2.122)

we finish the proof by setting v = u − Rc [F (u)]. First we show that there is a constant C = C(p, ν, n) such that ∗

F (u)(t)L2 (Rn ) ≤ C[uX(c,ν)]p t−p

−1

for

t ≥ 0.

(2.123)

The left-hand side of (2.123) raised the power 2 is estimated by [uX(c,ν)]2p times
I(t) := t + |x|−2pm ct − |x|−2pν dx. Rn

Switching to the polar co-ordinates and recalling (2.25), we get
∞ I(t) = cn t + r−2pm ct − r−2pν r2m dr 0
∞ ∗ t + r−2(p +1) ct − r−2pν dr. ≤ cn 0

148

H. Kubo and M. Ohta

Since pν > 1 according to the choice of ν, we obtain (2.123). Analogously we see that F (u)(t) ∈ C(R; L2 (Rn )), hence (2.120) in Proposition 2.7 yields
∞ Rc [F (u)](t)E(c) ≤ (n + 1) F (u)(τ )L2 (Rn ) dτ t
∞ ∗ p τ −p −1 dτ, ≤ C[uX(c,ν) ] t

by (2.123). Since p∗ > 0, we arrive at (2.122). This completes the proof.




Thanks to the fact that if F (x, t) ∈ C([0, ∞) × Rn ), then
∞

∞
dt Rc [F ](t, x)(∂ ∂t2 − ∆)φ(t, x)dx = dt F (t, x)φ(t, x)dx 0

Rn ∞ C0 ((0, ∞)× Rn ),

0

Rn

for all φ ∈ the result obtained in Theorem 2.5 can be formulated in a slightly different way as follows: Theorem 2.6. Let n = 2, 3 and let p > p0 (n). Suppose (ϕ, ψ) ∈ Y (ν) with ν the number chosen in Theorem 2.2. Let u ∈ X(c, ν) be a solution to (2.7) obtained in Theorem 2.2. Then there is a unique weak solution v ∈ X(c, ν) to (2.94) in the distributional sense satisfying (2.121). In the rest of this subsection, we discuss the so-called nonlinear scattering. The map W+ from H˙ 1 × L2 to itself defined as follows is called wave operator: Let u and v+ be solutions to (2.1)–(2.2) and (2.94) respectively such that u(t) − v+ (t)E(c) → 0

as t → +∞.

(2.124)

Then we put W+ [(v+ (0), (∂ ∂t v+ )(0))] := (u(0), ∂t u(0)). Similarly, when u and v− are solutions to (2.1)–(2.2) and (2.94) in (−∞, 0] × Rn respectively such that u(t) − v− (t)E(c) → 0 as t → −∞,

(2.125)

we define W− as ∂t v− )(0))] := (u(0), ∂t u(0)). W− [(v− (0), (∂ Moreover, if W+ is bounded, invertible and the range of W+ contains the range of W− , then one can define a scattering operator as S := (W W+ )−1 ◦ W− , or (v− (0), ∂t v− (0))
−→ (v+ (0), (∂ ∂t v+ )(0)).

(2.126)

Unfortunately, we do not find from Theorem 2.5 or its proof if it is possible to define W± even on some dense set of a neighborhood of the origin in H˙ 1 × L2 . Nevertheless, we can directly construct a densely defined nonlinear scattering operator S in the following manner. Let us consider u = Kc [ϕ− , ψ− ] + Tc [F (u)] in

R × Rn ,

(2.127)

Coupled Systems of Semilinear Wave Equations where Kc [ϕ, ψ] is defined by (2.8) and we put
t Tc [F ](t, x) = Jc [F (s, ·)](t − s, x) ds.

149

(2.128)

−∞

Roughly speaking, it suffices to carry out the following two steps: 1st step. For given v− (t, x) := Kc [ϕ− , ψ− ](t, x), find a unique solution u(t, x) of (2.127) satisfying (2.125). 2nd step. Find a unique solution v+ (t, x) of the homogeneous wave equation (2.94) satisfying (2.124). Then the scattering operator S related to (2.1) is defined by (2.126). To state the result obtained in this framework, we introduce vV (c,κ) =

sup (t,x)∈R×Rn

|t| + |x|m c|t| − |x|κ |v(t, x)|.

(2.129)

Then we have the following statements. ∂xα ϕ− , ∂xα ψ− ) ∈ Theorem 2.7. Let n = 2, 3 and let p > p0 (n). For |α| ≤ 2 suppose (∂ Y (ν) with ν the number chosen in Theorem 2.2. (i) Then there are constants ε0 = ε0 (p, ν, c, n) > 0 and C0 = C0 (ν, c, n) > 0 such that if ∂xα ϕ− , ∂xα ψ− )Y (ν) ≤ ε for 0 < ε ≤ ε0 , max (∂ |α|≤2

then there is a unique classical solution u of (2.1) in {v ∈ C(R × Rn ) : vV (c,ν) ≤ 2C C0 ε}. Moreover, we have ∂xα u ∈ V (c, ν) (|α| = 1, 2), and ∗

u(t) − v− (t)E(c) ≤ C[uV (c,ν) ]p t−p

for t ≤ 0,

(2.130)

where v− := Kc [ϕ− , ψ− ](t, x). (ii) Let u be as in the part (i) of the theorem. Then there exists a unique classical solution v+ (t, x) to (2.94) such that ∂xα v+ ∈ V (c, ν) (|α| ≤ 2), and ∗

u(t) − v+ (t)E(c) ≤ C[uV (c,ν) ]p t−p

for t ≥ 0.

(2.131)

Proof. First we consider the part (i). Since the procedure is similar to that of Theorem 2.2, we concentrate on establishing the following basic estimate for Tc [F ](t, x). Proposition 2.8. Let n = 2, 3 and let a, c, µ, κ > 0. Suppose F ∈ C(R × Rn ), µ > m + 1 and κ > 1. Then there exists a constant C1 = C1 (a, c, µ, κ) > 0 such that sup (t,x)∈R×Rn

≤ C1

|t| + |x|m ct − |x|κ0 |T Tc [F ](t, x)|

sup (t,x)∈R×Rn

|t| + |x|µ a|t| − |x|κ |F (t, x)|,

where we put κ0 = µ − m − 1, provided µ < 2 when n = 2.

(2.132)

150

H. Kubo and M. Ohta

Proof. First suppose t ≤ 0. Since Jc [g](−t, x) = −J Jc [g](t, x) by the definitions (2.10) and (2.11), it follows from (2.128) and (2.98) that Tc [F ](t, x) = Rc [F (−s, ·)](|t|, x). Therefore (2.100) implies |t| + |x|m ct − |x|κ0 |T Tc [F ](t, x)| m+κ0 |Rc [F (−s, ·)](|t|, x)| ≤ C|t| + |x| ≤C

sup (t,x)∈[0,∞)×Rn

t + |x|µ at − |x|κ |F (−t, x)|,

which is bounded by the right-hand side of (2.132). Next suppose t ≥ 0. Then we have from (2.128) and (2.9) Tc [F ](t, x) = Lc [F ](t, x) + L− c [F ](t, x), where we put L− c [F ](t, x)




0

= −∞

Jc [F (s, ·)](t − s, x) ds.

(2.133)

Since (2.49) implies t + |x|m ct − |x|κ0 |Lc [F ](t, x)| t + |x|µ at − |x|κ |F (t, x)|, ≤C sup (t,x)∈[0,∞)×Rn

it is enough to show t + |x|m ct − |x|κ0 |L− c [F ](t, x)| ≤C sup |t| + |x|µ a|t| − |x|κ |F (t, x)|.

(2.134)

(t,x)∈R×Rn

In what follows, we assume c = 1. When n = 3, we have from (2.133) and (2.36) |L− 1 [F ](t, x)| ≤ I ×

sup (t,x)∈R×Rn

|t| + |x|µ a|t| − |x|κ |F (t, x)|,

where we put I=

1 2r




0

−∞




t−s+r

|t−s−r|

λ − s−µ+1 λ + as−κ dλ ds.

Changing the variables by (2.51), we have
∞
t+r 1 1 − a −κ 1+a I= α+ β dβ, dα β−µ+1  4r t−r 2 2 α∨|r−t| where a ∨ b = max{a, b}.

(2.135)

Coupled Systems of Semilinear Wave Equations

151

When a = 1, we have 1 4r

≤

I




t+r

t−r

α−κ dα




∞

|r −t|

β−µ+1 dβ

≤ Ct + r−1 r − t−µ+2 , since κ > 1 and µ > 2. Here we have used (2.102). Thus we get I ≤ Ct + r−1 r − t−κ0 .

(2.136)

When a = 1, we consider two cases separately. First suppose t ≥ 2r > 0 or 0 < r ≤ 1. It follows that

∞ 1 − a −κ 1+a r − t−µ+1 t+r α+ β dβ. dα  I ≤ 4r 2 2 t−r −∞ Cr − t−µ+1 ,

≤

since µ > 1 and κ > 1. This estimate yields (2.136) in this case. Next suppose 2r ≥ t > 0 and r ≥ 1. Since κ > 1, we have
C ∞ I≤ β−µ+1 dβ, r |r−t| which implies (2.136) in this case. Thus we see that (2.136) is also valid for the case a = 1. Therefore (2.134) for n = 3 follows from (2.135). Next we prove (2.134) for n = 2 by dividing the argument into two cases. Case 1. a = 1 or t ≥ 2r > 0 or 0 < r ≤ 1. It follows from (2.133) and (2.37) that |L− 1 [F ](t, x)| ≤ (I1 + I2 )

sup (t,x)∈(−∞,0]×Rn

|t| + |x|µ a|t| − |x|κ |F (t, x)|,

(2.137)

where we put √ 1 λ − sλ − s−µ+ 2 λ + as−κ  dλ ds (λ − s + t + r)(λ + s + r − t) −∞ |t−s−r| √ 1

∞ 1−a −κ ββ−µ+ 2  1+a 1 t+r 2 α + 2 β √ √ dα dβ, 2 t−r β+t+r α+r−t α∨|r−t|


I1

= =




0

t−s+r

(2.138)

and
I2

(t−r)∧0




t−s−r

= −∞

=

1 2




0




t−r

∞

dα −∞

|α|

√ 1 λ − sλ − s−µ+ 2 λ + as−κ  dλ ds (λ − s + t + r)(t − r − λ − s)

√ 1 1−a −κ ββ−µ+ 2  1+a 2 α + 2 β √ √ dβ, β+t+r t−r−α

(2.139)

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H. Kubo and M. Ohta

where we have changed the variables by (2.51). Moreover, we divide I2 into two integrals J1 and J2 defined as follows:
t−r
β √ 1 1−a −κ ββ−µ+ 2  1+a 2 α + 2 β √ √ 2J J1 = H(t − r) dα (2.140) dβ β+t+r t−r−α 0 −β 1
∞
t−r √ 1−a −κ ββ−µ+ 2  1+a 2 α + 2 β √ √ 2J J2 = dβ dα, (2.141) β+t+r t−r−α |r −t| −β where H(t) is the Heaviside function, i.e., H(t) = 1 for t > 0, H(t) = 0 for t ≤ 0. We shall show I1 + J1 + J2 ≤ Ct + r− 2 r − t−µ+ 2 , 1

3

provided κ > 1 and 3/2 < µ < 2. Since it holds that for β, r, t > 0 β Cβ β + r + t ≤ t + r , we get for t > r 1 2

t + r J1




t−r

≤ C

−µ+1

β 0


≤ C

t−r

−∞




β

dβ −β

(2.142)

(2.143)

1−a −κ  1+a 2 α + 2 β √ dα t−r−α

(2.144)

β−µ+1 (t − r − β)− 2 dβ 1

≤ Cr − t− 2 r − t[2−µ]+ , 1

by (2.105). This estimate gives (2.142) for J1 . When a = 1, we use (2.143) to get
t+r
∞ −κ β−µ+1  1+a α + 1−a 1 2 β √ 2 t + r 2 I1 ≤ C dβ, dα α+r−t t−r |r −t|
∞
t−r −κ β−µ+1  1+a α + 1−a 1 2 β 2 √ 2 dβ t + r J2 ≤ C dα. t−r−α |r −t| −β It follows that
∞
∞ 1 1 − a −κ 1+a − 12 α+ β (α + r − t) dα ≤ C  σ−κ (σ − β ∗ )− 2 dσ, 2 2 ∗ t−r β 1−a 1+a (t − r) + β. Since κ > 1, we see from (2.106) that the last 2 2 ∗ − 12 integral is estimated by Cβ  . Hence,
∞ 1 1 1+a 2 t + r I1 ≤ C (r − t)− 2 dβ. β−µ+1 β − 1 − a |r −t| where β ∗ =

1

Analogously, we find that t + r 2 J2 is estimated by the same integral as in the above. Since µ > 3/2, we obtain the desired estimate (2.142) by the following lemma.

Coupled Systems of Semilinear Wave Equations

153

Lemma 2.17. Let κ > 1/2 and b, d ∈ R. Then there exists a constant C = C(κ) > 0 such that
∞ 1 1 (1 + σ)−κ (1 + |σ − d|)− 2 dσ ≤ C(1 + |b|)−κ (1 + |b| + |d|) 2 . (2.145) |b|

Proof. Let H(t) be the Heaviside function. If we set
∞ σ 1 P1 = H(|σ − d| − )(1 + σ)−κ (1 + |σ − d|)− 2 dσ, 2 |b|
∞ 1 σ P2 = (1 − H(|σ − d| − ))(1 + σ)−κ (1 + |σ − d|)− 2 dσ, 2 |b| then we see that the left-hand side of (2.145) is estimated by P1 + P2 . It is easy to see that
∞ 1 1 P1 ≤ C (1 + σ)−κ− 2 dσ ≤ C(1 + |b|)−κ+ 2 .

(2.146)

|b|

On the one hand, since |σ − d| ≤ σ/2 is equivalent to 2d/3 ≤ σ ≤ 2d, we see that I2 = 0 if d ≤ 0 and that for d > 0
2d 1 P2 ≤ C(1 + |b|)−κ (1 + |σ − d|)− 2 dσ 2 3d

≤

−κ

C(1 + |b|)

1

(1 + |d|) 2 .

Combining this estimate with (2.146), we get (2.145). The proof is complete.




When a = 1, it follows from (2.138) and (2.141) that
t+r
∞ 1 β−µ+ 2 α−κ √ 2I1 ≤ dβ, dα α+r−t t−r |r −t|
∞
t−r 1 β−µ+ 2 α−κ √ dβ 2J J2 ≤ dα. t−r−α |r −t| −β Since µ > 3/2 and κ > 1, we have
µ− 32 (I1 + J2 ) ≤ C r − t

∞ t−r

α−κ √ dα + α+r−t − 12

≤ Cr − t




t−r

−∞

α−κ √ dα t−r−α

,

by virtue of (2.105) and (2.106). When t ≥ 2r > 0 or 0 < r ≤ 1, this estimate implies (2.142). Thus we have shown (2.134) for n = 2 in this case. Case 2. a = 1, 0 < t ≤ 2r and r ≥ 1. If we set
1 b dσ √ √ K(a, b, d) = √ π a b−σ σ−a d−σ

(2.147)

154

H. Kubo and M. Ohta

for a < b < d, then it follows from (2.133) and (2.23) that I3 + I4 ) |L− 1 [F ](t, x)| ≤ (I where we put


I3




0

t−s+r

= −∞


I4

sup (t,x)∈(−∞,0]×Rn

|t−s−r|

|t| + |x|µ |t| − |x|κ |F (t, x)|,

(2.148)

√ 1 λλ − s−µ+ 2 λ + s−κ

×K((λ − r)2 , (t − s)2 , (λ + r)2 ) dλ ds, √ 1 λλ − s−µ+ 2 λ + s−κ

(t−r)∧0
t−s−r

= −∞

0

×K((λ − r)2 , (λ + r)2 , (t − s)2 ) dλ ds. Using Lemma 2.18 below, we get
t+r
∞ β−µ+1+θ α−κ √ √ I3 ≤ C dα dβ, t + r − α β + t − r(α + r − t)θ t−r |r −t|
I4

≤




β

dβ −β

0




∞

+C

t−r

dβ |r −t|

≡




t−r

CH(t − r)

−β

√ 1 ββ−µ+ 2 α−κ √ √ dα β+t+r t−r−α

(2.149)

(2.150)

β−µ+ 2 +θ α−κ √ dα r(t − r − α)θ 1

J3 + J4 ,

where we divided I4 into two integrals as we have done it in (2.140) and (2.141). Note that J3 is the same form as in (2.140) with a = 1, and that it was 1 3 evaluated by Ct + r− 2 r − t−µ+ 2 in (2.144) without using the assumption a = 1. Therefore we have only to estimate I3 and J4 . We fix θ as 0 < θ < min{µ − (3/2), 1/2}. Then we have
t−r √ −µ+ 32 +θ rJ J4 ≤ Cr − t α−κ (t − r − α)−θ dα. −∞

By (2.105) we get (2.142) for J4 , since κ > 1, and we have assumed 0 < t ≤ 2r and r ≥ 1. Finally, we evaluate I3 . By (2.149) and (2.104) we have
t+r 3 1 α−κ (t + r − α)− 2 (α + r − t)−θ dα. (2.151) r − tµ− 2 −θ I3 ≤ C t−r

Since (t + r)/3 ≥ t − r for 0 < t ≤ 2r, we may split the integral at α = (t + r)/3. Then we see that the right-hand side is estimated by
∞ 1 C(t + r)− 2 α−κ (α + r − t)−θ dα +Cr + t−κ

t−r
t+r t−r

(t + r − α)− 2 (α + r − t)−θ dα. 1

Coupled Systems of Semilinear Wave Equations

155

Since κ > 1, (2.106) yields
∞ α−κ (α + r − t)−θ dα ≤ Cr − t−θ . t−r

Moreover, since θ < 1/2, we have
t+r 1 1 (t + r − α)− 2 (α + r − t)−θ dα ≤ π(2r) 2 −θ . t−r

Therefore we see that the α–integral in (2.151) is estimated by Cr +t− 2 r −t−θ , because r ≥ 1, t > 0 and κ > 1. Hence we get (2.142) for I3 . This completes the proof of Proposition 2.8, assuming Lemma 2.18 below is valid.
1

Lemma 2.18. Let θ > 0, s < 0 and |t − s − r| < λ < t − s + r. Then there is a constant C = C(θ) > 0 such that C βθ √ , K((λ − r)2 , (t − s)2 , (λ + r)2 ) ≤ √ t + r − α β + t − r (α + r − t)θ where α, β are defined by (2.51). Let θ > 0 and 0 < λ < t − s − r. Then it holds that 1 √ K((λ − r)2 , (λ + r)2 , (t − s)2 ) ≤ √ , t−r−α t+r+β and that if β > t − r, then βθ C K((λ − r)2 , (λ + r)2 , (t − s)2 ) ≤ √ . rλ (t − r − α)θ

(2.152)

(2.153)

(2.154)

Proof. First we prove (2.153). It is easy to see from (2.24) that 1 K((λ − r)2 , (λ + r)2 , (t − s)2 ) ≤  , (t − s)2 − (λ + r)2 which implies (2.153). To prove (2.152) and (2.154), we prepare the following inequality: For θ > 0 there is a constant C = C(θ) > 0 such that  θ d−a 1 √ K(a, b, d) ≤ C . (2.155) d−b b−a In fact, splitting the integral at σ = (a + b)/2, we have from (2.147)
d
b √ dσ dσ √ √ √ b − a K(a, b, d) ≤ + . √ σ−a d−σ b−σ d−σ a a We see that the first term of the right-hand side is equal to π. While, by using the integration by parts, the second one is estimated by √  θ
b d−a d−a dσ 2 b−a √ ≤ 2 + log ≤C + d−b d−b d−a a d−σ for all θ > 0. Thus we obtain (2.155).

156

H. Kubo and M. Ohta Now it follows from (2.155) that K((λ − r)2 , (t − s)2 , (λ + r)2 )  θ 1 (λ + r)2 − (λ − r)2  ≤C 2 (λ + r)2 − (t − s)2 (t − s) − (λ − r)2  θ 4λr 1 √ √ , =C (β + r + t)(α + r − t) t+r−α β+t−r

which implies (2.152), since λ < β and β + r + t > r for s < 0. Moreover, we have K((λ − r)2 , (λ + r)2 , (t − s)2 )  θ (t − s)2 − (λ − r)2 1  ≤C 2 (t − s)2 − (λ + r)2 (λ + r) − (λ − r)2  θ (t − r + β)(t + r − α) 1 √ =C , (t − r − α)(t + r + β) 4rλ which yields (2.154), because t − r < β and −α < β. This completes the proof.
Next we consider the part (ii). Setting v+ (t, x) = u(t, x) − Rc [|u|p ](t, x)

for (t, x) ∈ R × Rn ,

one can verify that v+ has the desired properties as in the proof of Theorem 2.7.


3. Semilinear system, I In this section we consider the following system:

2 (t, x) ∈ [0, ∞) × Rn , (∂ ∂t − c21 ∆)u1 = |u2 |p , 2 2 q (∂ ∂t − c2 ∆)u2 = |u1 | , (t, x) ∈ [0, ∞) × Rn ,

(3.1)

where p, q > 1 and n ≥ 2. The initial value problem for (3.1) was studied by Del Santo, Georgiev and Mitidieri in [18]. They found the critical curve F (p, q) = 0 in the p-q plane when c1 = c2 . Here critical curve means that if F (p, q) < 0, then small data global existence holds, and blow-up occurs if F (p, q) > 0. The function F (p, q) is defined as follows:

& q + 2 + p−1 p + 2 + q −1 n−1 , . (3.2) F (p, q) = max − pq − 1 pq − 1 2 The blow-up part was also established by Deng [20] independently (see also [17]). The critical case where F (p, q) = 0 was treated independently by [3], [19] for n = 3 and by [65] for n = 2, 3. In these works the blow-up result was obtained. Next the authors studied the case c1 = c2 in [66]. This work is motivated by the recent results established by Kovalyov [57], Agemi and Yokoyama [5], Hoshiga and Kubo [36] and Yokoyama [114]. In those papers, the small data global existence

Coupled Systems of Semilinear Wave Equations

157

for systems of nonlinear wave equations with different propagation speeds has been well developed when the nonlinear terms depend only on the derivatives of the unknown functions but not on the unknown functions themselves (see also [96] and [2] for related results on nonlinear elastic wave equations, and [86] on KleinGordon-Zakharov equations). It was shown in [66] that even if c1 = c2 , the critical curve is the same as in the case c1 = c2 for n = 3. In Theorem 3.1 below we extend the result to the two dimensional case. Therefore we conclude that the unequal propagation speeds do not have a major effect on the system (3.1). Moreover, in the case where F (p, q) ≥ 0, the lifespan of the solution was studied by [67] for n = 3 and by Di Flaviano [16] for n = 2. It is interesting that the discrepancy of p and q has an effect on the order of ε in the estimate of the lifespan only for the critical case (see Theorem 3.3 below). On the one hand, in the case where F (p, q) < 0, the existence of self-similar solution was treated by Kubo and Tsugawa [70], and asymptotic behaviors of the global solution was considered by Kubo and Kubota [62, 63, 64]. This section is organized as follows. In the next subsection, we prove the blow-up result in the same line as in Subsection 2.1. The argument improves the proof in earlier works [65, 66] by employing Lemma 3.3. Subsection 3.2 is devoted to the existence part which can be handled by (2.49), since it is applicable also to the case where the propagation speeds are distinct. In Subsection 3.3 we consider the self-similar solution by making use of (2.92). Subsection 3.4 is concerned with asymptotic behaviors. We shall establish analogous result to those from Subsection 2.5 by using (2.100) and (2.132). 3.1. Blow-up As is well known, a solution (u1 , u2 ) of the system (3.1) with the initial data uj (0, x) = ϕj (x),

∂t uj (0, x) = ψj (x),

x ∈ Rn

(j = 1, 2),

(3.3)

is obtained by solving the following system of integral equations: u1 u2

= =

Kc1 [ϕ1 , ψ1 ] + Lc1 [|u2 |p ] in [0, ∞) × Rn , Kc2 [ϕ2 , ψ2 ] + Lc2 [|u1 |q ] in [0, ∞) × Rn ,

(3.4) (3.5)

where Kc [ϕ, ψ], Lc [F ] are defined by (2.8), (2.9) respectively. Throughout this section, we assume 1 < p ≤ q, and for convenience we set n−1 , (3.6) p∗ = mp − m − 1, q ∗ = mq − m − 1, m = 2 α = pq ∗ − 1, β = qp∗ − 1, Γ ≡ Γ(p, q) = α + pβ. (3.7) We note that β ≤ α and Γ(p, q) = −p(pq − 1)F (p, q) when 1 < p ≤ q. We take the initial data in such a way that ϕj (x) = 0,

ψj (x) = εgj (x)

for j = 1, 2,

(3.8)

where ε > 0 and gj ∈ C(R ) satisfies n

gj (x) ≥ 0 for all x ∈ Rn ,

gj (0) > 0

for j = 1, 2.

(3.9)

158

H. Kubo and M. Ohta

Theorem 3.1. Let n = 2, 3, 1 < p ≤ q and Γ(p, q) ≤ 0. Suppose that ε ∈ (0, 1] and g1 , g2 ∈ C(Rn ) satisfies (3.9). Then the solution of (3.4)–(3.5) with (3.8) blows up in a finite time T ∗ (ε). Moreover, there exists a positive constant C ∗ independent of ε such that   (3.10) if Γ(p, q) = 0, p < q, T ∗ (ε) ≤ exp Cε−p(pq−1)   T ∗ (ε) ≤ exp Cε−p(p−1) (3.11) if Γ(p, q) = 0, p = q, T ∗ (ε) ≤ Cεp(pq−1)/Γ(p,q)

if

Γ(p, q) < 0.

(3.12)

Remark 3.1. We remark that the orders of ε appearing in the estimates (3.10)– (3.12) are optimal, thanks to (3.40)–(3.42) below. In order to prove Theorem 3.1, we first prove the following lemma. Lemma 3.2. Let a, c > 0, µ∗ > 0, ν ≥ 0, µ ≤ µ∗ and α > 0. Then, there exists a positive constant C depending only on a, c, µ∗ , ν such that  

cs − λ λν f dλ ds (3.13) µ c Da (r,t)∩Σ(c,α) (s + λ) ν+1
(at−r)/a  Cr aη f (η) dη ≥ 1− (at + r)(at − r)µ−ν−1 α at − r holds for any non-negative function f on [α, ∞) and any (r, t) ∈ Σ(a, α), where Da (r, t) and Σ(c, α) are defined by (2.22) and (2.31). Proof. Let (r, t) ∈ Σ(a, α). We denote the left-hand side of (3.13) by I(r, t), and change the variables by cs − λ . c We divide the argument into two cases; a ≤ c and a > c. First, we consider the case a ≤ c. Then, by Lemma 2.2, we have 
(at−r)/a 

at+r (ξ − aη)ν I(r, t) ≥ C dξ f (η) dη ξµ α at−r
at+r
dξ (at−r)/a ≥ C (at − r − aη)ν f (η) dη µ at−r ξ α
(at−r)/a Cr ≥ (at − r − aη)ν f (η) dη (at + r)(at − r)µ−1 α ν
(at−r)/a  Cr aη = f (η) dη, 1 − (at + r)(at − r)µ−ν−1 α at − r ξ = as + λ,

η=

which implies (3.13). Next, we consider the case a > c. We divide further into two cases, (r, t) ∈ Σ(c, α) and (r, t) ∈ Σ(a, α) \ Σ(c, α). When (r, t) ∈ Σ(c, α), we have

Coupled Systems of Semilinear Wave Equations I(r, t) ≥ C{I1 (r, t) + I2 (r, t)}, where
(ct−r)/c
I1 (r, t) =

at+r

α at−r (at−r)/a
ξ ∗ (η)


I2 (r, t) =

(ct−r)/c

at−r

159

(ξ − aη)ν f (η) dξ dη, ξµ (ξ − aη)ν f (η) dξ dη. ξµ

While, if (r, t) ∈ Σ(a, α) \ Σ(c, α), we have I(r, t) ≥ CII3 (r, t), where
(at−r)/a
ξ∗ (η) (ξ − aη)ν f (η) I3 (r, t) = dξ dη. ξµ α at−r In the definitions of I2 (r, t) and I3 (r, t), we put ξ ∗ (η) =

2ac a+c (at − r) − η. a−c a−c

As in the case a ≤ c, we have Cr I1 (r, t) ≥ (at + r)(at − r)µ−ν−1




(ct−r)/c 

aη 1− at − r

α

On the other hand, for j = 2, 3, we have
(at−r)/a
(at − r − aη)ν f (η) Ij (r, t) ≥ C ηj∗

where we put

η2∗

η3∗

= (ct − r)/c and

ν

ξ ∗ (η) at−r

f (η) dη.

(3.14)

dξ dη, ξµ

= α. Since

a+c (at − r), a−c 2c ξ ∗ (η) − (at − r) = (at − r − aη), a−c at − r ≤ ξ ∗ (η) ≤

we have




ξ ∗ (η)

dξ at − r − aη ≥C . ξµ (at − r)µ

at−r

Thus, for j = 2, 3, we have Ij (r, t)




(at−r)/a 

≥

C (at − r)µ−ν−1

≥

Cr (at + r)(at − r)µ−ν−1

ηj∗

ν+1 aη f (η) dη (3.15) 1− at − r ν+1
(at−r)/a  aη f (η) dη. 1− at − r ηj∗

From (3.14) and (3.15), we see that (3.13) is also valid for the case a > c. This completes the proof.
By Lemma 3.2, we have the following proposition.

160

H. Kubo and M. Ohta

Proposition 3.1. Let n = 2, 3, a, c > 0, p ≥ 1 and κ ∈ R. Then there exists a constant C = C(a, c, p) > 0 such that b
y η [f c,κ (η)]p dη, y ≥ 1, (3.16) 1− La [|f |p ]a,p∗ (y) ≥ C y η pκ 1 where b = m + (1 − m)p + 1. ˜ y), by (2.21) and (3.13), we have Proof. Let y ≥ 1. For any (t, x) ∈ Σ(a, La [|f |p ](t, x)   

cs − λ λm+(1−m)p 1 ≥ f c,κ p pκ 2arm c Da (r,t)∩Σ(c,1) (cs + λ) (cs − λ)  
b (at−r)/a [f c,κ (η)]p Cr1−m aη dη. ≥ 1 − ∗ (at + r)(at − r)p 1 at − r η pκ Since the function




y

 1−

y
→ 1

η y

b

p

dλ ds

[f c,κ (η)]p dη η pκ

˜ y), we have is non-decreasing, for any (t, x) ∈ Σ(a, ∗

|x|−(1−m) (at + |x|)(at − |x|)p |La [|f |p ](t, x)| ≥ C


1

y

 b [f c,κ (η)]p η dη, 1− y η pκ


which implies (3.16). Now, we shall give the proof of Theorem 3.1.

Step 1. As in the proof of Theorem 2.1, from (3.8) and (3.9), we see that there exist constants δ ∈ (0, 1) and C0 > 0 such that uj (t, x) ≥ εJ Jcj [gj ](t, x) ≥

C0 ε (t + r)m

holds for |cj t − r| ≤ δ/2, cj t + r ≥ δ and j = 1, 2. Step 2. There exists a constant C1 > 0 such that u1 c1 ,p∗ (y) ≥ C1 εp ,

u2 c2 ,q∗ (y) ≥ C1 εq

for y ≥ 1.

˜ 1 , 1), by (2.21), Step 1 and (3.13), we have Indeed, for any (t, x) ∈ Σ(c u1 (t, x) ≥ ≥ ≥ ≥

Lc1 [|u2 |p ](t, x)  

c2 s − λ (C C0 ε)p λm χ dλ ds mp [δ/4,δ/2] 2c1 rm c2 Dc1 (r,t) (s + λ) m+1
δ/2  Cεp r1−m c1 η dη 1− (c1 t + r)(c1 t − r)mp−m−1 δ/4 c1 t − r
δ/2 Cεp r1−m (1 − η)m+1 dη. (c1 t + r)(c1 t − r)p∗ δ/4

(3.17)

Coupled Systems of Semilinear Wave Equations

161

Here χ[δ/4,δ/2] (λ) denotes the characteristic function on [δ/4, δ/2]. Thus we have u1 c1 ,p∗ (y) ≥ Cεp

for y ≥ 1.

In the same way, we have u2 c2 ,q∗ (y) ≥ Cε for y ≥ 1. Step 3. Since q

u1 c1 ,p∗ (y) ≥ Lc1 [|u2 |p ]c1 ,p∗ (y)

and u2 c2 ,q∗ (y) ≥ Lc2 [|u1 |q ]c2 ,q∗ (y),

by Proposition 3.1, there exists a constant C2 > 0 such that b
y η 1 [u2 c2 ,q∗ (η)]p 1− dη u1 c1 ,p∗ (y) ≥ C2 y η pq∗ 1 b
y η 2 [u1 c1 ,p∗ (η)]q u2 c2 ,q∗ (y) ≥ C2 dη 1− y η qp∗ 1

for y ≥ 1, (3.18) for y ≥ 1, (3.19)

where we put b1 = m + (1 − m)p + 1 and b2 = m + (1 − m)q + 1. Step 4. We put U1 (y) = u1 c1 ,p∗ (y) and U2 (y) = u2 c2 ,q∗ (y) for y ≥ 1. Then, by (3.17), (3.18) and (3.19), we can show that if Γ(p, q) ≤ 0, (U1 (y), U2 (y)) blows up in a finite time. Indeed, when Γ(p, q) = 0 and p = q, we put U (y) = min{U Uj (y) : j = 1, 2}. Then, we have b
y [U (η)]p η dη for y ≥ 1, 1− U (y) ≥ C1 εp , U (y) ≥ C2 y η pp∗ 1 where we put b = m + (1 − m)p + 1. Thus (3.11) follows from Lemma 2.3, since pp∗ = 1 in this case. Next, we consider the case where Γ(p, q) = 0 and 1 < p < q. Note that β < 0 < α and α + pβ = 0 in this case. We put f1 (z) = ε−p U1 (eλz ),

f2 (z) = ε−pq eβλz U2 (eλz ),

λ = ε−p(pq−1) .

Then, for z ≥ 0 we have f1 (z) ≥ C1 , f2 (z) ≥ 0, and
z b1 1 − e−(z−ζ) [ff2 (ζ)]p dζ, f1 (z) ≥ C2
0 z  b2 f2 (z) ≥ C2 1 − e−λ(z−ζ) eβλ(z−ζ) [f1 (ζ)]q λ dζ, 0

where we used the fact that z
→ (1 − e−z )b1 is increasing on [0, ∞). Since f1 (z) is non-decreasing and λ ≥ 1, for any z ≥ 1 and 0 < h ≤ 1 we have
z  b2 f2 (z) ≥ C2 [f1 (z − h)]q 1 − e−λ(z−ζ) eβλ(z−ζ) λ dζ z −h/λ


≥ C2 [f1 (z − h)]q

h

 b2 βτ 1 − e−τ e dτ.

0

Since 1 − e

−τ

−1

≥ (1 − e )τ for 0 ≤ τ ≤ 1, we have
h  b2 βτ (1 − e−1 )b2 eβ b2 +1 h 1 − e−τ e dτ ≥ . b2 + 1 0

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Thus, we see that
f1 (z) ≥ C1 ,

z

f1 (z) ≥ C3 hp(b2 +1)



1 − e−(z−ζ)

b1

[f1 (ζ − h)]pq dζ

(3.20)

1

holds for any z ≥ 1 and 0 < h ≤ 1. Thus (3.10) follows from Lemma 3.3 below. Finally, we consider the case Γ(p, q) < 0. Note that Γ(p, q) = α + pβ < 0 and β < 0 in this case. We put f1 (z) = ε−p U1 (ε−µ ez ),

f2 (z) = ε−pq (ε−µ ez )β U2 (ε−µ ez ),

µ=−

p(pq − 1) . α + pβ

Then, for z ≥ 0, we have f1 (z) ≥ C1 , f2 (z) ≥ 0, and
z b1 1 − e−(z−ζ) f1 (z) ≥ C2 [ff2 (ζ)]p dζ, 0
z b2 1 − e−(z−ζ) f2 (z) ≥ C2 eβ(z−ζ) [f1 (ζ)]q dζ. 0

Thus, as in the previous case, we see that (3.20) holds for any z ≥ 1 and 0 < h ≤ 1. Thus (3.12) follows from Lemma 3.3 below. Lemma 3.3. Let C1 , C2 > 0, a, b ≥ 0 and p > 1. Suppose that f (t) satisfies
f (t) ≥ C1 ,

f (t) ≥ C2 ha

t

(1 − e−(t−τ ) )b [f (τ − h)]p dτ

(3.21)

1

for any t ≥ 1 and 0 < h ≤ 1. Then, f (t) blows up in a finite time. Proof. By (3.21) with h = 1, for t ≥ 2, we have
f (t) ≥

t

(1 − e

C2 2

=







−(t−τ ) b

) [f (τ − 1)] dτ ≥ p

C1p C2

t

(1 − e−(t−τ ))b dτ

2 t−2

C1p C2

(1 − e−σ )b dσ ≥ C1p C2

0

(1 − e−1 )b (t − 2)b+1 . b+1

Thus, there exists T1 > 2 such that f (t) ≥ A1 for any t ≥ T1 , where we put ⎞ ⎛ ∞  b+1 log γ log j ⎠ , γ = max{ + 2(a + b + 1) A1 = exp ⎝1 + , 1}. j p−1 p C2 (1 − e−1 )b j=1 (3.22) Tk } by Here, we define sequences {Ak } and {T Ak+1 =

Apk γk 2(a+b+1)

,

Tk+1 = Tk +

2 , k2

k ∈ N.

(3.23)

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163

Then, for any k ∈ N, we see that f (t) ≥ Ak for t ≥ Tk . Indeed, by (3.21) with h = k −2 , for t ≥ Tk + 2k −2 , we have
t −2a f (t) ≥ C2 k (1 − e−(t−τ ) )b [f (τ − k −2 )]p dτ t−k−2

≥ C2 Apk k −2a




k−2

(1 − e−σ )b dσ ≥

0

C2 (1 − e−1 )b p −2(a+b+1) Ak k . b+1

Moreover, by (3.22) and (3.23), we have ⎛ ⎞ k k   log γ log j ⎠ log Ak+1 = pk ⎝log A1 − − 2(a + b + 1) j p pj j=1 j=1 ⎛ ⎞ ∞  log j ⎠ log γ − 2(a + b + 1) ≥ pk ⎝log A1 − ≥ pk , j p−1 p j=1 Tk+1

=

T1 +

k  2 j2 j=1

for any k ∈ N. Therefore, f (t) blows up in a finite time.




3.2. Small data global existence In this subsection we mainly consider the case where Γ ≡ Γ(p, q) > 0,

(3.24)

where Γ is defined by (3.7). One can verify that the condition F (p, q) < 0 with F defined by (3.2) is equivalent to (3.24) when p ≤ q. In addition, we assume the following condition for the sake of simplicity: 0 < p∗ ≤ q ∗ ,

i.e., (n + 1)/(n − 1) < p ≤ q.

(3.25)

Let the initial data (ϕ1 , ψ1 ) for u1 (t, x) and (ϕ2 , ψ2 ) for u2 (t, x) be in Y (ν) and Y (κ), respectively. In order to apply the basic estimate (2.49), we take ν, κ > 0 as follows. Lemma 3.4. If (3.25) and (3.24) hold, then there are ν and κ satisfying 0 < ν ≤ p∗ ,

(3.26)

∗

0 1,

q ∗ − κ + qν > 1.

(3.27) (3.28)

Moreover, when n = 2, we can choose them such that ν < 1/2,

κ < 1/2.

(3.29)

Proof. To find κ satisfying (3.27) and (3.28) for some ν, we need to assure that (1 − p∗ + ν)/p < q ∗ + qν − 1,

(1 − p∗ + ν)/p < q ∗ ,

0 < q ∗ + qν − 1.

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Equivalently, ν > (1 − p∗ − pq ∗ + p)/(pq − 1),

ν < pq ∗ + p∗ − 1,

ν > (1 − q ∗ )/q.

(3.30)

Note that (1 − q ∗ )/q ≤ 0 if q ∗ ≥ 1, and 1 − q∗ 1 − p∗ − pq ∗ + p q ∗ − 1 + q(p∗ − 1) − = ≤0 q pq − 1 q(pq − 1) if q ∗ ≤ 1. Therefore, to take ν satisfying (3.30) together with (3.26), it suffices to assure (1 − p∗ − pq ∗ + p)/(pq − 1) < pq ∗ + p∗ − 1, (1 − p∗ − pq ∗ + p)/(pq − 1) < p∗ , 0 < pq ∗ + p∗ − 1. Simplifying the above relations by the aid of (3.7), we get β + qα > 0,

α + pβ > 0,

α + p∗ > 0.

(3.31)

Notice that β≤

Γ ≤α p+1

for 1 < p ≤ q.

(3.32)

So β + qα ≥ α + pβ = Γ for 1 < p ≤ q. Therefore (3.24) implies (3.31). In addition, to choose ν and κ so that (3.29) holds, we need (1 − p∗ + ν)/p < 1/2,

i.e., ν < p∗ + (p − 2)/2,

hence (1 − p∗ − pq ∗ + p)/(pq − 1) < ∗

∗

(1 − p − pq + p)/(pq − 1)
0 when n = 2, which follows from (3.25). Thus we finish the proof.
Proposition 3.2. Assume that (3.24) and (3.25) hold. Let u ∈ X(c1 , ν) and v ∈ X(c2 , κ). If ν and κ satisfy (3.26), (3.27) and the first inequality in (3.28), and also (3.29) when n = 2, then there is a constant K0 = K0 (c1 , c2 , n, p, q, ν, κ) such that Lc1 [|v|p ]X(c1 ,ν) ≤ K0 [vX(c2 ,κ) ]p .

(3.35)

Moreover, if ν and κ satisfy (3.26), (3.27) and the second inequality in (3.28), and also (3.29) when n = 2, then we have Lc2 [|u|q ]X(c2 ,κ) ≤ K0 [uX(c1 ,ν) ]q .

(3.36)

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165

Proof. We shall only prove (3.35), since the other inequality can be shown in an analogous way. We make use of (2.49) by taking c = c1 , a = c2 , µ = m + 1 + ν and κ = p∗ − ν + pκ (> 1) (recall (3.28)). Then we get Lc1 [|v|p ]X(c1 ,ν) ≤ C1

sup (t,x)∈[0,T )×Rn

∗

t + |x|m+1+ν c2 t − |x|p

−ν+pκ

|v(t, x)|p . (3.37)

Since mp = p∗ + m + 1 by (3.6), we have |v(y, s)|p

∗

≤ [vX(c2 ,κ) ]p λ + s−m−1−p λ − c2 s−pκ ∗

≤ C[vX(c2 ,κ) ]p λ + s−m−1−ν λ − c2 s−pκ+ν−p

for (y, s) ∈ Rn × R with λ = |y|, where C is a constant depending only on c2 and
ν − p∗ . Therefore we obtain (3.35) from (3.37). Thus the proof is complete. We are now in position to state the main results of this subsection. Theorem 3.2. Let n = 2, 3 and let (3.24), (3.25) hold. Let ν and κ satisfy (3.26), (3.27) and (3.28), and also (3.29) when n = 2. Suppose (ϕ1 , ψ1 ) ∈ Y (ν) and (ϕ2 , ψ2 ) ∈ Y (κ). Then there is a positive constant ε0 = ε0 (c1 , c2 , n, p, q, ν, κ) such that if we assume (ϕ1 , ψ1 )Y (ν) + (ϕ2 , ψ2 )Y (κ) ≤ ε for 0 < ε ≤ ε0 , there exists a unique solution (u1 , u2 ) ∈ X(c1 , ν) × X(c2 , κ) of (3.4)–(3.5). Proof. For ε > 0 we put X := {(u, v) ∈ C([0, ∞) × Rn ) × C([0, ∞) × Rn ) : C0 ε}, (u, v) ≡ uX(c1 ,ν) + vX(c2 ,κ) ≤ 2C where ν, κ are the numbers in the theorem, and C0 is the number in (2.41). We have only to prove that M defined by M [u, v] = (Kc1 [(ϕ1 , ψ1 ], Kc2 [ϕ2 , ψ2 ]) + (Lc1 [|v|p ], Lc2 [|u|q ]) is a contraction in X for sufficiently small ε. This can be done by making use of (2.41) and Proposition 3.2. We omit the details.
At the end of this subsection, we state the estimate for the lifespan from below. This can be shown by combining the argument presented in the Subsection 2.3 and the computation made in [67] and [16]. Theorem 3.3. Let n = 2, 3 and let (3.25) hold. Suppose that (ffj , gj ) ∈ C 1 (Rn ) × C(Rn ) satisfies fj (x) = gj (x) = 0 for |x| ≥ R (3.38) for some R > 0 and j = 1, 2. Let us take the initial data as ϕj (x) = εffj (x),

ψj (x) = εgj (x)

for

x ∈ Rn .

(3.39)

Then there is a constant ε0 = ε0 (c1 , c2 , n, p, q, R) > 0 such that there exists a unique solution (u1 , u2 ) ∈ C([0, T ∗ (ε)) × Rn ) × C([0, T ∗ (ε)) × Rn ) of (3.4)–(3.5).

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Moreover, there is a constant C ∗ = C ∗ (c1 , c2 , n, p, q, R) > 0 such that for any ε with 0 < ε ≤ ε0 we have   if Γ(p, q) = 0, p = q, (3.40) T ∗ (ε) ≥ exp C ∗ ε−p(pq−1)   T ∗ (ε) ≥ exp C ∗ ε−p(p−1) if Γ(p, q) = 0, p = q, (3.41) T ∗ (ε) ≥ C ∗ εp(pq−1)/Γ(p,q)

if Γ(p, q) < 0.

(3.42)

3.3. Self-similar solution We study the existence of the self-similar solution to (3.1). For λ > 0, let us define uλ1 (t, x) = λ

2(p+1) pq−1

uλ2 (t, x) = λ

u1 (λt, λx),

2(q+1) pq−1

u2 (λt, λx).

If (u1 , u2 ) satisfies (3.1), then (uλ1 , uλ2 ) also solves (3.1). We call (u1 , u2 ) a selfsimilar solution, if it satisfies (3.1) and uλ1 (t, x) ≡ u1 (t, x),

uλ2 (t, x) ≡ u2 (t, x)

(3.43)

for (t, x) ∈ [0, ∞) × R and λ > 0. Additionally, we put n

p∗ =

2(p + 1) − m, pq − 1

q∗ =

2(q + 1) − m, pq − 1

m=

n−1 . 2

(3.44)

Our main result of this subsection reads as follows. Theorem 3.4. Let n = 2, 3 and let (3.24), (3.25) and p∗ > 0 hold. Suppose (ϕ1 , ψ1 ) ∈ Y˙ (p∗ ) and (ϕ2 , ψ2 ) ∈ Y˙ (q∗ ). Then there is a positive constant ε0 = ε0 (c1 , c2 , p, q, n) such that if we assume (ϕ1 , ψ1 )Y˙ (p∗ ) + (ϕ2 , ψ2 )Y˙ (q∗ ) ≤ ε for ˙ 1 , p∗ ) × X(c ˙ 2 , q∗ ) of 0 < ε ≤ ε0 , there exists a unique solution (u1 , u2 ) ∈ X(c (3.4)–(3.5). Remark 3.5. We can understand the choice of p∗ and q∗ in the connection with the limiting case of (3.28). Namely, for ν = p∗ and κ = q∗ we have p∗ − ν + pκ = 1,

q ∗ − κ + qν = 1.

Corollary 3.1. Let n = 2, 3 and let (3.24), (3.25) and p∗ > 0 hold. Let the initial data take the form ϕ1 (x) = f1 (ω)|x|−

2(p+1) pq−1

− 2(q+1) pq−1

ϕ2 (x) = f2 (ω)|x|

ψ1 (x) = g1 (ω)|x|−

, ,

2(p+1) pq−1 −1

,

(3.45)

− 2(q+1) pq−1 −1

(3.46)

ψ2 (x) = g2 (ω)|x|

with ω = x/|x|, fj ∈ C 1 (S 2 ), gj ∈ C(S 2 ) (j = 1, 2). If (ϕ1 , ψ1 ) ∈ Y˙ (p∗ ) and (ϕ2 , ψ2 ) ∈ Y˙ (q∗ ) is sufficiently small in Y˙ (p∗ ) and Y˙ (q∗ ) respectively, then there exists a unique self-similar solution of (3.4)–(3.5). To prove Theorem 3.4, the following proposition is crucial. Once we establish it, a standard argument yields the conclusion.

Coupled Systems of Semilinear Wave Equations

167

˙ 1 , p∗ ) and v ∈ Proposition 3.3. Assume that (3.24) and (3.25) hold. Let u ∈ X(c ˙ X(c2 , q∗ ). Then there is a constant K0 = K0 (c1 , c2 , n, p, q) such that p Lc1 [|v|p ]X(c ˙ 1 ,p∗ ) ≤ K0 [vX(c ˙ 2 ,q∗ ) ] ,

(3.47)

Lc2 [|u| ]X(c ˙ 2 ,q∗ ) ≤ K0 [uX(c ˙ 1 ,p∗ ) ] .

(3.48)

q

q

Proof. First we note that (3.24) is equivalent to pq∗ < 1. Moreover, (3.25) implies pq∗ ≥ qp∗ , hence qp∗ < 1 holds. We shall only prove (3.47), since the other inequality can be shown in an analogous way. We make use of Proposition 2.5 by taking c = c1 , a = c2 , µ = mp and κ = pq∗ , so that mp + pq∗ − m − 2 = p∗ . When n = 2, we have mp + pq∗ = (5/2) + p∗ . Therefore, p∗ < 1/2 assures mp + pq∗ < 3. Since 3 < p ≤ q, we get p∗ < 1/2, hence we can apply the proposition. In this way we obtain (3.47). Thus the proof is complete.
3.4. Asymptotic behavior In this subsection we consider asymptotic behaviors of solutions to (3.1) from the same point of view as in the Subsection 2.5. The first result claims that the classical solution of (3.1) with (3.3) tends to a solution of the system of homogeneous wave equations

2 (∂ ∂t − c21 ∆)v1 = 0, (t, x) ∈ [0, ∞) × Rn , (3.49) 2 2 (t, x) ∈ [0, ∞) × Rn , (∂ ∂t − c2 ∆)v2 = 0, as t → +∞ in the sense of the energy. Since this result can be shown by following the proof of Theorem 2.5, we omit the proof. Theorem 3.5. Let n = 2, 3 and let (3.24), (3.25) hold. Let ν and κ satisfy (3.26), (3.27) and (3.28), and also (3.29) when n = 2. For |α| ≤ 2 suppose (∂ ∂xα ϕ1 , ∂xα ψ1 ) ∈ α α Y (ν) and (∂ ∂x ϕ2 , ∂x ψ2 ) ∈ Y (κ). Let (u1 , u2 ) be a classical solution to (3.1) with (3.3) such that (∂ ∂xα u1 , ∂xα u2 ) ∈ X(c1 , ν) × X(c2 , κ) for |α| ≤ 2. Then there is a unique classical solution (v1+ , v2+ ) of (3.49) satisfying ∂xα v1+ ∈ X(c1 , ν), ∂xα v2+ ∈ X(c2 , κ) (|α| ≤ 2), and ' ( 12 ∗ u1 (t) − v1+ (t)E(c1 ) ≤ C[u2 X(c2 ,κ) ]p t−p t[1−2pκ]+ for t ≥ 0, (3.50) u2 (t) − v2+ (t)E(c2 ) ≤ C[u1 X(c1 ,ν) ]q t−q

∗

' ( 12 t[1−2qν]+

for t ≥ 0. (3.51)

Remark 3.6. 1) In particular, if we take ν as ν = p∗ , we have κ > 1/p from (3.28). /1 . Hence we can drop the factor (1 + |t|)[1−2pκ]+ 2 in (3.50). 2) If p and q satisfy β = qp∗ − 1 > 0, (3.52) then α = pq ∗ −1 > 0 by (3.32), hence p∗ > 1/q, q ∗ > 1/p. Note that (3.26) through (3.28) holds for all ν and κ satisfying 1/q < ν ≤ p∗ ,

1/p < κ ≤ q ∗ .

(3.53)

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Therefore, choosing ν and κ verifying (3.53), we can drop not only the same factor /1 . as in the item 1) but also the factor (1 + |t|)[1−2qν]+ 2 in (3.51). 3) Even in the case where 2pκ ≤ 1 (resp. 2qν ≤ 1), the right-hand side of (3.50) (resp. (3.51)) tends to zero as t → −∞, since (3.28) implies −p∗ +

1 1 − pκ < − − ν, 2 2

−q ∗ +

1 1 − qν < − − κ. 2 2

(3.54)

The second result is concerned with the nonlinear scattering related to (3.1). We consider the following integral equation − − − (u1 , u2 ) = (Kc1 [(ϕ− Tc1 [|u2 |p ], Tc2 [|u1 |q ]) 1 , ψ1 ], Kc2 [ϕ2 , ψ2 ]) + (T

(3.55)

in R × R , where Tc [F ] was defined by (2.128). Then we can show the result by making use of (2.132). n

Theorem 3.6. Let the assumption of Theorem 3.5 be fulfilled. (i) Then there is a constant ε0 = ε0 (p, q, ν, κ, c1 , c2 , n) > 0 such that if α − α − ∂xα ϕ− ∂xα ϕ− max {(∂ 1 , ∂x ψ1 )Y (ν) + (∂ 2 , ∂x ψ2 )Y (κ) } ≤ ε

|α|≤2

for 0 < ε ≤ ε0 ,

then there is a unique classical solution (u1 , u2 ) of (3.1) in {(v1 , v2 ) ∈ C(R × Rn ) × C(R × Rn ) : C0 ε}, v1 V (c1 ,ν) + v2 V (c2 ,κ) ≤ 2C where C0 = C0 (ν, κ, c1 , c2 , n) > 0 is a constant. Moreover, we have ∂xα u1 ∈ V (c1 , ν), ∂xα u2 ∈ V (c2 , κ) (|α| ≤ 2), and ' ( 12 ∗ for t ≤ 0, (3.56) u1 (t) − v1− (t)E(c1 ) ≤ C[u2 V (c2 ,κ) ]p t−p t[1−2pκ]+ u2 (t) − v2− (t)E(c2 ) ≤ C[u1 V (c1 ,ν) ]q t−q

∗

'

t[1−2qν]+

( 12

for t ≤ 0, (3.57)

− where vj− (t, x) := Kcj [ϕ− j , ψj ](t, x) (j = 1, 2). (ii) Let (u1 , u2 ) be as in the part (i) of the theorem. Then there exists a unique classical solution (v1+ , v2+ ) to (3.49) such that ∂xα v1+ ∈ V (c1 , ν), ∂xα v2+ ∈ V (c2 , κ) (|α| ≤ 2), and (3.50), (3.51) holds good.

4. Semilinear system, II In this section we study the following system:

2 (∂ ∂t − c21 ∆)u1 = λ1 |u1 |p1 |u2 |p2 , (∂ ∂t2 − c22 ∆)u2 = λ2 |u1 |q1 |u2 |q2 ,

(t, x) ∈ [0, ∞) × Rn , (t, x) ∈ [0, ∞) × Rn ,

(4.1)

where λ1 , λ2 ∈ R and n ≥ 2. If p1 = 0 and q2 = 0, then the system (4.1) reduces to the system (3.1). For this reason, we assume p1 , p2 , q1 , q2 ≥ 1.

(4.2)

Coupled Systems of Semilinear Wave Equations

169

This means that the nonlinearities are involved by a product of u1 and u2 and that they are Lipschitz continuous with respect to u1 and u2 . Based on this structure, we shall derive the effect of the discrepancy of the propagation speeds. For such a purpose, we may assume that the degree of the nonlinearity of the first equation is the same as that of the second one. Namely, we consider the case where there is α ≥ 2 such that p1 + p2 = q1 + q2 ≡ α.

(4.3)

When c1 = c2 , it follows from the result about the single wave equation (2.1) that small data global existence holds if α > p0 (n) and that blow-up occurs if 2 ≤ α ≤ p0 (n). Here p0 (n) is the positive root of (2.3). (For the details about (2.1), see Section 2.) Next we turn our attention to the case c1 = c2 . When n = 3, Kubo and Tsugawa [70] firstly proved small data global existence for all α > 2. Then the authors [68] showed that the same is true for α = 2. Let √us compare these results with those for the case c1 = c2 . Since p0 (3) = 1 + 2,√we find that there is a significant difference among them when 2 ≤ α ≤ 1 + 2. Actually, for such α we have a global solution if c1 = c2 , while blow-up occurs if c1 = c2 . This observation shows the effect of the discrepancy between the propagation speeds, which comes from the way of interaction in the nonlinearities (recall that we do not have such an effect for the system (3.1)). In fact, since the right-hand side of the equations in (4.1) are involved by a product of u1 and u2 , one can compensate the deficiency of the pointwise decaying order for the powers of u1 and u2 each other, based on the unequal propagation speeds. Recently the extension to the two dimensional case n = 2 was √ done by the authors [69]. The result implies that for 3 < α ≤ p0 (2) = (3 + 17)/2, we also have such an effect of the unequal propagation speeds as in case n = 3. (See also Remark 4.1 below.) Concerning the existence and non-existence of self-similar solution for the system (4.1), the following result was established by [70]. Let n = 3. Then in contrast to the condition on α for the existence of the smooth solution mentioned in the above, it is really necessary to assume α > 2 in order to assure the existence of self-similar solution. Actually, if α = 2, then we can show a counter-example which implies that self-similar solution does not exist in general (see Theorems 4.3 and 4.7 below). On the one hand, the asymptotic behavior of the smooth solution for the system (4.1) was considered by [63]. This section is organized as follows. We shall establish the existence result of the global solution and the self-similar solution in Subsections 4.1 and 4.2 by using (2.49) and (2.92), respectively. As for the peculiar case where α = 2 and n = 3, we show a blow-up result for slowly decaying initial data in Theorem 4.4. In Subsection 4.3 we extend these results to the case where the number of the equations is more than 2.

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4.1. Small data global existence As usual, we treat the system (4.1) in the following integral form: u1 u2

= Kc1 [ϕ1 , ψ1 ] + λ1 Lc1 [|u1 |p1 |u2 |p2 ] in [0, ∞) × Rn , = Kc2 [ϕ2 , ψ2 ] + λ2 Lc2 [|u1 |q1 |u2 |q2 ] in [0, ∞) × Rn ,

(4.4) (4.5)

where Kc [ϕ, ψ], Lc [F ] are defined by (2.8), (2.9) respectively. For convenience, we put n−1 2 − m, m = , (4.6) α∗ = α−1 2 where α is defined by (4.3). The main result of this subsection is as follows. Theorem 4.1. Let n = 2, 3 and let (4.2) and (4.3) hold. Assume that c1 = c2 and that either α > 2,

ν ≥ α∗ ,

ν > 0,

(4.7)

or α = 2,

ν>1

(4.8)

for n = 3, and that 1 (4.9) 2 for n = 2. Suppose (ϕj , ψj ) ∈ Y (ν) (j = 1, 2). Then there is a positive constant ε0 = ε0 (c1 , c2 , n, p1 , p2 , q1 , q2 , ν) such that if we assume (ϕ1 , ψ1 )Y (ν) + (ϕ2 , ψ2 )Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ X(c1 , ν) × X(c2 , ν) of (4.4)–(4.5). α > 3,

ν ≥ α∗ ,

0 3 in the theorem is sharp. We also remark that one can take ν in such a way that α∗ = (5−α)/2(α−1) < ν < 1/2, since α > 3 when n = 2. To prove Theorem 4.1, the following proposition is crucial. Once we establish it, a standard argument yields the conclusion. Proposition 4.1. Let the assumption in Theorem 4.1 be fulfilled. Let u ∈ X(c1 , ν) and v ∈ X(c2 , ν). Then there is a constant K0 = K0 (c1 , c2 , n, p1 , p2 , ν) such that Lc1 [|u|p1 |v|p2 ]X(c1 ,ν) ≤ K0 [uX(c1 ,ν) ]p1 [vX(c2 ,ν) ]p2 .

(4.10)

Proof. First we consider the case where α > (n + 1)/(n − 1), ν ≥ α∗ and ν > 0. We use the key assumption c1 = c2 in the following way: If we take (s, y) near the light cone c2 s = |y|, then the weight c1 s − |y| behaves like s + |y|. On the contrary, if (s, y) is close to the other light cone c1 s = |y|, then the weight c2 s − |y| behaves

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like s + |y|. In conclusion, one can extract some additional decay in s + |y|, provided c1 = c2 . This observation leads us to |u(s, y)|p1 |v(s, y)|p2

(4.11) −mα−p1 ν

≤ C[uX(c1 ,ν) ] [vX(c2 ,ν) ] {λ + s p1

p2

−mα−p2 ν

+λ + s

−p1 ν

λ − c1 s

−p2 ν

λ − c2 s

}

for (s, y) ∈ R × R with λ = |y|. For j = 1, 2 we put n

ρj = m(α − 1) − 1 + (pj − 1)ν. Since pj ≥ 1, ν > 0, and m(α − 1) > 1 for α > (n + 1)/(n − 1), we have ρj > 0. Therefore, (4.11) implies |u(s, y)|p1 |v(s, y)|p2 ≤ C[uX(c1 ,ν) ]p1 [vX(c2 ,ν) ]p2 {λ + s−m−1−ν λ − c2 s−ρ1 −p2 ν +λ + s−m−1−ν λ − c1 s−ρ2 −p1 ν . Now we consider two different cases. First suppose ν > α∗ . We shall make use of (2.49) by taking µ = m + 1 + ν,

κ = (m + ν)(α − 1) − 1.

Then we have κ > 1 by ν > α∗ and ν > 0. Since ρ1 + p2 ν = ρ2 + p1 ν = κ, (2.49) yields (4.10). Next suppose ν = α∗ . Then we have α∗ > 0 and for n = 2, α∗ < 1/2. Moreover, we have κ = 1. Since ρj > 0, we see that pj ν < 1 for j = 1, 2. Therefore, (4.11) and (2.49) yield (4.10), since mα + p1 ν + p2 ν − m − 2 = α∗ for ν = α∗ . It remains to deal with the case where n = 3, α = 2 and ν > 1. By (4.11) we have |u(s, y)|p1 |v(s, y)|p2 ≤ C[uX(c1 ,ν) ]p1 [vX(c2 ,ν) ]p2 {λ + s−2−ν λ − c2 s−ν +λ + s−2−ν λ − c1 s−ν } for (s, y) ∈ R × R3 with λ = |y|, since p1 = p2 = 1 for α = 2. Therefore (2.49) implies (4.10), because of ν > 1. This completes the proof.
At the end of this subsection, we give a unified statement of Theorems 3.2 and 4.1 except for the limiting case where n = 3 and α = 2. We assume in the system (4.1) that either p1 = q2 = 0,

p2 , q1 > 1,

(4.12)

or p1 , p2 , q1 , q2 ≥ 1.

(4.13)

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Note that in this situation, the assumption (4.3) does not hold in general. On the one hand, the system (4.1) keeps the scaling invariance related to the following exponents: k1 =

2{(q2 − 1) − p2 } − m, (p1 − 1)(q2 − 1) − p2 q1

k2 =

2{(p1 − 1) − q1 } − m. (p1 − 1)(q2 − 1) − p2 q1

Here we assumed (p1 − 1)(q2 − 1) − p2 q1 = 0. Actually, in the case where (4.12) holds, k1 and k2 reduce to k1 =

2(p2 + 1) − m = (p2 )∗ , p2 q1 − 1

k2 =

2(q1 + 1) − m = (q1 )∗ , p2 q1 − 1

(4.14)

in view of (3.44). While, when (4.13) and (4.3) hold, we see from (4.6) that 2 − m = α∗ . α−1 Then we have the following generalization of the theorems. k1 = k2 =

Theorem 4.2. Let n = 2, 3 and let pj , qj (j = 1, 2) satisfy (p1 − 1)(q2 − 1) = p2 q1 and either (4.12) or (4.13). In addition, for j = 1, 2 we assume that kj > 0 and that when n = 2, kj < 1/2. Suppose that for c1 = c2 : k1 p1 + k2 p2 < 1,

k1 q1 + k2 q2 < 1,

(4.15)

k1 q1 < 1,

(4.16)

and for c1 = c2 : k1 p1 < 1,

k2 p2 < 1,

k2 q2 < 1.

Let (ϕj , ψj ) ∈ Y (kj ). Then there is a positive constant ε0 = ε0 (c1 , c2 , pj , qj , n) such that if we assume max (ϕj , ψj )Y (kj ) ≤ ε for 0 < ε ≤ ε0 , then there exists j=1,2

a unique solution (u1 , u2 ) ∈ X(c1 , k1 ) × X(c2 , k2 ) of (4.4)–(4.5). Proof. First suppose that (4.12) holds. Putting p = p2 and q = q1 , we see that both conditions (4.15) and (4.16) are equivalent to k2 p < 1,

k1 q < 1.

We proceed as in the proof of Proposition 3.2 by using (2.49) with µ = mp, κ = pk2 and recalling (4.14). Then we get Lc1 [|u2 |p ]X(c1 ,k1 ) ≤ C[u2 X(c2 ,k2 ) ]p ,

(4.17)

Lc2 [|u1 | ]X(c2 ,k2 ) ≤ C[u1 X(c1 ,k1 ) ] .

(4.18)

q

q

We omit the further details. Next we consider the case where (4.13) and c1 = c2 hold. Similarly to the above, we get Lc1 [|u1 |p1 |u2 |p2 ]X(c1 ,k1 ) ≤ C[u1 X(c1 ,k1 ) ]p1 [u2 X(c2 ,k2 ) ]p2 .

(4.19)

In the application of (2.49), we take µ = m(p1 + p2 ), κ = p1 k1 + p2 k2 . Then we have µ + κ − m − 2 = k1 . We omit the further details.

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Finally we treat the case where (4.13) and c1 = c2 hold. Analogously to the proof of Proposition 4.1, we again get (4.19). We omit the further details. This completes the proof.
4.2. Self-similar solution When (4.3) holds, we see that the system (4.1) is invariant with respect to the scale transform uj −→ uλj (j = 1, 2), where 2

uλj (t, x) = λ α−1 uj (λt, λx)

(λ > 0).

(4.20)

We call (u1 , u2 ) a self-similar solution, if it satisfies (4.1) and uλj (t, x) ≡ uj (t, x),

(4.21)

for (t, x) ∈ [0, ∞) × R , j = 1, 2, and λ > 0. Our main result of this subsection reads as follows. n

Theorem 4.3. Let n = 2, 3 and let (4.2) and (4.3) hold. Assume that c1 = c2 and that n+3 n+1 0 if α < (n+3)/(n−1).) Besides, when n = 2, we have α∗ < 1/2 since α > 3. Thus µ + κ < 3. Moreover, for j = 1, 2 we have   2 − m < 1. (4.24) pj α∗ = pj α−1 In fact, since α > (n + 1)/(n − 1) is equivalent to (α − 1)m > 1, we have pj (2 − (α − 1)m) < pj . On the one hand, (4.2) and (4.3) imply pj ≤ α − 1. Hence (4.24) holds. Thus Proposition 2.5 yields (4.23). This completes the proof.
Remark 4.2. We give a remark about the assumption on α for n = 3 in Theorems 4.1 and 4.3. In the former, α = 2 is admitted in (4.8). While it is excluded in the latter. This can be understood as follows. The global solution obtained in Theorem ˙ 1 , 1) × X(c ˙ 2 , 1), since α∗ = 1 for α = 2. On the contrary, we 4.3 belongs to X(c need to assume ν > 1 for α = 2 in Theorem 4.1 for showing the global solution exists in X(c1 , ν) × X(c2 , ν). Moreover, we can show that blow-up actually occurs for the case where α = 2 and ν = 1 as in Theorem 4.4 below. Theorem 4.4. Let n = 3 and let p1 = p2 = q1 = q2 = 1 (that is α = 2). Assume that c1 = c2 and that (ϕj , ψj ) ∈ Y (1) (j = 1, 2). Then there is a positive constant ε0 = ε0 (c1 , c2 ) such that if we assume (ϕ1 , ψ1 )Y (1) +(ϕ2 , ψ2 )Y (1) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ C([0, T ∗ (ε))×R3 )×C([0, T ∗ (ε))×R3 ) of (4.4)–(4.5). Moreover, there is a constant C > 0, independent of ε, such that T ∗ (ε) ≥ exp(Cε−1 ).

(4.25)

Furthermore, when the coefficients λ1 and λ2 in (4.4) and (4.5) are positive, there exist (ϕj , ψj ) ∈ Y (1) with (ϕ1 , ψ1 )Y (1) + (ϕ2 , ψ2 )Y (1) = ε(> 0) such that there is a constant C ∗ > 0, independent of ε, such that T ∗ (ε) ≤ exp(C ∗ ε−1 ).

(4.26)

Proof. First we prove (4.25). Similarly to (4.11) we have |u(s, y)||v(s, y)|

≤ CuZ(c1 ,1) vZ(c2 ,1) {λ + s−3 λ − c2 s−1 +λ + s−3 λ − c1 s−1 }

for (s, y) ∈ R × R3 with λ = |y|, where  · Z(c,κ) is defined by (2.76). Therefore (2.65) with µ = 3 implies Lc1 [|u||v|]Z(c1 ,1) + Lc2 [|u||v|]Z(c2 ,1) ≤ C log(2 + T )uZ(c1,1) vZ(c2 ,1) .

(4.27)

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If we define a map M [u, v] = (Kc1 [(ϕ1 , ψ1 ], Kc2 [ϕ2 , ψ2 ]) + (Lc1 [|u||v|], Lc2 [|u||v|]), then we see from (2.41) and (4.27) that M is a contraction in {(u, v) ∈ C([0, T ) × R3 ) × C([0, T ) × R3 ) : C0 ε} (u, v) ≡ uZ(c1 ,1) + vX(c2 ,1) ≤ 2C with C0 is the number in (2.41), provided that ε is sufficiently small and that ε log(2 + T ) is bounded by some suitable constant. This will give us (4.25). Next we prove (4.26). Since (0, x−3 ) ∈ Y (1) when n = 3, we can choose ε ϕj (x) ≡ 0, ψj (x) = x−3 . 2 Then it follows from (2.8), (2.20) and Lemma 2.2 that
cj t+r ε uj (t, x) ≥ Kcj [0, ψj ](t, x) = ρρ−3 dρ 4cj r |cj t−r| ≥ C1 ε(cj t + r)−1 (cj t − r)−1 for cj t > r and j = 1, 2. Recalling (2.30), we have uj cj ,1 (y) ≥ C1 ε for y ≥ 1.

(4.28)

Moreover, by Proposition 4.3 below, we have 2
y η u1 c1 ,1 (η)u2 c2 ,1 (η) dη 1− Lcj [|u1 ||u2 |]cj ,1 (y) ≥ C y η 1 for y ≥ 1. Setting U (y) = min{u1 c1 ,1 (y), u2 c2 ,1 (y)}, and recalling (4.28), we arrive at
U (y) ≥ C1 ε,

U (y) ≥ C2 1

y

 2 U (η)2 η dη 1− y η

for y ≥ 1.

Hence we see from Lemma 2.3 with α = 1, β = 0, κ = 1 and p = 2 that the lifespan of U (y) is bounded by exp(C ∗ ε−1 ), which gives (4.26).
Proposition 4.3. Let n = 3, and let a0 > 0, 0 < a1 < a2 , p, q ≥ 1, κ∗ > 0, κ1 ∈ R and κ2 ≤ κ∗ . Then, there exists a constant C = C(a0 , a1 , a2 , p, q, κ∗ ) > 0 such that La0 [|f |p |g|q ]a0 ,p+q+qκ2 −2 (y) 2
y [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q η ≥C dη, 1− y η pκ1 1

(4.29) y ≥ 1.

Proof. From the definition of f a1 ,κ1 (y), we have |f (t, x)| ≥

f a1 ,κ1 ((a1 t − r)/a1 ) , (a1 t + r)(a1 t − r)κ1

˜ 1 , 1). (t, x) ∈ Σ(a

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˜ 1 , 1), we have (t, x) ∈ Σ(a ˜ 2 , (a1 t − r)/a1 ). Thus, from Since a1 < a2 , if (t, x) ∈ Σ(a the definition of ga2 ,κ2 (y), we have |g(t, x)| ≥

ga2 ,κ2 ((a1 t − r)/a1 ) , (a2 t + r)(a2 t − r)κ2

˜ 1 , 1). (t, x) ∈ Σ(a

Moreover, since 2a1 (a2 t − r) = (a2 − a1 )(a1 t + r) + (a1 + a2 )(a1 t − r), we have a2 − a1 a2 (a1 t + r) ≤ a2 t − r ≤ (a1 t + r) 2a1 a1 ˜ 1 , 1). Thus, we have for (t, x) ∈ Σ(a |f (t, x)|p |g(t, x)|q ≥ C

[f a1 ,κ1 ((a1 t − r)/a1 )]p [ga2 ,κ2 ((a1 t − r)/a1 )]q (t + r)p+q+qκ2 (a1 t − r)pκ1

˜ 1 , 1). Let y ≥ 1. By Proposition 2.1 and Lemma 3.2, we have for (t, x) ∈ Σ(a C La0 [|f |p |g|q ](t, x) ≥ (a0 t + r)(a0 t − r)p+q+qκ2 −2 2
(a0 t−r)/a0  [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q a0 η × dη 1− a0 t − r η pκ1 1 ˜ 0 , y). Since the function for (t, x) ∈ Σ(a 2
y η [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q dη y
−→ 1− y η pκ1 1 ˜ 0 , y), we have is non-decreasing, for any (t, x) ∈ Σ(a (a0 t + |x|)(a0 t − |x|)p+q+qκ2 |La0 [|f |p |g|q ](t, x)| 2
y [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q η ≥C dη, 1− y η pκ1 1


which implies (4.29).

Similarly to Theorem 4.2, we can state Theorems 3.4 and 4.3 in a unified way as follows. Since the proof is done by a straightforward computation, we omit it. Theorem 4.5. Let n = 2, 3 and let pj , qj , kj (j = 1, 2) satisfy the assumptions in Theorem 4.2. Let (ϕj , ψj ) ∈ Y˙ (kj ). Then there is a positive constant ε0 = ε0 (c1 , c2 , pj , qj , n) such that if we assume max (ϕj , ψj )Y˙ (kj ) ≤ ε for 0 < ε ≤ ε0 , j=1,2

˙ 1 , k1 ) × X(c ˙ 2 , k2 ) of (4.4)–(4.5). then there exists a unique solution (u1 , u2 ) ∈ X(c 4.3. Generalization To make clear the essential structure of the system (4.1) we consider the following generalized version of it: N 

(∂ ∂t2 − c2j ∆)uj =

pjkl Akl |ul |qjkl , j |uk |

(t, x) ∈ [0, ∞) × Rn , (4.30)

k,l=1

uj (0, x) = ϕj (x),

∂t uj (0, x) = ψj (x),

x ∈ Rn ,

(4.31)

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177

where 1 ≤ j ≤ N , pjkl , qjkl ≥ 1, and Akl j are real constants. We restrict ourselves to the case where there is a constant α ≥ 2 such that pjkl + qjkl ≡ α

for all j, k, l = 1, · · · , N,

(4.32)

since the general case can be formulated as in Theorems 4.2 and 4.5. Moreover, we assume that the propagation speeds are distinct, that is ck = cl if k = l. The crucial point of the argument in the preceding subsections is to make use of the interaction between uk and ul with k = l. Thus we need to exclude self-interaction terms from the nonlinearities. To do so, we pose the non-resonant assumption on them, i.e., Akk j = 0 for all j, k = 1, · · · , N.

(4.33)

We treat the problem (4.30)–(4.31) in the following integral form: uj = Kcj [ϕj , ψj ] +

N 

pjkl Akl |ul |qjkl ] in j Lcj [|uk |

[0, ∞) × Rn ,

(4.34)

k,l=1

where 1 ≤ j ≤ N . We look for solutions of (4.34) in the following spaces: X = {u = (u1 , u2 , · · · , uN ) ∈ C([0, ∞) × Rn )N :

(4.35)

uX ≡ max uj X(cj ,ν) < ∞}, 1≤j≤N

and X˙ = {u = (u1 , u2 , · · · , uN ) ∈ C([0, ∞) × Rn )N : uX˙ ≡ max uj X(c ˙ j ,ν) < ∞}.

(4.36)

1≤j≤N

Similarly to the proof of Theorem 4.1, one can show the following. Theorem 4.6. Let n = 2, 3 and let (4.32) and (4.33) hold. Assume that c1 , . . . , cN are different from each other and that either (4.7) or (4.8) holds for n = 3, and that (4.9) holds for n = 2. Suppose (ϕj , ψj ) ∈ Y (ν) (1 ≤ j ≤ N ). Then there is a positive constant ε0 = ε0 (c1 , c2 , n, pjkl , qjkl , ν) such that if we assume max (ϕj , ψj )Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution 1≤j≤N

(u1 , . . . , uN ) ∈ X of (4.34). Next we consider the self-similar solution (u1 , . . . , uN ) ∈ X˙ which is a solution of (4.30) satisfying (4.21) for all (t, x) ∈ [0, ∞) × Rn , j = 1, 2, . . . , N , and λ > 0. The following analogue to Theorem 2.4 is concerned with the existence and nonexistence of the self-similar solution. Theorem 4.7. Let n = 2, 3 and let (4.32) and (4.33) hold. Assume that c1 , . . . , cN are different from each other and that (ϕj , ψj ) ∈ Y˙ (α∗ ) (j = 1, . . . , N ). (i) Suppose n+3 n+1 0 (see also the proof of Theorem 2.4). While, the part (i ) can be proved in a similar way to the proof of Theorem 4.3.


5. Semilinear system, III As we have seen in the Subsection 4.3, the assumption (4.33) plays an essential role in the analysis of the system (4.30). The aim of this section is to study the system for the case where (4.33) is not satisfied. To be more precise, we consider the following one as a simplest example:

2 (t, x) ∈ [0, ∞) × R3 , (∂ ∂t − c21 ∆)u1 = |u1 |p1 |u2 |p2 , (5.1) 2 2 q (∂ ∂t − c2 ∆)u2 = |u1 | , (t, x) ∈ [0, ∞) × R3 , where p1 , p2 ≥ 1, q > 1. This system was studied by the authors in [68]. We can regard (5.1) as an intermediate case between (3.1) and (4.1). The point is that the right-hand side of the first equation in (5.1) is involved by a product of u1 and u2 , while that of the second one does not. For simplicity, we focus on the case where p1 = p2 = 1.

(5.2)

The exposition for the general case where p1 ≥ 1, p2 ≥ 1 is complicated, although the real proof for large values of p1 and p2 is easier because of the “smallness” of solutions under our consideration. For this reason, we prefer to take p1 = p2 = 1. The main result of this section is roughly stated as follows. Theorem 5.1. Suppose that c1 = c2 and that (5.2) holds. Then for the initial value problem of (5.1) we have: (i) If 1 < q < 3, then blow-up occurs. (ii) If q > 3, then small data global existence holds. (iii) Let q = 3. If c1 > c2 , then blow-up occurs. While, when c1 < c2 , small data global existence holds. It is remarkable that when q = 3, the result depends on the order of the speeds of propagation. We mainly consider this peculiar case and prove the blowup part in Subsection 5.1 and the existence part in Subsection 5.2. To carry out the proof, the argument prepared in Section 2 is not sufficient. In other words, we need additionally a something new approach for the case q = 3.

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5.1. Blow-up First of all, we precisely state the blow-up part of Theorem 5.1. Let us consider the system

2 (∂ ∂t − c21 ∆)u1 = |u1 ||u2 |, (t, x) ∈ [0, ∞) × R3 , (5.3) 2 2 q (t, x) ∈ [0, ∞) × R3 (∂ ∂t − c2 ∆)u2 = |u1 | , with the initial data uj (0, x) = 0,

∂t uj (0, x) = εgj (x),

x ∈ R3 (j = 1, 2).

(5.4)

Here q > 1, cj > 0, ε > 0, and gj ∈ C(R3 ) (j = 1, 2) satisfies gj (x) ≥ 0 for all x ∈ R3 ,

g1 (0) > 0.

(5.5)

Then we have the following. Theorem 5.2. Let c1 = c2 and 1 < q ≤ 3. Suppose that g1 , g2 ∈ C(R3 ) satisfy (5.5). Then for sufficiently small ε > 0 the solution (u1 , u2 ) of (5.3)–(5.4) blows up in a finite time T ∗ (ε), if either q = 3 and c1 > c2 or 1 < q < 3. Moreover, there exists a constant A > 0, independent of ε, such that ⎧ −3 if q = 3 and c1 > c2 ⎨ exp(Aε ) ∗ −q(2+q)/(3−q)2 Aε if 1 < q < 3 and c1 > c2 . T (ε) ≤ (5.6) ⎩ −2q/(3−q)2 Aε if 1 < q < 3 and c1 < c2 Remark 5.1. As for the case where q = 3 and c1 > c2 , Katayama and Matsumura [45] recently proved that there is a constant B > 0, independent of ε, such that T ∗ (ε) ≥ exp(Bε−3 ).

(5.7)

Proof. We treat the problem (5.3)–(5.4) in the integral form: u1 = εJ Jc1 [g1 ] + Lc1 [|u1 ||u2 |] in Jc2 [g2 ] + Lc2 [|u1 | ] u2 = εJ q

[0, ∞) × R3 ,

(5.8)

in [0, ∞) × R .

(5.9)

3

Basically we follow the proof of Theorem 4.4. In particular, the proof for the case where 1 < q < 3 can be done analogously and less hard. For this reason, we concentrate on the case where q = 3 and c1 > c2 . It is the most delicate one in the sense that the result depends not only on the exponent q but also on the propagation speeds c1 and c2 . By (5.8), (5.9) and (5.5), we have u1 (t, x) ≥ εJ Jc1 [g1 ](t, x),

(t, x) ∈ [0, ∞) × R3 ,

u1 (t, x) ≥ Lc1 [|u1 ||u2 |](t, x), u2 (t, x) ≥ Lc2 [|u1 |3 ](t, x),

(5.10)

(t, x) ∈ [0, ∞) × R , 3

(t, x) ∈ [0, ∞) × R3 .

(5.11) (5.12)

As in the proof of (2.26), from (5.5), we see that there exists a constant C > 0 such that u1 (t, x) ≥ Cεr−1 for (t, x) ∈ E. (5.13)

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Here we put E := {(t, x) ∈ [0, ∞) × R3 : |c1 t − |x|| ≤ δ/2, c1 t + |x| ≥ δ}. Based on this estimate, we shall show u1 c1 ,2 (y) ≥ C1 ε4 ,

u2 c2 ,1 (y) ≥ C2 ε3

for y ≥ 1,

(5.14)

provided 0 < δ ≤ min{c2 , 2c1 (c1 − c2 )/(5c1 + c2 )}. Since δ ≤ c2 , by (5.12), (2.21) and (5.13), we have
c2 t+r
dξ Cε3 δ/2 Cε3 u2 (t, x) ≥ dη ≥ 2 r −δ/2 (c2 t + r)(c2 t − r) c2 t−r ξ ˜ 2 , 1). Thus the second inequality in (5.14) holds true. for (t, x) ∈ Σ(c To prove the first one, we prepare the following estimate: u2 (t, x) ≥

Cε3 (c1 t − r) , (t + r)3

(t, x) ∈ Ω,

(5.15)

where we set Ω = {(t, x) ∈ [0, ∞) × R3 : c1 t − |x| ≥ 0, |x| − c2 t ≥ δ}. By (5.12), (2.21) and (5.13), we have
λ2 (r,t)
Cε3 0 dλ dη , (t, x) ∈ Ω, u2 (t, x) ≥ 2 r −δ/2 λ1 (r,t) λ where we put λ1 (r, t) =

c1 (r − c2 t), c1 − c2

λ2 (r, t) =

c1 (r + c2 t). c1 + c2

Since λ2 (r, t) − λ1 (r, t) = 2c1 c2 (c1 t − r)/(c21 − c22 ), we get (5.15). By (5.13) and (5.15), we have |u1 (t, x)||u2 (t, x)| ≥

Cε4 (c1 t − r) , r(c1 t + r)3

(t, x) ∈ E ∩ Ω.

Since δ ≤ 2c1 (c1 − c2 )/(5c1 + c2 ), by (5.11) and (2.21), we have
c1 t+r
dξ Cε4 δ/2 Cε4 η dη ≥ u1 (t, x) ≥ 3 r 0 (c1 t + r)(c1 t − r)2 c1 t−r ξ ˜ 1 , 1), which implies the first inequality in (5.14). for (t, x) ∈ Σ(c Unfortunately, the first estimate in (5.14) is not enough to show the blow-up result because of the fast decay with respect to (c1 t − r). Thus our next step is to improve it. To this end, for 0 ≤ κ ≤ 2 we set U1,κ (y) = u1 c1 ,κ (y),

U2 (y) = u2 c2 ,1 (y).

Then (5.14) implies U1,2 (y) ≥ C1 ε4 ,

U2 (y) ≥ C2 ε3 ,

y ≥ 1.

(5.16)

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Moreover, since c2 < c1 , by Propositions 3.1 and 4.3, we see that there exist positive constants C3 = C3 (c1 , c2 ) and C4 = C4 (c1 , c2 ) such that for any κ ∈ [0, 2] 2
y U1,κ (η)U U2 (η) η dη, y ≥ 1, (5.17) U1,κ (y) ≥ C3 1− y η 1 2
y U1,κ (η)3 η U2 (y) ≥ C4 dη, y ≥ 1. (5.18) 1− y η 3κ 1 Note that C3 and C4 do not depend on κ ∈ [0, 2]. Now (5.16) and (5.17) yield 2
y U1,κ (η) η dη, y ≥ 1, (5.19) U1,κ (y) ≥ 16b 1− y η 1 where b = C2 C3 ε3 /16. Especially (5.16) and (5.19) with κ = 2 give 2
y η U1,2 (η) dη, y ≥ 1 U1,2 (y) ≥ a, U1,2 (y) ≥ 16b 1− y η 1

(5.20)

with a = C1 ε4 . One can show that U1,2 (y) grows in y, by using the following lemma. Lemma 5.2. Let a > 0 and 0 < b ≤ 1. Assume that f (y) satisfies 2
y f (η) η dη, y ≥ 1. 1− f (y) ≥ a, f (y) ≥ 16b y η 1 Then we have f (y) ≥

a b y , 4

y ≥ 1.

Proof. Put g(y) = (a/4)y b . Then we have g(y) < f (y) for any y ∈ [1, 41/b ). Moreover, since 0 < b ≤ 1 and & 2


y
1 y/2 b−1 1  y b η b−1 η dη ≥ η dη = −1 , 1− y 4 1 4b 2 1 we have

 2 g(η) η dη, y ≥ 41/b . 1− y η 1 By the comparison argument, we see that f (y) ≥ g(y) holds for any y ≥ 1. This completes the proof.



y

g(y) ≤ 16b

Applying Lemma 5.2 to (5.20), we get a U1,2 (y) ≥ y b , 4

y ≥ 1.

(5.21)

˜ 1 , y) with y ≥ 1, we have By (5.21) and the definition of U1,κ (y), for (t, x) ∈ Σ(c   b  c1 t − |x| a c1 t − |x| 2 . (c1 t + |x|)(c1 t − |x|) |u1 (t, x)| ≥ U1,2 ≥ c1 4 c1

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Again by the definition of U1,κ (y), we have U1,2−b (y) ≥

a , 4c1 b

y ≥ 1.

Repeating this procedure 2n times, we obtain U1,2−2nb (y) ≥

a , (4cb1 )2n

y ≥ 1,

which implies (c1 t + |x|)(c1 t − |x|)2−2nb |u1 (t, x)| ≥ U1,2−2nb (y) ≥

a (4cb1 )2n

˜ 1 , y). Combining this with c1 t − |x| ≥ c1 y, we obtain for y ≥ 1 and (t, x) ∈ Σ(c U1,2−nb (y) ≥

a y nb , 42n c1 nb

y ≥ 1.

(5.22)

Let  be the smallest natural number satisfying 3(2 − b) ≤ 1. Since a = C1 ε4 and b = C2 C3 ε3 /16, we see that  = O(ε−3 ) and a ≥ exp(−C C5 ε−3 ) 42 c1 b

(5.23)

with a positive constant C5 . By (5.22), (5.23) and b ≥ 1, we have C5 ε−3 )y b ≥ exp(−C C5 ε−3 )y, U1,2−b (y) ≥ exp(−C

y ≥ 1.

(5.24)

Thus, we see that U1,2−b (y) ≥ 1 for y ≥ α1 := exp(C C5 ε−3 ). Moreover, by (5.18) with κ = 2 − b, we have 2
y U1,2−b (η)3 η U2 (y) ≥ C4 dη 1− y η 3(2−b) α1 2
y/2  C4 y 1 η ≥ C4 dη ≥ log , y ≥ 2α1 . 1− y η 4 2α1 α1 C4 ). Thus, U2 (y) ≥ 1 for y ≥ α∗ := 2α1 exp(4/C Finally, rescaling as U (z) = min{U1,2−b (α∗ z), U2 (α∗ z)} and using 3(2−b) ≤ 1, we find from (5.17) and (5.18) that 2
z U (ζ)2 ζ dζ, z ≥ 1, U (z) ≥ 1, U (z) ≥ C6 1− z ζ 1 where C6 = min{C C3 , C4 }. By Lemma 2.3, we see that U (z) blows up in a finite time. Hence, the classical solution of (5.3)–(5.4) blows up in a finite time. Moreover, the lifespan T ∗ (ε) is estimated from above by exp(C ∗ ε−3 ) with a suitable positive
constant C ∗ , by the choice of α∗ and α1 . This completes the proof.

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5.2. Small data global existence Let us consider the following integral version of the initial value problem (5.3): u1 = Kc1 [ϕ1 , ψ1 ] + Lc1 [|u1 ||u2 |] in [0, ∞) × R3 ,

(5.25)

u2 = Kc2 [ϕ2 , ψ2 ] + Lc2 [|u1 |q ]

(5.26)

in [0, ∞) × R3 .

To deal with the case q = 3, we need the following modified L∞ -space: W (c, κ) = {u ∈ C([0, ∞) × R3 ) : uW (c,κ) < ∞}, uW (c,κ) =

sup (t,x)∈[0,∞)×R3

t + |x|ct − |x|κ {w(|x|, t)}−1 |u(t, x)|,

where κ > 0, and where we defined ⎧ ⎪ 1, ⎨ c2 t − r w(r, t) = log(2 + c2 t − r), 1+ ⎪ r ⎩ {log(2 + |c1 t − r|)}2 ,

(r, t) ∈ Ω1 , (r, t) ∈ Ω2 ,

(5.27)

(r, t) ∈ Ω3 ,

with Ω1 = {(r, t) ∈ [0, ∞)2 : c2 t < r}, Ω2 = {(r, t) ∈ [0, ∞)2 : (c1 + c2 )t/2 < r < c2 t}, Ω3 = {(r, t) ∈ [0, ∞)2 : r < (c1 + c2 )t/2}. Now we precisely state the existence part of Theorem 5.1. Theorem 5.3. Let c1 = c2 . (i) Let q > 3 and (ϕj , ψj ) ∈ Y (q − 2) (j = 1, 2). Then there is a positive constant ε0 = ε0 (c1 , c2 , q) such that if we assume max (ϕj , ψj )Y (q−2) ≤ ε for 0 < j=1,2

ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ X(c1 , q−2)×X(c2 , q−2) of (5.25)–(5.26). (ii) Let q = 3, 0 < c1 < c2 and 1/3 < κ < 1. Suppose (ϕ1 , ψ1 ) ∈ Y (κ), (ϕ2 , ψ2 ) ∈ Y (1). Then there is a positive constant ε0 = ε0 (c1 , c2 ) such that if we assume (ϕ1 , ψ1 )Y (κ) + (ϕ2 , ψ2 )Y (1) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ W (c1 , κ) × X(c2 , 1) of (5.25)–(5.26). Proof. First we consider the part (i). Since c1 = c2 , analogously to the proof of Proposition 4.1, we get Lc1 [|u1 ||u2 |]X(c1 ,q−2) ≤ Cu1 X(c1 ,q−2) u2 X(c2 ,q−2) .

(5.28)

In the application of (2.49), it suffices to take µ = q, κ = q − 2 (> 1). While, using (2.49) with µ = q, κ = q(q − 2) (> 3), we get Lc2 [|u1 |q ]X(c2 ,q−2) ≤ C[u1 X(c1 ,q−2) ]q .

(5.29)

We omit the further details. Next we consider the part (ii). Since X(c1 , κ) →  W (c1 , κ), (2.41) implies Kc1 [ϕ1 , ψ1 ]W (c1 ,κ) ≤ C0 (ϕ, ψ)Y (κ) .

(5.30)

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Therefore, once we establish Propositions 5.1 and 5.2 below, the standard contraction argument gives the conclusion. Proposition 5.1. Let 0 < c1 < c2 and 0 < κ < 1. Then there is a constant C = C(c1 , c2 , κ) > 0 such that Lc1 [f g]W (c1 ,κ) ≤ Cf W (c1 ,κ) gX(c2 ,1) .

(5.31)

Proof. We see from (2.38) that |Lc1 [f g](t, x)| ≤ f W (c1 ,κ) gX(c2,1) I(r, t) for any (t, x) ∈ [0, ∞) × R3 with r = |x|, where

λw(λ, s) 1 dλ ds, I(r, t) = 2 c s − λκ c s − λ 2c1 r s + λ 1 2 Dc1 (r,t)

(5.32)

Dc (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ s ≤ t, |c(t − s) − r| ≤ λ ≤ c(t − s) + r}. Thus it suffices to show that I(r, t) ≤ Ct + r−1 c1 t − r−κ w(r, t)

(5.33)

for any (r, t) ∈ [0, ∞) . In the following, we use the notation: 2

ξ = c1 s + λ,

ηj = cj s − λ,

(5.34)

where j = 1, 2. First we suppose (r, t) ∈ Ω1 . In this case we have Dc1 (r, t) ⊂ Ω1 . Moreover, r − c1 t is equivalent to r + t. Hence, c1 s + λ ≥ λ − c1 s ≥ C(t + r)

for

(λ, s) ∈ Dc1 (r, t).

Therefore from (5.32) and (5.27) we have

dλ ds C 1+κ . t + r I(r, t) = r λ − c2 s Dc1 (r,t)

(5.35)

(5.36)

Changing the variables by (5.34) with j = 2 and putting c1 + c2 2c1 ξ ∗ (η2 ) = (r − c1 t) + η2 , c2 − c1 c2 − c1 we see that λ = r − c1 (t − s) is equivalent to ξ = ξ ∗ (η2 ). Thus from (5.36) we have

C c2 t−r dη2 r+c1 t 1+κ I(r, t) ≤ dξ t + r r −(r+c1 t) η2  ξ∗ (η2 )
2c1 C c2 t−r c2 t − r − η2 = dη2 c2 − c1 r −(r+c1 t) η2 
|η2 | C 0 dη2 , ≤ r −(r+c1 t) η2  since c2 t < r. Therefore t + r1+κ I(r, t) is bounded. This means that (5.33) is valid for (r, t) ∈ Ω1 .

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Next we consider the case (r, t) ∈ Ω2 . Again, r − c1 t is equivalent to r + t, hence (5.35) is still valid. Therefore from (5.32) and (5.27) we have

λ dλ ds C (5.37) t + r1+κ I(r, t) ≤ r s + λc2 s − λ Dc1 (r,t)∩Ω2

(c2 s − λ) log(2 + c2 s − λ) C 1 dλ ds + r r + t c2 s − λ Dc1 (r,t)
C c2 t−r c2 t − r − η2 ≤ dη2 r −(r+c1 t) η2 
c2 t−r C 1 + (c2 t − r − η2 ) log(2 + η2 ) dη2 , r r + t 0 where we have changed the variables by (5.34) with j = 2. It is easy to see that the second term of the right-hand side is estimated by some constant times (c2 t − r)2 log(c2 t − r) ≤ w(r, t) rr + t for (r, t) ∈ Ω2 . While, since 0 < c2 t − r − η2 < 2(c2 t − r) for |η2 | < c2 t − r, and 0 < c2 t − r − η2 < 2|η2 | for η2 < −(c2 t − r) < 0, we see that the first term of the right-hand side of (5.37) is estimated by some constant times c2 t − r r




c2 t−r

−(c2 t−r)

1 dη2 + η2  r




−(c2 t−r)

−(c1 t+r)

dη2 ≤ 2w(r, t)

for (r, t) ∈ Ω2 (recall c2 > c1 ). In conclusion, we obtained (5.33) for (r, t) ∈ Ω2 . Finally, we treat the case (r, t) ∈ Ω3 . Note that w(r, t) ≤ C log(2 + |c2 t − r|)

for

(r, t) ∈ Ω1 ∪ Ω2 ,

(5.38)

since 0 < c2 t − r < (c2 − c1 )r/(c2 + c1 ). Moreover, c1 s − λ is equivalent to s + λ if (λ, s) ∈ Ω1 ∪ Ω2 , and c2 s − λ is so if (λ, s) ∈ Ω3 . Having these observations in mind, we see from (5.32) and (5.27) that

log(2 + |c2 s − λ|) C dλ ds I(r, t) ≤ r s + λ1+κ c2 s − λ Dc1 (r,t)

C {log(2 + |c1 s − λ|)}2 + dλ ds 2 κ r Dc1 (r,t) s + λ c1 s − λ

c2 ξ/c1 C c1 t+r 1 log(2 + |η2 |) ≤ dη2 dξ 1+κ r |c1 t−r| ξ η2  −ξ
ξ
{log(2 + |η1 |)}2 C c1 t+r 1 + dξ dη1 , r |c1 t−r| ξ2 η1 κ −ξ

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where we have changed the variables by (5.34) with j = 2 in the first term, and with j = 1 in the second term. Since κ < 1, we have
C c1 t+r {log(2 + ξ)}2 dξ I(r, t) ≤ r |c1 t−r| ξ1+κ ≤

C{log(2 + |c1 t − r|)}2 . t + rc1 t − rκ

At the last step, we have used (2.71). Thus we get (5.33) for (r, t) ∈ Ω3 . This completes the proof.
Proposition 5.2. Let 0 < c1 < c2 and κ > 1/3. Then there exists C = C(c1 , c2 , κ) > 0 such that Lc2 [f 3 ]X(c2 ,1) ≤ C[f W (c1 ,κ) ]3 .

(5.39)

Proof. We see from (2.38) that |Lc2 [f 3 ](t, x)| ≤ [f W (c1 ,κ) ]3 I(r, t) for any (t, x) ∈ [0, ∞) × R3 with r = |x|, where

1 λ{w(λ, s)}3 I(r, t) = dλ ds. 3 3κ 2c2 r Dc2 (r,t) s + λ c1 s − λ

(5.40)

Thus it suffices to show that I(r, t) ≤ Ct + r−1 c1 t − r−1

(5.41)

for any (r, t) ∈ [0, ∞)2 . By (5.38) and the fact that c1 s−λ is equivalent to s+λ if (λ, s) ∈ Ω1 ∪Ω2 , we get

λ{log(2 + |c2 s − λ|)}3 C dλ ds I(r, t) ≤ r s + λ3+3κ Dc2 (r,t)

C λ{log(2 + |c1 s − λ|)}6 + dλ ds 3 3κ r Dc2 (r,t) s + λ c1 s − λ
ξ
1 C c2 t+r dξ {log(2 + |η2 |)}3 dη2 ≤ r |c2 t−r| ξ2+3κ −ξ
c1 ξ/c2
C c2 t+r 1 {log(2 + |η1 |)}6 + dξ dη1 , r |c2 t−r| ξ2 η1 3κ −ξ where we have changed the variables by (5.34) with j = 2 in the first term, and with j = 1 in the second term. Since 3κ > 1, we have
C c2 t+r dξ I(r, t) ≤ , r |c2 t−r| ξ2 which implies (5.41). This completes the proof.




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By (5.30), Propositions 5.1 and 5.2 we can prove the part (ii). This completes the proof of Theorem 5.3.


6. Nonlinear system In the preceding sections we have studied the hyperbolic system for the case where the nonlinearity depends only on the unknowns themselves. From now on, we shall consider the system whose nonlinearity is involved by the derivatives of the unknowns. In this case the following Fujita type exponent comes into play: 2 (n ≥ 2). (6.1) pc (n) = 1 + n−1 Notice that pc (n) < p0 (n), since Φ(n, pc (n)) = −1 with Φ(n, p) the function in (2.3). The number can be introduced by the following heuristic argument. Let us consider the single wave equation ∂t u|p , (∂ ∂t2 − c2 ∆)u = |∂

(t, x) ∈ [0, ∞) × Rn ,

(6.2)

where c > 0 and p > 1. Suppose that we a priori know that the solution to (6.2) is asymptotically free. Namely, it is close to a solution v(t, x) of the homogeneous wave equation (2.94) in the sense of the energy for large values of t. Then, in virtue of Proposition 2.7, we can expect
∞ u(t) − v(t)E(c) ≤ (n + 1) F (∂ ∂t u)(τ )L2 (Rn ) dτ −→ 0, (6.3) t

as t → +∞, where F (u) = |u|p . As it is well known, v(t)E(c) is conserved and n−1 |∂ ∂t v(t, x)| ≤ Ct− 2 for (t, x) ∈ [0, ∞) × Rn . Therefore we have
∞
∞ n−1 F (∂ ∂t v)(τ )L2 (Rn ) dτ ≤ Cv(0)E(c) τ − 2 (p−1) dτ. t

t

It is clear that when p > pc (n), the last integral tends to zero as t → +∞. Hence the condition p > pc (n) seems to ensure (6.3). We come back to the systems of nonlinear wave equations of the form (∂ ∂t2 − c2i ∆)ui = Fi (u, ∂u, ∂∇u), (t, x) ∈ [0, ∞) × Rn (1 ≤ i ≤ N ),(6.4) (1 ≤ i ≤ N ),(6.5) ui (x, 0) = εffi (x), ∂t ui (x, 0) = εgi (x), x ∈ Rn ∂t , ∇), ∇ = (∂ ∂x1 , . . . , ∂xn ), ε > 0 is a small parawhere ∂ = (∂0 , ∂1 , . . . , ∂n ) = (∂ meter, ci is a positive constant, fi , gi are real-valued functions which belong to C0∞ (Rn ), and u = (u1 , . . . , uN ) is a RN -valued unknown function of (t, x). In what follows, the nonlinear term F = (F F1 , . . . , FN ) is supposed to be a smooth function of (u, ∂u, ∂∇u). Moreover, we assume that F vanishes to the p-th order near the origin with some integer p ≥ 2. Namely, there exist positive constants C and δ such that N  |F Fi (u, v, w)| ≤ C(|u|p + |v|p + |w|p ) for |u| + |v| + |w| ≤ δ, (6.6) i=1

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where the variables v = (vaj ; 0 ≤ a ≤ n, 1 ≤ j ≤ N ) ∈ R(1+n)×N , w = (warj ; 0 ≤ a ≤ n, 1 ≤ r ≤ n, 1 ≤ j ≤ N ) ∈ R(1+n)×n×N correspond to (∂a uj ), (∂a ∂r uj ), respectively. In the sequel, we use the abbreviation F (u, ∂u, ∂∇u) = O(|u|p + |∂u|p + |∂∇u|p ). Without loss of generality, we may assume that F is affine with respect to ∂∇u (see, e.g., Courant and Hilbert [13], Chapter I, Section 7). This means F = (F F1 , · · · , FN ) has the following form: Fi (u, v, w) =

N n  

αabj i (u, v)wabj + βi (u, v)

(6.7)

a,b=0 j=1

with some p−1 αabj + |v|p−1 ) and βi (u, v) = O(|u|p + |v|p ), i (u, v) = O(|u|

where a, b = 0, . . . , n, i, j = 1, . . . , N . In order to guarantee the existence of the local solution for the system, we assume baj abi αabj i (u, v) = αi (u, v) = αj (u, v).

(6.8)

Besides, we denote by F (p) the p-th order term of Taylor’s expansion for F around the origin (u, v, w) = (0, 0, 0). First we recall known results for the case where the propagation speeds coincide each other, that is, c1 = c2 = · · · = cN . It has been proved by John [39] that the blow-up occurs for the single equations (6.2) with p = pc (3) = 2, n = 3, and (∂ ∂t2 − c2 ∆)u = u ∂t u,

(t, x) ∈ [0, ∞) × R3 .

(6.9)

Agemi [1] extended these results to the case n = 2 (see also [93, 90, 58]). Therefore, when n = 2, 3, we see that the problem (6.4)–(6.5) with p = pc (n) in (6.6) does not admit the global classical solution in general, however small the initial data is. Nevertheless, Christodoulou [12] and Klainerman [54] independently established small data global existence for the case where n = 3 and p = 2 in (6.6) by different approaches, provided that F (2) has some algebraic structure called null condition which will be defined in the Definition 6.1 below. A corresponding result for the case where n = 2 and p = pc (2) = 3 in (6.6) was obtained by Katayama [42]. Moreover, when F is independent of the unknowns themselves, Alinhac [8] showed that small data global existence holds for the case where n = 2 and p = 2 in (6.6), provided that both F (2) and F (3) satisfy the null condition (also see [27, 42] for the scalar case N = 1). For the super critical case p > pc (n), we refer to [94, 53, 106, 74, 75, 31, 113] and references cited therein.

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Definition 6.1. For µ, ν ∈ RN and X = (X0 , X1 , . . . , Xn ) ∈ Rn+1 we put V (µ, X) = (Xa µj ; 0 ≤ a ≤ n, 1 ≤ j ≤ N ), W (ν, X) = (Xa Xr νj ; 0 ≤ a ≤ n, 1 ≤ r ≤ n, 1 ≤ j ≤ N ). We say F (u, v, w) = (F F1 , . . . , FN )(u, v, w) satisfies the null condition if Fi (λ, V (µ, X), W (ν, X)) = 0

(6.10)

holds for all i ∈ {1, . . . , N }, λ, µ, ν ∈ RN and X ∈ Rn+1 satisfying X02 = c2i (X12 + · · · + Xn2 ). On the contrary, when p = 2, the null condition is necessary to ensure small data global existence if F (u, v, w) ≡ F (v, w). In fact, a blow-up result was obtained by Alinhac [6] for n = 2 and by [7] for the cases where either n = 3 or n = 2 and F (2) satisfies the null condition, together with precise estimates for the lifespan. (See also [40, 32, 27, 33].) However, when F depends on u itself, the situation is different. Indeed, small data global existence holds for the single wave equation (∂ ∂t2 − c2 ∆)u = u ∆u,

(t, x) ∈ [0, ∞) × R3 .

(6.11)

Such an interesting result was shown by Lindblad [77] for the radially symmetric case and by Alinhac [9] for the general case. Next we turn our attention to the case where the propagation speeds are distinct. To our knowledge, the effect of the discrepancy of the propagation speeds was firstly pointed out by Kovalyov [57] for the case where n = 2, 3, p = pc (n) and F (u, v, w) = F (v). Namely there is a possibility to show small data global existence for such a nonlinearity that does not satisfy the null condition. Firstly let us consider the case n = 3. For the sake of simplicity of the exposition, we shall assume that αabj i (u, v) ≡ 0 in (6.7). In other words, we consider the case where the nonlinear function F is independent of the second derivatives of the unknowns. Then F verifying (6.6) with p = 2 can be written as N 

Fi (u, ∂u) =

Aj,k i uj uk +

Bia,j,k uj ∂a uk

(6.12)

a=0 j,k=1

j,k=1

+

n  N 

N n  

Dia,b,j,k ∂a uj ∂b uk + Hi (u, ∂u),

a,b=0 j,k=1

where

Aj,k i ,

Bia,j,k ,

Dia,b,j,k

are constants and

Hi (u, v) = O(|u|3 + |v|3 ).

(6.13)

In order to regard Hi (u, ∂u) as a higher-order perturbation, we need the assumption = 0 for all i, j, k = 1, . . . , N, Aj,k i

(6.14)

because of Theorem 5.2 with q = 3 and c1 > c2 . Moreover we consider the following two cases separately; the second term of the right-hand side in (6.12) vanishes and it does not.

190

H. Kubo and M. Ohta First we consider the former case, namely Bia,j,k = 0

for all i, j, k = 1, . . . , N and a = 0, . . . , n.

(6.15)

For the case where Hi (u, ∂u) = Hi (∂u), that is, F is independent of the unknowns themselves, Yokoyama [114] proved that the small data global existence holds for (6.4)-(6.5) if (6.14), (6.15) and the following assumption on the quadratic part of F (u, v, w) are valid: For each i ∈ {1, . . . , N }, ˜ (ν, X)) = 0 Fi (λ, V˜ (µ, X), W

(6.16)

holds for all λ, µ, ν ∈ RN and X = (X0 , X1 , . . . , Xn ) ∈ Rn+1 satisfying X02 = c2i (X12 + · · · + Xn2 ). Here we put V˜ (µ, X) = (Xa δij µj ; 0 ≤ a ≤ n, 1 ≤ j ≤ N ), ˜ (ν, X) = (Xa Xr δij νj ; 0 ≤ a ≤ n, 1 ≤ r ≤ n, 1 ≤ j ≤ N ), W and δij is the Kronecker delta. We remark that Fi satisfying the condition (6.16) does not always verify (6.10). For instance, not only the nonlinearity of the system ∂t u1 )(∂ ∂t u2 ), (∂ ∂t2 − c21 ∆)u1 = a(∂ (∂ ∂t2

−

c22 ∆)u2

(t, x) ∈ [0, ∞) × R3 ,

(6.17)

(t, x) ∈ [0, ∞) × R ,

(6.18)

3

= b(∂ ∂t u1 )(∂ ∂t u2 ),

but also that of the system ∂t u2 )2 , (∂ ∂t2 − c21 ∆)u1 = a(∂ (∂ ∂t2

−

c22 ∆)u2

2

= b(∂ ∂t u1 ) ,

(t, x) ∈ [0, ∞) × R3 ,

(6.19)

(t, x) ∈ [0, ∞) × R3 ,

(6.20)

does not satisfy (6.10) unless a = b = 0, while they enjoy (6.16) for all a, b ∈ R. Thus the above result implies the existence theorem for these systems if c1 = c2 . Another proof based on the work of Klainerman and Sideris [56] was given by Sideris and Tu [98], Sogge [99] and Hidano [30]. Without such a restriction on Hi (u, v), Kubota and Yokoyama [73] firstly treated the problem, and then Katayama [43] obtained the existence result under the same assumption as in [114]. In conclusion, comparing this with the results for semilinear systems obtained in Sections 4 and 5, we see that the effect of the discrepancy of the propagation speeds strongly appears in the case where the nonlinearity depends only on the derivatives of the unknowns. Next we consider the case where (6.15) does not hold. Katayama [44] tried to relax the condition (6.15), and its extension was obtained by Katayama and Yokoyama [46]. Their result reads as follows: let (6.14) and Bia,i,i = 0 for all i = 1, . . . , N and a = 0, . . . , n, be valid. If Nij (∂uj ) =

n 

(6.21)

Dia,b,j,j ∂a uj ∂b uj , which is a part of Fi expressed in

a,b=1

(6.12), satisfies Nij (V (µ, X)) = 0

(6.22)

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191

for all i, j ∈ {1, · · · , N }, µ ∈ R and X ∈ Rn+1 satisfying X02 = c2j (X12 + · · · + Xn2 ), then the small data global existence holds. One may suppose that the condition on Nij for j = i is not necessary, in view of the result in [43]. But it is not true, because of a counter-example obtained by Ohta [85]. In fact, when c1 < c2 , we have a blow-up result for the system (∂ ∂t2 − c21 ∆)u1 = u2 ∂t u1 , (∂ ∂t2

−

c22 ∆)u2

2

= (∂ ∂t u1 ) ,

(t, x) ∈ [0, ∞) × R3 ,

(6.23)

(t, x) ∈ [0, ∞) × R3 .

(6.24)

We discuss the detail in Subsection 6.1 below. Finally, we consider the case n = 2. When F is independent of the unknowns themselves, Agemi and Yokoyama [5] refined the result in [57], and Hoshiga and Kubo [36] extended their result in such a way that the small data global existence holds if F satisfies (6.6) with p = pc (2) = 3 and its cubic part verifies (6.16). The novelty of the latter work is to provide an idea to deal with the so-called null form Q0 (φ, ψ) = ∂t φ ∂t ψ − c2 ∇φ · ∇ψ.

(6.25)

(For the detail, see Remark 6.7 below.) As for the lifespan, Hoshiga [34, 35] handled the case where F satisfies (6.6) with p = 2 and either its quadratic or cubic part does not verify (6.16). The case where F depends on the unknowns themselves is complicated especially when n = 2, because we do not have suitable L2 -bound for the unknowns themselves. Despite of the difficulty, the following partial result was obtained by Hoshiga and Kubo [37]. Let Fi admit the following decomposition: Fi (u, ∂u, ∂∇u) =

N 

Nij (u, ∂uj , ∂∇uj )+Ri (u, ∂u, ∂∇u)+H Hi (u, ∂u, ∂∇u), (6.26)

j=1

where Hi is a higher-order term satisfying Hi (u, v, w) = O(|u|4 + |v|4 + |w|4 ),

(6.27)

Nij is a homogeneous polynomial only in (u, ∂uj , ∂∇uj ) of degree 3, and Ri is a homogeneous polynomial in (u, ∂u, ∂∇u) of degree 3 being explicitly written as   ijkl Ri (u, ∂u, ∂∇u) = qαβγ (∂ α uj )(∂ β uk )(∇γ ∂ul ). (6.28) j,k,l=1,··· ,N |α|≤1,|β|=1,|γ|≤1 k=  l ijkl is a constant. If Nij satisfies Here qαβγ

Nij (λ, V (µ, X), W (ν, X)) = 0

(6.29)

for all i, j ∈ {1, · · · , N }, λ ∈ RN , µ, ν ∈ R and X ∈ Rn+1 satisfying X02 = c2j (X12 + · · · + Xn2 ), then the small data global existence holds. The point is to make use of a kind of Hardy’s inequality obtaind in [14]: For s ∈ [0, 1/2), t ≥ 0 and v ∈ C0∞ (Rn ), there is a positive constant C = C(s, n) such that     v s   (6.30)  || · | − t|s  2 n ≤ C|ξ| vˆL2 (Rn ) L (R )

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Then we have the loss of decay with respect to |x| − ci t. But we can absorb such a loss by means of the point-wise decay estimates. 6.1. Blow-up The aim of this subsection is to establish the counter-example mentioned in the above. More precisely, we consider the system

2 (∂ ∂t − c21 ∆)u1 = u2 ∂t u1 , (t, x) ∈ [0, ∞) × R3 , (6.31) 2 2 2 ∂t u1 ) , (t, x) ∈ [0, ∞) × R3 , (∂ ∂t − c2 ∆)u2 = (∂ with the following initial condition:

u1 (x, 0) = 0, ∂t u1 (x, 0) = εψ1 (|x|), u2 (x, 0) = 0, ∂t u2 (x, 0) = 0,

x ∈ R3 , x ∈ R3 ,

(6.32)

where ε > 0, ψ1 (|x|) ∈ C0∞ (R3 ). In addition, we assume that there exists a constant δ > 0 such that ψ1 (r) > 0 for r ∈ [0, δ),

ψ1 (r) = 0 for r ∈ [δ, ∞).

(6.33)

The main result in the present subsection reads as follows. Theorem 6.1. Let c1 < c2 and ε ∈ (0, 1]. Suppose that (6.33) holds. Then the classical solution (u1 , u2 ) of (6.31)–(6.32) blows up in a finite time T ∗ (ε). Moreover, there exists a positive constant C ∗ , which is independent of ε, such that T ∗ (ε) ≤ exp(C ∗ ε−2 ). Since both equations and initial data in (6.31) and (6.32) are radially symmetric, by the uniqueness of classical solutions, the classical solution (u1 , u2 ) of (6.31)–(6.32) is also radially symmetric. For this reason, we shall write ui (r, t) for ui (t, x) (i = 1, 2) in what follows. Moreover, for v = v(r, t) and c > 0, we denote ∂t2 (rv) − c2 ∂r2 (rv)}.
c v = r−1 {∂ Then (6.31)–(6.32) is rewritten in the following form:

(r, t) ∈ [0, ∞)2 ,
c1 u1 = u2 ∂t u1 , 2
c2 u2 = (∂ ∂t u1 ) , (r, t) ∈ [0, ∞)2 , with



u1 (r, 0) = 0, ∂t u1 (r, 0) = εψ1 (r), u2 (r, 0) = 0, ∂t u2 (r, 0) = 0, The following lemma is well known.

r ∈ [0, ∞), r ∈ [0, ∞).

Lemma 6.2. Let v(r, t) be the classical solution of


c v = f (r, t), (r, t) ∈ [0, ∞) × [0, T ), r ∈ [0, ∞). v(r, 0) = 0, ∂t v(r, 0) = g(r), Then, for (r, t) ∈ [0, ∞) × [0, T ), we have 


r+c(t−τ ) 1 r+ct 1 t rv(r, t) = ρg(ρ) dρ + ρf (ρ, τ ) dρ dτ. 2c |r−ct| 2c 0 |r −c(t−τ )|

(6.34)

(6.35)

(6.36)

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193

Moreover, if r ≥ ct ≥ 0, then we have r∂ ∂t v(r, t) =

1 {(r + ct)g(r + ct) + (r − ct)g(r − ct)} 2
1 t + {(r + c(t − τ ))f (r + c(t − τ ), τ ) 2 0 +(r − c(t − τ ))f (r − c(t − τ ), τ )} dτ.

Using the above lemma, we have the following result. Lemma 6.3. Assume (6.33). Let T ∗ (ε) be the lifespan of the classical solution (u1 , u2 ) of (6.34)–(6.35). Then we have u2 (r, t) ≥ 0 for all (r, t) ∈ [0, ∞) × [0, T ∗ (ε)). Moreover, we have ∂t u1 (r, t) > 0 if 0 < r − c1 t < δ, and ∂t u1 (r, t) = 0 if r − c1 t ≥ δ. Proof. By Lemma 6.2, for (r, t) ∈ [0, ∞) × [0, T ∗ (ε)), we have 
t 
r+c2 (t−τ ) 1 2 ru2 (r, t) = ρ∂ ∂t u1 (ρ, τ ) dρ dτ ≥ 0. 2c2 0 |r −c2 (t−τ )| Moreover, since u1 (r, 0) = ∂t u1 (r, 0) = 0 for r ≥ δ, by the uniqueness of classical solutions of (6.34)–(6.35), we have ∂t u1 (r, t) = 0 for r − c1 t ≥ δ. Finally, we show that ∂t u1 (r, t) > 0 if 0 < r − c1 t < δ. We put Ω = {(ρ, τ ) ∈ [0, ∞) × (0, T ∗ (ε)) : 0 < ρ − c1 τ < δ}. Moreover, for (r, t) ∈ Ω, we put D(r, t) = {(ρ, τ ) ∈ [0, ∞)2 : 0 ≤ τ ≤ t, |ρ − r| ≤ c1 (t − τ )}. By (6.33) and the continuity of ∂t u1 , there exists (r1 , t1 ) ∈ Ω such that ∂t u1 (r, t) > 0 for (r, t) ∈ D(r1 , t1 ) ∩ Ω. Suppose that there exists (r0 , t0 ) ∈ Ω such that ∂t u1 (r0 , t0 ) ≤ 0. Then, by the continuity of ∂t u1 , there exists (r2 , t2 ) ∈ Ω such that ∂t u1 (r2 , t2 ) = 0 and ∂t u1 (r, t) ≥ 0 for (r, t) ∈ D(r2 , t2 ). Since ∂t u1 (r, t) ≥ 0 and u2 (r, t) ≥ 0 for (r, t) ∈ D(r2 , t2 ), by Lemma 6.2, we have ε 0 = r2 ∂t u1 (r2 , t2 ) ≥ (r2 − c1 t2 )ψ1 (r2 − c1 t2 ). 2 On the other hand, since (r2 , t2 ) ∈ Ω, by (6.33), we have (r2 −c1 t2 )ψ1 (r2 −c1 t2 ) > 0. This is a contradiction. Hence, we obtain that ∂t u1 (r, t) > 0 if 0 < r − c1 t < δ.
Let 0 < δ1 < δ2 < δ, and we put Σ = {(r, t) ∈ [0, ∞)2 : δ1 ≤ r − c1 t ≤ δ2 },

Σ(t) = {r ∈ [0, ∞) : (r, t) ∈ Σ}.

Moreover, for the classical solution (u1 , u2 ) of (6.34)-(6.35), we define ∂t u1 (r, t) : r ∈ Σ(t)}, U1 (t) = inf{∂

U2 (t) = inf{ru2 (r, t) : r ∈ Σ(t)}.

Then we see from Lemma 6.3 that U1 (t) ≥ 0 and U2 (t) ≥ 0 for t ∈ [0, T ∗ (ε)). Moreover, we have the following lemma.

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Lemma 6.4. Under the assumptions in Theorem 6.1, there exist positive constants C1 , C2 , C3 such that
t U1 (τ )U U2 (τ ) U1 (t) ≥ C1 ε + C2 dτ, t ≥ 1, (6.37) τ 1
t  τ  U1 (τ )2 δ2 1− dτ, t ≥ U2 (t) ≥ C3 . (6.38) t τ c − c1 2 (c2 −c1 )t/c2 Proof. First we show (6.37). Let (r, t) ∈ Σ. By Lemmas 6.2 and 6.3, we have r∂ ∂t u1 (r, t) ≥

ε (r − c1 t)ψ1 (r − c1 t) 2
1 t + λ(τ )∂ ∂t u1 (λ(τ ), τ ) u2 (λ(τ ), τ ) dτ, 2 0

where we put λ(τ ) = (r − c1 t + c1 τ ). Setting C1 = inf{ρψ1 (ρ)/2 : δ1 ≤ ρ ≤ δ2 }, we get
U2 (τ ) 1 t U1 (τ )U dτ, r∂ ∂t u1 (r, t) ≥ C1 ε + 2 0 c1 τ + δ 2 since λ(τ ) ≤ c1 τ + δ2 for (r, t) ∈ Σ. (Note that C1 > 0 by (6.33).) Therefore (6.37) holds. Next we show (6.38). Let (r, t) ∈ Σ with t ≥ δ2 /(c2 − c1 ). Then we have 0 ≤ (c2 t − r)/c2 ≤ (c2 − c1 )t/c2 and c2 t + r ≥ δ2 . By Lemma 6.2, we have 

t r+c2 (t−τ ) (ρ∂ ∂t u1 (ρ, τ ))2 χΣ(τ ) (ρ) dρ dτ (6.39) ru2 (r, t) ≥ ρ (c2 t−r)/c2 |r −c2 (t−τ )|
t 2 ¯ τ ) U1 (τ ) dτ, ≥ (t, c1 τ + δ 2 (c2 −c1 )t/c2 where χΣ(τ ) denotes the characteristic function of Σ(τ ) and we put
¯ τ ) = inf{(r, t, τ ) : r ∈ Σ(t)}, (t,

r+c2 (t−τ )

(r, t, τ ) = |r −c2 (t−τ )|

χΣ(τ ) (ρ) dρ.

¯ τ ) ≥ (δ2 − δ1 )(1 − τ /t) for (c2 − c1 )t/c2 ≤ τ ≤ t. By (6.39), for Then, we have (t, any (r, t) ∈ Σ with t ≥ δ2 /(c2 − c1 ), we have
t  τ  U1 (τ )2 1− ru2 (r, t) ≥ (δ2 − δ1 ) dτ, t c1 τ + δ 2 (c2 −c1 )t/c2 which implies (6.38). This completes the proof.




Once we establish the following lemma, we can conclude that Theorem 6.1 is valid.

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195

Lemma 6.5. Let C1 , C2 , C3 > 0, α ≥ β > 1, ε ∈ (0, 1]. Assume that (f (t), g(t)) satisfies
t f (τ )g(τ ) dτ, t ≥ 1, (6.40) f (t) ≥ C1 ε, f (t) ≥ C2 τ 1
t  τ  f (τ )2 1− dτ, t ≥ α. (6.41) g(t) ≥ C3 t τ t/β Then, (f (t), g(t)) blows up in a finite time T∗ (ε). Moreover, there exists a positive constant C∗ , which is independent of ε, such that T∗ (ε) ≤ exp(C∗ ε−2 ). Proof. We define F (s) = ε−1 f (exp(ε−2 s)),

G(s) = ε−2 g(exp(ε−2 s)).

Then we have


s F (s) ≥ C1 , F (s) ≥ C2 F (σ)G(σ) dσ, s ≥ 0, (6.42) 0
s . / 1 − exp(−ε−2 (s − σ)) F (σ)2 dσ, s ≥ log α. G(s) ≥ C3 ε−2 s−ε2 log β

Suppose we have found that F (s) ≥ A > 0 for s ≥ S ≥ 0. Let h ∈ (0, 1]. Then we have for s ≥ max{S + h log β, log α}
s . / G(s) ≥ C3 ε−2 A2 1 − exp(−ε−2 (s − σ)) dσ (6.43)


s−ε2 h log β h log β −σ

(1 − e

= C3 A2

) dσ ≥

0

C3 (β − 1) log β 2 2 h A , 2β

where we used the fact that β−1 1 − exp(− log β) 1 − e−σ ≥ σ= σ, log β β log β

0 ≤ σ ≤ log β.

By (6.42) and (6.43), we have for s ≥ max{S + h(1 + log β), log α + h}
s C1 C2 C3 (β − 1) log β 3 2 F (s) ≥ C1 C2 h A . G(σ) dσ ≥ 2β s−h Moreover, since F (s) ≥ C1 for s ≥ 0, we have
s F (s) ≥ C1 C2 G(σ) dσ

(6.44)

log α

≥

C13 C2 C3 (β − 1) log β (s − log α), 2β

Now we define constants γ and A1 by 2η }, γ = max{1, C1 C2 C3 (β − 1) log β

s ≥ log α.



A1 = γ exp 1 + 6

∞  k=1

 2

−k

log k .

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Then, by (6.44), there exists a constant S1 ≥ log α such that F (s) ≥ A1 for s ≥ S1 . Furthermore we define sequences {An } and {Sn } by An+1 =

A2n , γn6

Sn+1 = Sn +

1 + log β , n2

n ∈ N.

Then, for any n ∈ N, we have F (s) ≥ An for s ≥ Sn , and Sn = S1 + (1 + log β)

k=1

 log An+1 = 2

n

n−1 

1 , k2

log A1 − (1 − 2

−n

) log γ − 6

n 

 2

−k

log k

≥ 2n .

k=1

Therefore, (F (s), G(s)) blows up at some s = S∗ satisfying S∗ ≤ lim Sn < ∞. n→∞ This completes the proof.
6.2. Null condition In this subsection we first derive an estimate (6.54) below for a function that verify the null condition after preliminary steps. Then we consider asymptotic behavior of the solution to the problem (6.4)–(6.5) in Theorem 6.2 below as an application of (6.54). It has been observed by Kubota and Yokoyama [73] that the null condition is closely related to the radiation operators which are defined by Ta = ∂a − ωa ∂r where ∂r =

x · ∇, r

ω0 = −ci ,

(0 ≤ a ≤ n), ωl =

(6.45)

xl (1 ≤ l ≤ n). r

Explicitly we have T0 = ∂t + ci ∂r ,

Tl = ∂l −

xl ∂r (0 ≤ l ≤ n). r

Besides, we denote T = (T T0 , T1 , · · · , Tn ). Then we have the following lemma based on the special structure coming from the null condition. Lemma 6.6. Let F (u, v, w) be a homogeneous polynomial of degree 2 in its arguments and u(t, x) be a RN -valued smooth function. If Fi satisfies (6.10) for all λ, µ, ν ∈ RN and X ∈ Rn+1 satisfying X02 = c2i (X12 + · · · + Xn2 ), then there is a positive constant C, independent of t, x and u(t, x), such that  |F Fi (u, ∂u, ∂∇u)(t, x)| ≤ C( |∇α ∂u(t, x)||T ∇β u(t, x)| + |∂u(t, x)||T ∂r u(t, x)|) |α|+|β|≤1

(6.46) for (t, x) ∈ [0, ∞) × Rn with |x| ≥ 1.

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197

Proof. We see from (6.10) that F is independent of u. In fact, we easily have Fi (u, 0, 0) ≡ 0 for all u ∈ RN . Namely, Fi does not include terms which are quadratic in u. Moreover, we find that it is impossible to include a term which is linear in u (for the detail, see, e.g., the Appendix in [73]). Therefore Fi (1 ≤ i ≤ N ) is expressed as Fi (u, ∂u, ∂∇u)(t, x) =

n N  

Ai,j,k a,b ∂a uj (t, x)∂b uk (t, x)

(6.47)

j,k=1 a,b=0

+

n  n N  

i,j,k Ba,b,c ∂a uj (t, x)∂b ∂c uk (t, x),

j,k=1 a,b=0 c=1 i,j,k where Ai,j,k a,b , Ba,b,c are constants. Taking λ = u, µ = ∂r u, ν = ∂r2 u and X = (ω0 , ω1 , · · · , ωn ), we have from (6.10) Fi (λ, V (µ, X), W (ν, X)) = 0.

Therefore we get from (6.47) Fi (u, ∂u, ∂∇u) = =

Fi (u, ∂u, ∂∇u) − Fi (λ, V (µ, X), W (ν, X)) n N  

Ai,j,k Ta uj ∂b uk + ωa ∂r uj Tb uk ] a,b [T

(6.48)

j,k=1 a,b=0

+

n  n N  

i,j,k Ba,b,c [T Ta uj ∂b ∂c uk + ωa ∂r uj Tb ∂c uk

j,k=1 a,b=0 c=1

+ωa ωb ∂r uj ∂r Tc uk ]. As for the last term, we note that for 1 ≤ c ≤ n 1 [∂ ∂r , Tc ] = [∂ ∂r , ∂c ] = − Tc . r Here [ , ] denotes the usual commutator of linear operators, namely [A, B] = AB − BA. Hence, we obtain (6.46) from (6.48). This completes the proof.
In order to extract additional decay by using the above lemma, we introduce the vector fields: S = t∂ ∂t + x · ∇,

Ωjk = xj ∂k − xk ∂j (1 ≤ j < k ≤ n).

(6.49)

∂t , ∂1 , · · · , ∂n ), We denote by Γ = (Γ1 , · · · , Γn0 ) these fields together with ∂ = (∂ + n + 2. It is a part of the vector fields Λ which includes not where n0 = n(n−1) 2 only Γ but also xi Li = ct∂ ∂i + ∂t (1 ≤ i ≤ n) (6.50) c with c > 0. In [54] the fields Λ play an essential role to handle nonlinearities satisfying the null condition. One of the advantages to use Λ is contained in the

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following Sobolev type inequality which is well known as Klainerman’s inequality:  n−1 1 Λα u(t)L2 (Rn ) (6.51) t + |x| 2 ct − |x| 2 |u(t, x)| ≤ C |α|≤[ n 2 ]+1

(for the proof, see [55] or [41]). This inequality will give us a decay estimate for derivatives of the solution to (6.4)–(6.5), once the right-hand side of (6.51) is under the control. This can be realized via the energy estimate if the system (6.4) has a common propagation speed, because ∂, Ω are commute with
c for all c > 0, S is so modulo lower order term, i.e., [S,
c ] = −2
c , and 2 2 (c − c2 )∂ ∂t ∂j = 0 (1 ≤ j ≤ n; 1 ≤ i ≤ N ) (6.52) c i if we choose c as c = c1 = · · · = cN in (6.50). Clearly we cannot expect (6.52) when the propagation speeds are distinct. In the consequence, Lj is not applicable to the system (6.4) with multiple speeds, hence (6.51) is not available. To overcome the difficulty, we can make use of the point-wise estimates for the derivatives of the solution. Such estimates is also derived by the evaluation of the fundamental solution as we have done in Section 2, and can be found, for example, in [73] for n = 3 and [37] for n = 2 (see also [57, 5, 114, 43]). We do not go further into this direction. We come back to the estimation of functions verifying the null condition. To this end, we introduce notations. For a RN -valued smooth function v(t, x) we set [Lj ,
ci ] =

|v(t, x)|k =

N  

|Γα vi (t, x)|,

|α|≤k i=1

where k is a non-negative integer, α = (α1 , · · · , αn0 ) is a multi-index, Γα = αn0 1 Γα 1 · · · Γn0 and |α| = α1 + · · · + αn0 . The order of the application of Γj is not essential because of the following commutator relations: [S, ∂a ] = −∂a , [S, Ωjk ] = 0, [Ωjk , ∂a ] = ηka ∂j − ηja ∂k ,

(6.53)

[Ωjk , Ωlh ] = ηkl Ωjh + ηjh Ωkl − ηkh Ωjl − ηjl Ωkh , for a, b = 0, · · · , n and j, k, l, h = 1, · · · , n, where ∂0 = ∂t and η = (ηab ) = diag(−1, 1, · · · , 1). Besides we define
|v(t, x)|2k dx. v(t)2k = R

Then we have the following estimate. Proposition 6.1. Let the assumption of Lemma 6.6 be fulfilled. Then there is a positive constant C, independent of t, x and u(t, x), such that t + |x||F Fi (u, ∂u, ∂∇u)(t, x)| ≤ C|∂u(t, x)|1 (|x| − ci t|∂u(t, x)|1 + |u(t, x)|1 ) (6.54)

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for (t, x) ∈ [0, ∞) × Rn with |x|/2 ≤ ci t ≤ 2|x| and |x| ≥ 1. Proof. In view of (6.46), we see that it suffices to prove t + r|T u(t, x)| ≤ C(|ci t − r||∂u(t, x)| + |Γu(t, x)|),

(6.55)

provided r/2 ≤ ci t ≤ 2r and r ≥ 1. We easily have from (6.45) Tj u(t, x) = −

n 1  xk Ωjk u(t, x) r2

(6.56)

k=1

for 1 ≤ j ≤ n and r > 0. This implies |T Tj u(t, x)| ≤

n |Ωu(t, x)|. r

While we have ci t − r 1 ∂r u(t, x) + Su(t, x) (6.57) t t for t > 0. Thus, when r/2 ≤ ci t ≤ 2r and r ≥ 1, we get (6.55). This completes the proof.
T0 u(t, x) =

Remark 6.7. An analogous estimate to (6.54) for functions which are cubic and which satisfy the null condition has been obtained by [37]. The point of the proof is the relation (6.57) which is firstly adopted by [36]. On the one hand, if we use all of the fields Λ, then we can avoid |x| − ci t in the right-hand side of (6.54). In fact, by choosing c = ci in (6.50), we have n  ci T0 u(t, x) = (S + ωi Li )u(t, x), ci t + r j=1

(6.58)

instead of (6.57). This observation is useful for the case where the system (6.4) has common propagation speeds. The local existence of the classical solution to (6.4)–(6.5) is known (see for instance Kato [51], Majda [80]). Consequently what we need to do for proving the global existence theorem is nothing like to derive suitable a-priori estimates for the solution. This can be done by making use of the estimate (6.54) together with (6.59) below. If one would like to evaluate the derivatives of functions verifying the null condition, then the following lemma can be useful. Lemma 6.8. For a real-valued smooth function u(t, x) and a non-negative integer k, we have t + r|T u(t, x)|k ≤ Ck (ci t − r|∂u(t, x)|k + |u(t, x)|k+1 ), for (t, x) ∈ [0, ∞) × Rn with |x|/2 ≤ ci t ≤ 2|x| and |x| ≥ 1.

(6.59)

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Proof. When k = 0, (6.59) is nothing like (6.55). To consider the case k = 1, we notice that ci T0 , [Ωjk , T0 ] = 0, [∂l , T0 ] = Tl , [∂ ∂t , T0 ] = 0, [S, T0 ] = −T r 1 xl xk xk [S, Tk ] = −T Tk , [∂l , Tk ] = − δlk ∂r + 3 ∂r − 2 Tl , [∂ ∂t , Tk ] = 0, r r r and [Ωjk , Tl ] = ηkl Tj − ηjl Tk for j, k, l = 1, · · · , n. These relations can be checked by using (6.53) and [∂l , ∂r ] =

1 Tl , r

[Ωjk , ∂r ] = 0,

[S, ∂r ] = −∂ ∂r .

(6.60)

Thus we have |ΓT Ta u(t, x)| ≤ C(|T Ta Γu(t, x)| +

n  b=0

1 ∂r u(t, x)|). |T Tb u(t, x)| + |∂ r

In virtue of (6.55), this estimate implies (6.59) for k = 1. Moreover, noting that for all integer m 1 m mxl 1 1 ] = − m , [Ωjk , m ] = 0, [∂l , m ] = − m+2 , rm r r r r we obtain (6.59) for k ≥ 2, inductively. This completes the proof. [S,




As an application of Proposition 6.1, we consider the asymptotic behavior of the classical solution to (6.4)–(6.5). To this end, we restrict ourselves to the case where n = 3, and the nonlinearity Fi (u, v, w) is independent of u and admits the following decomposition: Fi (∂u, ∂∇u) =

N 

Nij (∂uj , ∂∇uj ) + Ri (∂u, ∂∇u),

(6.61)

j=1

where Nij (∂uj , ∂∇uj )

=

n 

Ai,j,j a,b ∂a uj (t, x)∂b uj (t, x)

(6.62)

a,b=0

+

n n  

i,j,j Ba,b,c ∂a uj (t, x)∂b ∂c uj (t, x),

a,b=0 c=1

Ri (∂u, ∂∇u)(t, x) =

n  

Ai,j,k a,b ∂a uj (t, x)∂b uk (t, x)

j=  k a,b=0

+

n  n   j=  k a,b=0 c=1

i,j,k Ba,b,c ∂a uj (t, x)∂b ∂c uk (t, x)

(6.63)

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i,j,k with some constants Ai,j,k a,b , Ba,b,c . We shall show that the solution to the problem verifying (6.65), (6.66) below tends to a solution of the system of homogeneous wave equations:

(∂ ∂t2 − c2i ∆)vi = 0

for (t, x) ∈ [0, ∞) × Rn

(1 ≤ i ≤ N ),

(6.64)

provided all Nij satisfy the null condition. We find from the part 2) of Theorem 1.2 in [73] that there exists a global solution to the problem under the above assumption on Fi . Theorem 6.2. Let n = 3 and let the propagation speeds be distinct. Let Fi (u, v, w) be independent of u, and be decomposed as in (6.61). Suppose that u(t, x) is a classical solution to (6.4)–(6.5) satisfying the following properties: There are constants M > 0 and 0 ≤ ν < 1 such that |ui (t, x)|1 t + |x|ci t − |x|ν + |∂ui (t, x)|1 xci t − |x|ν+1 ≤ M, ∂u(t)1 ≤ M,

(6.65) (6.66)

for (t, x) ∈ [0, ∞) × Rn and i = 1, · · · , N . If Nij satisfies (6.29) for all i, j ∈ {1, · · · , N }, µ, ν ∈ R and X ∈ Rn+1 satisfying X02 = c2j (X12 + · · · + Xn2 ), then there is a unique classical solution v(t, x) of (6.64) verifying N 

ui (t) − vi (t)E(ci ) ≤ CM 2 t−ν

for

t ≥ 0,

(6.67)

i=1

where the constant C is independent of t and M . Proof. If we set vi = ui − Rci [F Fi (u, ∂u, ∂∇u)] (recall that Rc [F ] is defined by (2.98)), then we see from Proposition 2.7 that vi satisfies (6.64) and
∞ F Fi (u, ∂u, ∂∇u)(τ )L2 (R3 ) dτ ui (t) − vi (t)E(ci ) ≤ C t

holds. Therefore, in view of (6.61), it suffices to show N Nij (∂uj , ∂∇uj )(t)L2 (R3 ) ≤ CM 2 t−ν−1 ,

(6.68)

−ν−1

(6.69)

Ri (∂u, ∂∇u)(t)L2 (R3 ) ≤ CM t 2

.

First we show (6.68). We divide the argument into two cases. When |x|/2 ≤ cj t ≤ 2|x| and |x| ≥ 1, it follows from (6.54) and (6.65) that t + |x||N Nij (∂uj , ∂∇uj )(t, x)| ≤ CM |∂u(t, x)|1 (x−1 |x| − cj t−ν + t + |x|−1 |x| − cj t−ν ) ≤ CM |∂u(t, x)|1 t + |x|−1 . While, in the other case, |x| − cj t is equivalent to t + |x|. Therefore, by (6.65) we have |N Nij (∂uj , ∂∇uj )(t, x)| ≤ CM |∂u(t, x)|1 t + |x|−ν−1 . Since ν < 1, the above estimate is valid for all (t, x) ∈ [0, ∞) × Rn . Thus (6.66) yields (6.68).

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Next we show (6.68). Since j = k, one of the terms |x| − cj t and |x| − ck t is equivalent to t + |x|, at worst. Therefore we have |Ri (∂u, ∂∇u)(t, x)| ≤ CM |∂u(t, x)|1 t + |x|−ν−1 , which implies (6.69) by (6.66). This completes the proof.




Remark 6.9. It is an interesting open problem to determine the condition on the nonlinearity under which the global solution to the problem (6.4)–(6.5) tends to a free solution, because of an example found by Alinhac [9] which admits a global solution whose energy grows as t → ∞. In other words, the example is not a nonlinear perturbation from the system of homogeneous wave equations at all. Many other examples of this type are known for different kind of equations. For instance, we refer to J.-M. Delort, D. Fang and R. Xue [15] and Sunagawa [103, 104, 105] for systems of nonlinear Klein-Gordon equations with multiple masses.

Appendix The aim of this appendix is to give an elementary proof of the domain of dependence property (see Theorem 4 in [39] or Appendix 1 in [41]). We consider (∂ ∂t2 − c2i ∆)ui = Fi (t, x, u, ∂u, ∂∇u) for (t, x) ∈ Γ(t0 , x0 ),

(A.1)

where 1 ≤ i ≤ N and Γ(t0 , x0 ) is the backward light cone Γ(t0 , x0 ) = {(t, x) ∈ (0, T ) × Rn : |x − x0 | < t0 − t} F1 , · · · , FN )(t, x, r, q, p) with (t0 , x0 ) ∈ (0, T ) × Rn and T > 0. We assume that (F is locally Lipschitz in (r, q), namely, for any M > 0 there is a number L > 0 such that if 2  (|rj | + |qqj | + |pj |) ≤ M, |t| + |x| + j=1

then for all i = 1, · · · , n, we have |F Fi (t, x, r1 , q1 , p1 ) − Fi (t, x, r2 , q2 , p2 )| ≤ L(|r1 − r2 | + |q1 − q2 |). Example A.1. We give examples of F (t, x, r, q, p) which are locally Lipschitz in (r, q). • Let bijk , cik and fi ∈ C((0, T ) × Rn ). If we set Fi (t, x, r, q, p) =

n  N  j=0 k=0

bijk (t, x)qqjk +

N 

cik (t, x)rk + fi (t, x),

k=0

then F = (F F1 , · · · , FN ) is locally Lipschitz in (r, q). F1 , · · · , FN ) is locally Lipschitz in • If we set Fi (t, x, r, q, p) = |r|αi , then F = (F (r, q) when αi ≥ 1.

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203

The first result in this appendix is concerned with the domain of dependence property. Theorem A.1. Let (t0 , x0 ) ∈ (0, T )× Rn and Fi (t, x, r, q, p) ∈ C((0, T )× Rn × RN × R(n+1)N × Rn(n+1)N ). Suppose F = (F F1 , · · · , FN ) is locally Lipschitz in (r, q) and Fi (t, x, 0, 0, p) = 0

(t, x) ∈ Γ(t0 , x0 ),

for

p ∈ Rn(n+1)N

(1 ≤ i ≤ N ).

If u is the classical solution to (A.1) satisfying u(0, x) = ∂t u(0, x) = 0

for

|x − x0 | < t0 ,

(A.2)

then u ≡ 0 in Γ(t0 , x0 ). Proof. Since u ∈ C 2 (Γ(t0 , x0 )), by the assumption on F , we have |F Fi (t, x, u(t, x), ∂u(t, x), ∂∇u(t, x))| ≤ L(|u(t, x)| + |∂u(t, x)|)

(A.3)

for all (t, x) ∈ Γ(t0 , x0 ). Note that ui satisfies (∂ ∂t2 − c2i ∆)ui + ui = ui + Fi (t, x, u, ∂u, ∂∇u)

(A.4)

for (t, x) ∈ Γ(t0 , x0 ). Setting
1 (|ui (t, x)|2 + |∂ ∂t ui (t, x)|2 + c2i |∇u(t, x)|2 ) dx, Ei (t) = 2 |x−x0 | 0},

x0 ∈ RN ,

(1.1)

where ν(x) is the outward normal at x ∈ ∂Ω. We note that V is star-shaped with respect to x0 if and only if Γ(x0 ) = ∅. We shall assume Hyp. A. a(x) > ε0 > 0 on a neighborhood ω of Γ(x0 ) for some x0 . a(x) ≡ 0 (no dissipation) is allowed if V is star-shaped. After the introductory and preliminary sections, in Section 3 we are concerned with the decay of local energy to the linear problem: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(1.2)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0. (1.3) 0 We shall prove an algebraic decay of the local energy ER (t) = ΩR |ut (t)|2 + |∇u(t)|2 )dx, ΩR = Ω ∩ BR for the finite energy solutions u(t) of (1.2)–(1.3), where BR denotes the ball centered at the origin with the radius R. When N is odd we can further apply the method due to Morawetz [35] to conclude the exponential decay of ER (t). These results will be applied in the Section 5 to the derivation of Lp estimates of solutions. When a(x) ≡ 0 and V is not star-shaped we can not expect any uniform decay rate like ER (t) ≤ C(E(0))g(t) with limt→∞ g(t) = 0 (Ralston [58]). But, in our case, due to the dissipation a(x)ut we need not assume any geometrical condition on V . Our result is a natural extension of the classical one due to Morawetz [35] to a general domain. The ingredients of Section 3 are taken from Nakao [40]. The same result has been proved by Aloui and Khenissi [1] by a different method based on Lax-Phillips Theory. For a dog-bone type obstacle see Bloom and Kazarinoff [4]. For some special cases admitting exponential local energy decay under a certain derivative-loss see [14], [64]. Iwasaki [15] considered the local energy decay for a general hyperbolic system with a general boundary condition which yields a contraction semigroup. But, no decay is given there. In Section 4 we shall derive (total) energy decay like E(t) ≤ CII02 (1 + t)−1 , where I0 = u0 H 1 + u1 , for the same problem (1.2)–(1.3). For this, however, we must assume further Hyp. A . a(x) ≥ ε0 > 0 for |x|

1.

The result is well-known when Ω = RN and a(x) ≥ ε0 > 0 on RN and our result extends it to a more delicate situation. This result is easily generalized also to the

Decay and Global Existence

215

case of variable coefficients. Indeed, in the next section we treat the wave equation with variable coefficients and prove a unique continuation property as well as a basic inequality concerning energy E(t). General theorems on unique continuation properties based on micro-local analysis are well studied (cf. Tataru [66]). But, here we prove a simple case by use of multiplier method, which makes our argument in this paper more self-contained. The argument here follows Nakao [42], [43]. In Section 6, following Nakao [44], we apply the local energy decay proved in the first section to derive Lp estimates for the linear wave equation in exterior domains. For this, we use the so-called ‘cut-off’ method as in Shibata and Tsutsumi [63]. In Section 7 we apply the estimates established in Sections 2 and 4 to the existence problem of small amplitude global solutions for the semilinear equations: utt − ∆u + a(x)ut = f (u) in Ω × [0, ∞),

(1.4)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(1.5)

where f (u) is a nonlinear resource term like f (u) = |u| u, α > 0. We consider this problem under two types of assumption on a(x) (1) Hyp. A and Hyp. A , (2) Hyp. A. For the first case we require only the regularity on the initial data as (u0 , u1 ) ∈ H01 × L2 or (u0 , u1 ) ∈ H 2 ∩ H01 × H01 , while for the second case we need much more regularity as (u0 , u1 ) ∈ H 2M × H 2M−1 , M = [N/2] + 1. Note that the latter is applied to N ≥ 3. Our restrictions on the exponent α do not seem to be optimal compared with the non-dissipative case or the dissipative case as ut in the whole space (Pecher [54], Georgiev [9], Todorova-Yordanov [68], Nishihara [51], Narazaki [50] etc.) and it is desirable to refine our results by making appropriate additional assumptions on the initial data (cf. Ikehata [14], Ono [52]). The arguments of this section are taken from [43], [44]. In Section 8 we consider the quasilinear wave equation:   utt − div{σ |∇u|2 ∇u} + a(x)ut = 0 in Ω × [0, ∞), (1.6) α

u(x, 0) = u0 (x),

ut (x, 0) = u1 (x) and u|∂Ω = 0, (1.7) √ 2 where σ(v ) is a function like σ(v ) = 1/ 1 + v 2 . Under two types of assumptions considered in Section 3 and Section 4, respectively, we prove the existence of smooth global solutions for small initial data. The essential idea of the proof is the same as in Section 4. But, more careful analysis will be required. When a(x) ≡ 1 Matsumura [28] proved the global existence of smooth solutions for (1.6)–(1.7) with Ω = RN , Cauchy problem in the whole space, and this result was generalized by Shibata [62] to the exterior problems with N ≥ 3. Our first result establishes a global existence result under a weaker assumption on a(x) which admits a(x) to vanish in a large area. When a(x) ≡ 0 and N = 1, 2 we can not generally expect the global existence of smooth solutions of (1.6)–(1.7) even if the initial-data are small and smooth. 2

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Mitsuhiro Nakao

Indeed, when Ω = RN nonexistence was proved by Lax [23] and John [16] for the case N = 1 and by Hoshiga [11] for the case N = 2. For the case N ≥ 3, Klainerman and Ponce [21], Shatah [61] proved global existence of small amplitude solutions when Ω = RN and Shibata and Tsutsumi [63] proved similar results for exterior problems under the assumption that the obstacle V := RN \ Ω is non-trapping, especially convex. Recently, Keel, Smith and Sogge [18], [19] have developed the theory in this direction. However, if Ω is a general domain no result on global existence has been known. The reason is that when V is trapping the local energy never decays uniformly and hence it is difficult to expect global solutions for such an exterior domain. In the latter part of this section, by introducing a localized dissipative term a(x)ut as in Hyp. A we treat general exterior domains in odd dimensions and prove global existence theorem for the problem (1.6)–(1.7). Section 8 follows [45]. In the final section we consider the initial-boundary value problem (1.4)– (1.5) with the linear dissipation a(x)ut replaced by a nonlinear one ρ(x, ut ). It is an interesting problem to discuss the energy decay property for the nonlinear dissipation like ρ(x, ut ) = a(x)|ut |r ut . Indeed, such a problem has been fully studied for the case of bounded domains (see Nakao [36, 39], Tcheugou´ ´e T´ebou [67], Martinez [27] and the references cited there). Further, corresponding problem for the Klein-Gordon equation was studied by Nakao [37, 41], Mochizuki and Motai [33]. But, for the wave equation under consideration there seem to be almost no results. Mochizuki and Motai [33] treated the case ρ(x, ut ) = |ut |r ut and derived a logarithmic decay rate. Ono [52] treated the case ρ = ut + |ut |r ut and derived an algebraic decay of energy E(t). In [52] the existence of the linear term ut plays an essential role. Some related topics are discussed also in Matsuyama [29]. The difficulty for the whole or exterior domains comes from the facts that • Poincare’s ´ inequality fails and • we have very few means to control L2 (Ω) norm well. Here, in Section 9, we consider the case like ρ(x, ut ) = a(x)|ut |r ut on some bounded domain ΩR and ρ(x, ut ) = a(x)ut , linear, for large |x| with a(x) satisfying Hyp. A and Hyp. A . We also present a result on the global existence for the semilinear equations (1.4)–(1.5) with a source term f (u). Section 9 is taken from [47]. In this article we will not discuss the Kirchhoff type quasilinear wave equations in exterior domains. For this topic see Racke [57], Mochizuki [31], Yamazaki [69], Bae and Nakao [2] and the references cited in these papers. The material is organized as follows.

2. Preliminaries Let Ω be a domain in RN . We denote by C0∞ (Ω) and C0∞ (Ω) the sets of infinitely many times differentiable functions with compact supports in Ω and Ω, respectively. Lp (Ω), 1 ≤ p ≤ ∞, is the Banach space defined in a standard way. L2 (Ω)

Decay and Global Existence

217

norm is often denoted by  · . W m,p (Ω) and W0m,p (Ω) are the completions of C0∞ (Ω) and C0∞ (Ω), respectively, with respect to the norm  Dα up , |α|≤m

where for a multi-index α = (α1 , . . . , αN ) we set |α| = α1 + · · · + αN and Dα = Dxα11 · · · DxαNN , Dxi = ∂/∂xi . The completion of C0∞ (Ω) with respect to the norm * α ˙ m,p (Ω). We often use the notation W m,2 (Ω) = |α|=m D up is denoted by W ˙ m,2 (Ω) = H˙ m (Ω). The set of L2 (Ω)-valued continuous functions H m (Ω) and W on [0, ∞) is denoted by C([0, ∞); L2 (Ω)). Similar notation will be freely used throughout the article. We begin with the following well-known lemma. Lemma 2.1 (Gagliardo-Nirenberg). Let 1 ≤ r < p ≤ ∞, 1 ≤ q ≤ p and 0 ≤ k ≤ m. Then we have the inequality for v ∈ W0m,q (Ω) ∩ Lr (Ω) ||v||W k,p (Ω) ≤ C||Dm v||θq ||v||1−θ r * with some C > 0, where Dm uq = |α|=m Dα uq ,   −1 k 1 1 m 1 1 + − + − θ= N r p N r q and we assume 0 < θ ≤ 1 (0 < θ < 1 if p = ∞ and mq = N ). Let f ∈ L2loc ([0, ∞); L2 (Ω)) and let u ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) be a solution of the problem utt − ∆u + a(x)ut = f in Ω × [0, ∞),

(2.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0.

(2.2)

Let η(x) ∈ W 1,∞ (Ω) and h(x) = (h1 (x), . . . , hN (x)) ∈ W 1,∞ (Ω). Then multiplying the equation by ut , η(x)u and h(x) · ∇u and integrating by parts we obtain the following identities:

d 2 E(t) + a(x)|ut | dx = f ut dx, (A) dt Ω Ω


d η(x)ut udx − η(x)|ut |2 dx + ∇u · ∇(ηu)dx dt Ω Ω Ω

+ η(x)a(x)ut udx = f η(x)udx (B) Ω

Ω

and  



N   ∂hi ∂u ∂u 1 d ut h(x) · ∇udx + ∇·h(x) |ut |2 − |∇u|2 dx+ dx dt 2 ∂x j ∂xi ∂xj Ω Ω Ω i,j=1 1 − 2

2

∂u ν · h(x)dS + a(x)ut h(x) · ∇udx = f h(x) · ∇udx. ∂ Ω ∂ν Ω Ω




(C)

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Mitsuhiro Nakao

To derive the identity (C) we have used the following calculations:  

N
 ∂u ∂ ∂u ∂u hi dx − ∆uh · ∇udx = h · ∇udS − ∂x ∂x ∂x j j i Ω ∂ Ω ∂ν i,j=1 Ω 2
N
N
  ∂u ∂hi ∂u ∂u ∂u ∂2u dx + hi dx − h · ν dS. = ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj ∂ν ∂Ω i,j=1 Ω i,j=1 Ω Here,   N
 ∂u ∂u ∂u ∂2u ∂ hi dx = − hi dx ∂x ∂x ∂x ∂x ∂x ∂x j i j i j j i,j=1 Ω i,j=1 Ω 2
N
N
  ∂u ∂u ∂u ∂hi ∂u ∂u + νj hi dS − dx + h · ν dS ∂xj ∂xi ∂xj ∂xi ∂xj ∂ν ∂Ω i,j=1 ∂ Ω i,j=1 Ω N


and

N
 i,j=1

Ω

∂u ∂2u 1 hi dx = ∂xj ∂xi ∂xj 2


∂Ω

2 ∂u h · ν dS. ∂ν

Hence,
−

∆uh · ∇udx = Ω

2
N
 ∂u ∂hi ∂u ∂u 1 dx − h · ν dS. ∂xj ∂xi ∂xj 2 ∂Ω ∂ν i,j=1 Ω

We also use some variations of the above identities. These are the main tools throughout the paper.

3. Local energy decay 3.1. Problem and result In this section, following [40] we investigate the decay property of the local energy of the solutions to the initial-boundary value problem for the wave equation with a localized dissipation: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(3.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0. Concerning the initial data we assume that (u0 , u1 ) belongs to has a compact support, that is,

(3.1.2)

H01 (Ω) × L2 (Ω)

and

supp u0 ∪ supp u1 ⊂ BL = {x ∈ RN | |x| ≤ L} for some L > 0. As it was already stated we make the following hypothesis on a(x).

Decay and Global Existence

219

Hyp. A. a(·) is a nonnegative function on Ω belonging to L∞ (Ω), and there exist a relatively open set ω in Ω and x0 ∈ RN such that Γ(x0 ) ⊂ ω and a(x) ≥ ε0 > 0 on ω for some ε0 . ˜ > 0. Moreover, throughout this section we assume that supp a(·) ⊂ BL˜ for some L ˜ We may assume L ≤ L. Our main result reads as follows: Theorem 3.1. Under Hyp. A the solutions u(t) ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) of the problem (3.1.1)–(3.1.2) satisfy the estimate ε Eloc (t) ≤ Cε,δ E(0)(1 + t)−1+δ

(3.1.3)

with any 0 < ε, δ < 1, where we set
 2  1 ε Eloc (t) := |ut | + |∇u|2 dx. 2 Ω∩BL+εt The constant Cε,δ depends on ε, δ and L. Since a(x) has compact support we can apply the argument in Morawetz [35] to get the exponential decay for the case of odd dimensions. Corollary 3.1. Let N ≥ 3 be odd. Then, under the conditions of Theorem 3.1, we have further ε Eloc (t) ≤ Cε,δ E(0)e−λt for some λ = λ(ε, δ) > 0. Remark 3.1. When a(x) ≡ 1 Dan and Shibata [7] proved by a spectral method that Eloc (t) ≤ CE(0)(1 + t)−N , 0 where Eloc (t) := 12 ΩR (|ut |2 + |∇u|2 )dx for all R 1. Remark 3.2. When a(x) ≡ 0 and V consists of several convex bodies in some location Ikawa [12], [13] proved for the case N = 3 Eloc (t) ≤ C(||u0 ||H 2 + ||u1 ||H 1 )e−λt with some λ > 0. 3.2. Proof of Theorem 3.1. The proof is given by combining the ideas in Morawetz [35], Bloom and Kazarinoff [4] and Zuazua [70]. Let 0 ≤ T0 < T . First, we note that from (A)
T
a(x)|ut |2 dx dt = E(T T0 ). (3.2.1) E(T ) + T0

Similarly,




T

Ω







T

ta(x)|ut | dx dt = T0 E(T T0 ) + 2

T E(T ) + T0

Ω

E(t)dt. T0

(3.2.2)

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Mitsuhiro Nakao

From (B),
T


  η |∇u|2 − |ut |2 dx dt T0 Ω
T

= −(ηu, ut )|TT0 − ηa(x)ut u dx dt − T0

Ω

T

T0




(3.2.3) ∇η · ∇uu dx dt

Ω

for η ∈ W 1,∞ (Ω). Also from (C),
T


 2  N T |ut | − |∇u|2 dx dt + |∇u|2 dx dt + (ut , (x − x0 ) · ∇u)|TT0 2 T0 Ω T0 Ω 2 0T 0 0T 0 dx dS = − T0 Ω a(x)(x − x0 ) · ∇uut dx dt + 12 T0 ∂ Ω (x − x0 ) · ν(x) ∂u ∂ν (3.2.4) and ⎛ ⎞
T
   ∂hj ∂u ∂u ⎠ ⎝ 1 ∇ · h |ut |2 − |∇u|2 + dx dt 2 ∂xi ∂xi ∂xj T0 Ω i,j
T
(3.2.5) a(x)h · ∇uut dx dt − (ut , h · ∇u)|TT0 =− T0 Ω 2

∂u 1 T + h(x) · ν(x) dx dS 2 T0 ∂ Ω ∂ν for any W 1,∞ vector field h = (h1 , . . . , hN ). It follows from (3.2.2), (3.2.3) with η = 1 and (3.2.4) that 1 T E(T ) + ((x − x0 ) · ∇u(T ), ut (T )) + (N − 1)(u(T ), ut (T )) 2
T

T
2 + a(x)t|ut | dx dt + a(x)(x − x0 ) · ∇uut dx dt T0 Ω T0 Ω
1 + (N − 1) a(x)|u(T )|2 dx 2 Ω 1 = T0 E(T T0 ), ut (T T0 ) + ((x − x0 ) · ∇u(T T0 ), uT (T T0 )) + (N − 1)(u(T T0 )) 2 2

T
∂u 1 1 + (N − 1) a(x)|u(T T0 )|2 dx + (x − x0 ) · ν dx dS. 2 2 ∂ν Ω

T0

(3.2.6)

∂Ω

Here,

T a(x)(x − x0 ) · ∇uut dx dt T0 Ω
T


a(x) ε T 1 2 |(x − x0 ) · ∇u| dx dt + a(x)t|ut |2 dx dt ≤ 2ε T0 Ω t 2 T0 Ω

¯ 2 ||a(·)||∞
T
L ε T 2 ≤ |∇u| dx dt + a(x)t|ut |2 dx dt 2εT T0 2 T0 Ω T0 ΩL

(3.2.7)

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221

for any T0 > 0 and ε > 0, where we set ¯ = supx∈Ω(L) |x − x0 | = supx∈∂Ω∪∂B |x − x0 |. L L Thus, for any δ > 0, we can take T0 = T0 (δ) > 0 such that

T a(x)(x − x0 ) · ∇uut dx dt T0 Ω
T

T
≤δ |∇u|2 dx dt + δ a(x)t|ut |2 dx dt. T0

Ω(L)

T0

(3.2.8)

Ω

If a(x) ≡ 0, we can take δ = 0 in (3.2.8). To estimate the boundary integral in (3.2.6) we take h(x) such that h(x) = ν(x) on Γ, h(x) = 0 on ω ˜ c and h(x) · ν(x) ≥ 0 for all x ∈ RN , ˜ and ω ˜ ∩ Ω ⊂ ω. Then, by the where ω ˜ is an open set in RN such that Γ(x0 ) ⊂ ω definition of Γ(x0 ) and the identity (3.2.5), we have 2 2
T
∂u ∂u (x − x0 ) · ν dx dS ≤ (x − x0 ) · ν dx dS ∂ν ∂ν T0 ∂ Ω T0 Γ(x0 ) 2
T

T
∂u  2  ˜ dx dS ≤ C ≤L |ut | + |∇u|2 dx dt + CE(0), ∂ν







T

T0

Γ(x0 )

T0

ω ˜ ∩Ω

(3.2.9) where C is a constant depending on sup x∈ω (|h| + |∇Dh|). The kinetic local 0T 0 0T 0 energy T0 ω˜ ∩Ω |ut |2 dx dt in (3.2.9) will be absorbed into T0 Ω a(x)t|ut |2 dx dt in 0T 0 the left-hand side of (3.2.6), and we must estimate T0 ω˜ ∩Ω |∇u|2 dx dt. For this we take η ∈ W 1,∞ (Ω) such that √ η(x) = 1 on ω ˜ ∩ Ω, η(x) = 0 on ω c , 0 ≤ η(x) ≤ 1 on Ω and |∇η|/ η is bounded. Then, we have from (3.2.3) that


T

T0





ω ˜ ∩Ω

T




|∇u|2 dx dt ≤ C T0

 2  |ut | + |u|2 dx dt + CE(0),

(3.2.10)

ω

where we have used the Poincar´´e’s inequality ||u||L2 (ω) ≤ C||∇u||L2 (Ω(L)) for u ∈ H01 (Ω) and (3.2.1). Therefore, from (3.2.9) and (3.2.10) we have


T

T0

2
T
∂u C (x − x0 ) · ν dx dS ≤ ta(x)|ut |2 dx dt ∂ν ε0 T0 T0 Ω ∂Ω
T
+C |u|2 dx dt + CE(0).




T0

ω

(3.2.11)

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The term ((x − x0 ) · ∇u(T ), u(T )) in (3.2.6) is estimated in a standard manner as follows:
(|x0 | + |x|)|∇u(T )||ut (T )|dx |((x − x0 ) · ∇u(T ), ut (T ))| ≤ Ω(L+T )

= |x0 | |∇u(T )||ut (T )|dx + |x||∇u(T )||ut (T )|dx Ω(L+T ) Ω(εT )
+ |x||∇u(T )||ut (T )|dx BL+T \BεT

    T εT 2 2 ≤ (|x0 | + L)E(0) + |∇u| + |ut | dx + |∇u|2 + |ut |2 dx 2 Ω(εT ) 2 BL+T \BεT
  T ε = (|x0 | + L)E(0) + εT Eloc |∇u(T )|2 + |ut (T )|2 dx, (T ) + 2 BL+T \BεT (3.2.12) where we have assumed εT > L. We also note that |(u(T ), ut (T ))| ≤ ||u(T )||2 + E(0). Summarizing (3.2.6), (3.2.8), (3.2.11) and (3.2.12), we obtain  
T
C ε (1 − ε)T Eloc + 1−δ− a(x)t|ut |2 dx dt ε0 T 0 T Ω 0  



T

≤ (C + T0 )E(T T0 ) + C

T

||u(T )||2 +

|u(t)|2 dx dt + ε T0

|∇u|2 dx dt T0

ω

Ω(L)

(3.2.13) and, taking T0 large enough so that (1 − δ)ε0 T0 > C and εT T0 > L, we have  

T
T ε 2 2 ε (1−ε)T Eloc ≤ (C +T T0 )E(T T0 )+C ||u(T )|| + |u(t)| dx dt +δ Eloc (t)dt. T0

ω

T0

(3.2.14) Finally, we must show the boundedness of the second term in the right-hand side of (3.2.14). For this we use a device due to Morawetz [35]. Let h(x) be the unique solution of the elliptic exterior problem ∆h(x) = u1 (x) + a(x)u0 (x) in Ω 2−N

with h|∂Ω = 0 and h(x) = O(|x| Then, setting

(3.2.15)

) as x → ∞ (h(x) = O(log|x|) if N = 2).




w(t) =

t

u(s) ds + h(x) 0

we have wtt − ∆w + a(x)wt = 0 in Ω × (0, ∞), w(x, 0) = h(x), wt (x, 0) = u0 (x) and w|∂Ω = 0.

(3.2.16)

Concerning the solution h(x) of (3.2.15) it is known (cf. Meyers and Serrin [30]) 2 (Ω) and that h ∈ Hloc sup

N −2 |h(x)| |x|≥2L {|x|

+ |x|N −1 |∇h(x)|} ≤ C(L)(||u1 || + ||u0 ||),

(3.2.17)

Decay and Global Existence

223

where we note supp u0 ∩ supp u1 ⊂ BL (|x|N −2 |h(x)| in the above should be replaced by (log(L + |x|))−1 |h(x)| if N = 2). Thus, multiplying (3.2.16) by wt we can prove

T
  1 2 2 sup a(x)|wt |2 dx dt |wt (t)| + |∇w(t)| dx + 0≤t≤T 2 Ω(L+T ) 0 Ω ≤ CE(0) < ∞ (CE(0)log(2 + T ) if N = 2),

(3.2.18)

which implies the boundedness or logarithmic growth of the terms in the second bracket in the right-hand side of (3.2.14). Thus, changing the notation for δ, i.e., replacing δ/(1 − ε) by δ, we arrive at the inequality
T ε ε (T ) ≤ Cε,δ E(0) + δ Eloc (t)dt if N ≥ 3 (3.2.19) T Eloc T0

or




T

ε T Eloc (t) ≤ Cε,δ E(0)log(2 + T ) + δ

ε Eloc (t)dt if N = 2

(3.2.19)

T0

for any 0 < ε, δ < 1. We can easily solve the inequality (3.2.19) or (3.2.19)’ as in Bloom and Kazarinoff [4]. Indeed, setting
T −δ ε Eloc (t)dt φ(t) = T T0

we see that (3.2.19) and (3.2.19)’ are equivalent to φ (T ) ≤ Cε,δ E(0)T −δ−1 and φ (t) ≤ Cε,δ E(0)T −δ−1 log(2 + T ), respectively. Thus, we have φ(T ) ≤ Cε,δ E(0), T0 ≤ T < ∞. Returning to (3.2.19) or (3.2.19)’ we obtain ε (T ) ≤ Cε,δ E(0)(1 + δT δ ) T Eloc

for any 0 < ε, δ < 1, which implies the desired result in Theorem 3.1.




3.3. Proof of Corollary 3.1. The proof of Corollary 3.1 is given along the same line of the argument by Morawetz [35]. For convenience of the readers we sketch it briefly. We use the following notation for R > L:
  1 |fft (t)|2 + |∇f (t)|2 dx. E(f, R, t) = 2 ΩR The same notation will be employed when f (t) is defined on RN and ΩR is replaced by BR . We split the solution u(t) as u(t) = F0 (t) + W0 (t), where F0 is the free space solution of the wave equation F0 = 0 in [0, ∞) × RN F0tt − ∆F

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Mitsuhiro Nakao

with F0 (0) = u0 and F0t (0) = u1 . Here, we extend F0 (0) and F0t (0) to inside of the body V = RN \ Ω appropriately so that
E(0), µ > 0. E(F F0 , ∞, 0) ≤ (1 + µ) Ω

By Huygens’ principle we see F0 (x, t) = 0 if |x| ≤ t − L or |x| ≥ t + L. We also note that supp W0 (t) = supp(u(t) − F0 (t)) ⊂ B3L for t = 2L. We easily show that W0 + a(x)W W0t = −a(x)F F0t = 0 and W0 |∂Ω = 0 if t ≥ 2L W0tt − ∆W and E(W W0 , ∞, 2L) ≤ (4 + 2µ)E(0). Next, we take T ≥ 2L and split W0 (t) as W0 (t) = F1 (t) + W1 (t) for t ≥ T, where F1 is the free solution of the wave equation for t ≥ T with F1 (0) = W0 (T ) and F1t (T ) = W0t (T ). We see F1 (x, t) = 0 if |x| ≤ t − T − L or |x| ≥ t + T + L and W1 + a(x)W W1t = −a(x)F F1t = 0 if t ≥ T + 2L. W1tt − ∆W Noting that the value of F1 (t) on BL , T ≤ t ≤ T + 2L, depends only on the data of F1 (T ) on B3L , we take F˜1 such that F˜1tt − ∆F˜1 = 0 on [T, ∞) × RN with F˜1 (T ) = F1 (T ) and F˜t (T ) = F1t (T ) if |x| ≤ 3L and F˜1 (T ) = F˜1t (T ) = 0 if |x| ≥ 3L + ε0 , F1 , B3L , T ). Now, we 0 < ε0 < 1. We may assume E(F˜1 , B3L+ε0 , T ) ≤ (1 + µ)E(F set ˜ 0 = W1 + F˜1 = W0 − F1 + F˜1 . W Then, ˜ 0tt − ∆W ˜ 0 + a(x)W ˜ 0t = a(x)(F W F1t − F˜1t ) = 0 if T ≤ t ≤ T + 2L ˜ 0 (T ) = W0 (T ), ˜ 0 |∂Ω = W1 |∂Ω + F˜1 |∂Ω = u|∂Ω = 0. We further see that W and W ˜ 0t (T ) = W0t (T ) on B3L ∩ Ω and W ˜ 0t (T ) = W0t (T ) = 0 if |x| ≥ 3L + ε0 . Setting W

Decay and Global Existence

225

˜ 0 − F˜1 , we have W1 := W0 − F1 = W ˜ 0 , ∞, T + 2L) + 2E(F˜1 , ∞, T + 2L) E(W W1 , ∞, T + 2L) ≤ 2E(W ˜ 0 , ∞, T ) + 2E(F˜1 , ∞, T ) ≤ 2E(W ˜ ≤ 2E(W0 , B3L+ε0 , T ) + 2E(F˜1 , B3L+ε0 , T ) F1 , B3L , T )) ≤ 2(1 + µ)(E(W W0 , B3L , T ) + E(F ≤ kE(W W0 , B3L , T ), k := 2(1 + µ)(2 + µ). Repeating this argument we obtain the splitting: W0 =

n 

Fj + Wn ,

u = F0 + W0 ,

n = 1, 2, 3, . . . ,

(3.3.1)

j=1

where Fj is a free solution of the wave equation for t ≥ jT with suppF Fj (t) ⊂ Bt−jT −L ∩ Bt+jT +L and Wn satisfies Wntt − ∆W Wn + a(x)W Wnt = 0 for t > nT + 2L with the boundary condition Wn |∂Ω = 0. Further, for Wn , we have Wn−1 , B3L , nT ). E(W Wn , ∞, nT + 2L) ≤ kE(W

(3.3.2)

Applying the decay property of the local energy of Theorem 3.1 to Wn−1 (t) we know Wn−1 , ∞, (n − 1)T + 2L), E(W Wn−1 , B3L , nT ) ≤ p(T − 2L)E(W

(3.3.3)

where p(t) = (1 + t)−1+δ . From (3.3.2) and (3.3.3) we obtain W0 , ∞, 2L) ≤ k(kp(T −2L))n E(0). (3.3.4) E(W Wn , ∞, nT +2L) ≤ (kp(T −2L))n E(W Hence, we have E(W Wn , ∞, t) ≤ k(kp(T − 2L))n E(0) if nT + 2L ≤ t < (n + 1)T + 2L. For 0 < ε < 1 and for t > 2L/(1 − ε) we take n such that ((n + 1)T + 2L)/(1 − ε) > t ≥ (nT + 2L)/(1 − ε). Then, Fj = 0 on ΩL+εt , j = 1, 2, . . . , n, and ε (t) = E(u, ΩL+εt , t) = E(W Wn , ΩL+εt , t) Eloc

≤ E(W Wn , ∞, t) ≤ k(kp(T − 2L))n/(1−ε) kE(0). Thus, taking T sufficiently large so that kp(T − 2L) < 1, we conclude the expoε nential decay of Eloc (t).


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4. Total Energy decay for the wave equation with a localized dissipation 4.1. Problem and result Following [43], we derive L2 boundedness and (total) energy decay of solutions for the linear problem: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(4.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0. (4.1.2) In addition to Hyp. A we make the assumption Hyp. A . Summarizing these we state: ˜ Hyp. A (1) There exists x0 ∈ RN and a relatively open set ω ⊂ Ω such that Γ(x0 ) ⊂ ω and a(x) ≥ ε0 > 0 for x ∈ ω with some ε0 . (2) There exists L > 0 such that a(x) ≥ ε0 > 0 for |x| ≥ L.

Needless to say we may assume V := Ωc ⊂ BL = {x ∈ RN | |x| ≤ L}. Our result on the total energy decay reads as follows: ˜ Then, for each Theorem 4.1. Let a(·) belong to L∞ (Ω) and satisfy Hyp. A. 1 2 (u0 , u1 ) ∈ H0 (Ω) × L (Ω) there is a unique solution u(t) ∈ C 1 ([0, ∞); L2 (Ω)) ∩ C([0, ∞); H01 (Ω)) such that u(t)2 ≤ C0 I0 (4.1.3) and  1 ut (t)2 + ∇u(t)2 ≤ C0 I02 (1 + t)−1 , E(t) := (4.1.4) 2 where C0 is a positive constant independent of u and we set I0 := u0 H01 + u1 . For more regular solutions we can derive sharper decay estimates. The following is a typical one. ˜ and let (u0 , u1 ) ∈ Theorem 4.2. Let a(·) belong to C 1 (Ω) and satisfy Hyp. A H 2 (Ω) ∩ H01 (Ω) × H01 (Ω). Then, the solution u(t) in Theorem 4.1 satisfies further, utt (t)2 + ∇ut (t)2 ≤ C1 I12 (1 + t)−2

(4.1.5)

and ∆u(t)2 ≤ C1 I12 (1 + t)−1 , where C1 is a constant independent of u and I1 = u0 H 2 + u1 H 1 .

(4.1.6)

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227

4.2. Proof of Theorem 4.1 The existence and uniqueness part is well-known, and for the proof of Theorem 4.1 it suffices to derive the estimates (4.1.3)–(4.1.4). We note that for each (u0 , u1 ) ∈ H01 × L2 , the solution u(t) is given as the limit of smooth solutions um (t) with um (0) = um,0 ∈ C0∞ (Ω) and umt (0) = um,1 ∈ C0∞ (Ω) such that um,0 → u0 ∈ H01 and um,1 → u1 ∈ L2 . Therefore, in deriving estimates we may assume that u(t) is smooth (H 2 -valued is sufficient) and supp u(t) is compact in Ω for each t, which is due to the finite propagation property of the wave equation. For a simple proof of this property by multiplier technique see John [16]. From (A),
d E(t) + a(x)|ut (t)|2 dx = 0. (4.2.1) dt Ω From (B) with η(x) = 1,

  d 1 d a(x)|u|2 dx = 0. (ut , u) + |∇u|2 − |ut |2 dx + dt 2 dt Ω Ω

(4.2.2)

We define a Lipschitz continuous function φ(r) on R+ as follows: ⎧ ⎨ε 0 if 0 ≤ r ≤ L, φ(r) = ε0 L ⎩ if r ≥ L, r ˜ Setting h(x) = φ(r)(x−x0 ) where ε0 , L are positive constants appearing in Hyp. A. with r = |x − x0 | in (C) we see

  d 1 ut φ(x − x0 ) · ∇u dx + (N φ + φ r) |ut |2 − |∇u|2 dx dt Ω 2 Ω 
 
2 φ 1 2 2 ∂u ν · (x − x0 )φ dS + |(x − x ) · ∇u| + φ|∇u| − dx (4.2.3) 0 2
∂ Ω ∂ν r Ω a(x)ut φ(x − x0 ) · ∇u dx = 0, + Ω

and hence, combining this with (4.2.2) and (4.2.3), we have for α ≥ 0 and k ≥ 0,  

d α 2 ut φ(x − x0 ) · ∇u dx + α(ut , u) + a(x)|u| dx + kE(t) dt 2 Ω   
Ω
N φ + φ r N φ + φ r + − α + ka(x) |ut |2 dx + + φ + φ r |∇u|2 dx α− 2 2 Ω Ω 2

1 ∂u φν · (x − x0 )dS − ≤ aut φ(x − x0 ) · ∇u dx. 2 Γ(x0 ) ∂ν Ω (4.2.4) Noting that a(x) ≥ ε0 > 0 if |x| ≥ L, we easily see that there exists α > 0 such that N φ + φ r ka(x) + − α > ε1 > 0 (4.2.5) 2 2

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Mitsuhiro Nakao

and

N φ + φ r + φ + φ r > ε1 > 0 (4.2.6) 2 for some ε1 , 0 < ε1 < ε0 /2, and for any k ≥ N + 1. Indeed, we can choose, for example, ε0 (2N − 1)ε0 and ε1 = . α= 4 4 Further, we note that


C 2 a(x)ut φx · ∇u dx ≤ k a(x)|u | dx + |∇u|2 dx t 4 k α−

Ω

Ω

Ω

with some C > 0. Therefore, we obtain from (4.2.4) that  


d α ut φ(x − x0 ) · ∇u dx + α(ut , u) + a(x)|u|2 dx + kE(t) dt 2 Ω Ω 2

∂u k 1 φν · (x − x0 ) dS + a(x)|ut |2 dx + ε1 E(t) ≤ (4.2.7) 4 Ω 2 Γ(x0 ) ∂ν for any large k. In order to control the right-hand side of (4.2.7) we take a vector field h(x) as in (3.2.9). Then, from (C) we have 2


∂u   1 dS ≤ d u h · ∇u dx + ∇ · h |ut |2 − |∇u|2 dx (4.2.8) t ∂ν dt
Ω 2 Ω
Γ(x0 ) |a(x)ut h · ∇u|dx + |∇u|2 |Dh|dx
ω
Ω  2  d ≤ ut h · ∇u dx + C |ut | + |∇u|2 dx dt Ω ω ∩Ω +

with some C > 0. Thus, we have from (4.2.7) and (4.2.8) that  


d α 2 ut (φ(x − x0 ) − C0 h) · ∇u dx + α(ut , u) + a(x)|u| dx + kE(t) dt 2 Ω Ω

 2  k ε1 + a(x)|ut |2 dx + E(t) ≤ C |ut | + |∇u|2 dx (4.2.9) 4 Ω 2 ω ∩Ω for k ≥ N + 1 and some C0 , C > 0. The first term of the right-hand side of (4.2.9) is easily absorbed into the left-hand side if we take k large. To control the second term we choose a nonnegative function η ∈ W 1,∞ (Ω) as for getting (3.2.10). Then from (C), 


2 |∇u| dx ≤ η|∇u|2 dx ω ∩Ω Ω


d 1 d 2 ∇η · ∇uu dx − (ut , ηu) + η|ut | dx − a(x)η|u|2 dx. =− dt 2 dt Ω Ω Ω (4.2.10) Since


1 |∇η · ∇uu|dx ≤ η|∇u|2 dx + C |u|2 dx 2 Ω Ω ω

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229

we have from (4.2.10),  


 2  d |ut | + |u|2 dx − |∇u|2 dx ≤ C a(x)η|u|2 dx . 2(ut , ηu) + dt ω ∩Ω ω Ω (4.2.11) We obtain from (4.2.9) and (4.2.11) that  


d 1 2 ut (φ(x − x0 ) + h) · ∇u dx + α(ut , u) + a(x)(α + η)|u| dx + kE(t) dt 2 Ω Ω

k ε1 + a(x)|ut |2 dx + E(t) ≤ C |u|2 dx, (4.2.12) 8 Ω 2 ω where we have chosen a large k > 0 (for simplicity, we use notations h and η for Ch and Cη, respectively). Finally, we prepare the following proposition which shows the possibility of the right-hand side of (4.2.12) being controlled by the last two terms of the left-hand side of (4.2.12). Proposition 4.1. There exists T0 > 0 independent of u such that if T > T0 , the inequality
t+T


t+T
ε1 t+T 2 2 |u(x, s)| dx ds ≤ C a(x)|ut (x, s)| dx ds + E(s) ds, 4 t t t ω Ω (4.2.13) t > 0, holds, where C is a constant which may depend on T . Proof. When Ω is bounded the inequality (4.2.13) holds without the second term in the right-hand side as it is shown in Zuazua [70]. For the proof of (4.2.13) we modify the argument in [70]. If (4.2.13) would be not true, there would exist a sequence of numbers {tn } and a sequence of solutions {un } such that
tn +T

tn +T

ε1 tn +T |un |2 dx ds ≥ n a(x)|unt |2 dx ds + En (s) ds, 4 tn tn tn ω Ω (4.2.14) where En (t) is defined by E(t) with u replaced by un . Setting
tn +T
2 λn = |un |2 dx ds and vn (t) = u(· + tn )/λn tn

we have




ω





ε1 T n a(x)|vnt (t)| dx dt + En (t)dt ≤ 1, (4.2.15) 4 0 0 Ω where En (t) is defined by En (t) with un replaced by vn . Then, we note that
T
|vn (t)|2 dx dt = 1, (4.2.16) T

2


0

T


Ω

0

ω

  |vnt (t)|2 + |∇vn (t)|2 dx dt ≤ 8ε−1 1 < ∞

(4.2.17)

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Mitsuhiro Nakao


T


a(x)|vnt (t)|2 dx dt → 0 as n → ∞.

0

(4.2.18)

Ω

Therefore, along a subsequence, vn (t) is weakly convergent to a function v ∈ L2 ([0, T ]; H01,loc (Ω)) ∩ W 1,2 ([0, T ]; L2loc(Ω)) with ∇v, vt ∈ L2 ([0, T ]; L2(Ω)). Here we note that H01,loc (Ω) denotes the set of functions u such that φ(·)u(·) ∈ H01 (Ω) for any φ(·) ∈ C0∞ (Ω). We may assume, by Rellich’s Lemma, that vn (t) converges to v strongly in L2 ([0, T ] × ω). It is clear that the limit function v(t) solves the equation vtt − ∆v = 0 in [0, T ] × Ω and, further, it satisfies the conditions
T
|v|2 dx dt = 1 0

(4.2.19)

(4.2.20)

ω

and (4.2.21) vt (x, t) = 0 on [0, T ] × supp a(·). c Since supp a(·) includes ω and ΩL = {x ∈ Ω| |x| > L}, by a general result of unique continuation, we see that there exists T0 > 0 such that if T > T0 , vt (x, t) = 0 on Ω × [0, T ].

(4.2.22)

(For a convenient unique continuation property see Remark 4.1 below. A more general result will be proved in Section 5.) The fact (4.2.22) means that v(x, t) = v(x), independent of t, and by (4.2.19), −∆v(x) = 0 in Ω. Since v ∈ H01,loc (Ω) and ∇v ∈ L2 (Ω) we conclude from the above that v(x) ≡ 0 in Ω. This is a contradiction to (4.2.20).
Remark 4.1. Let us assume that Γ(x0 ) ⊂ ω and let v(t) ∈ C([0, T ]; H01 (Ω)) ∩ C 1 ([0, T ]; L2 (Ω)) be a solution of the problem (4.1.1)–(4.1.2). Further, we assume v(x, t) = 0 on ω ∪ ΩcL . Then, we know by the inequality (4.2.12),  

d vt φ(x − x0 ) · ∇vdx + α(vt , v) + kE(t) + ε1 E(t) ≤ 0, dt Ω which implies easily for large k > 0 that E(T ) + T E(T ) ≤ CE(0) = CE(T ) for some C > 0 independent of T , where we have used the inequality |(vt , v)| ≤ Cvt (t)∇v(t) which follows from the facts v(t) ∈ H01 (Ω) and supp v(t) ⊂ BL . Taking T > C we see E(0) = 0, that is, v(x, t) = 0 on [0, T ] × Ω. When v is a weaker solution

Decay and Global Existence

231

we should apply the above to vε (t) = (ρε  v)(t), where ρε (t) is the mollifier with respect to the t variable. Now, we return to (4.2.12). By Proposition 4.1 we have for large k > 0 and for T > T0 ,


ε1 t+T k t+T 2 a(x)|ut | dx ds + E(s) ds ≤ 0, (4.2.23) X(t + T ) − X(t) + 8 t 4 t Ω where we set

α ut (φ(x − x0 ) + h) · ∇u dx + α(ut , u) + a(x)(1 + η)|u(t)|2 dx + kE(t). X(t) = 2 Ω Ω (4.2.24) Noting that φ(r)(x − x0 ) is bounded, a(x) ≥ ε0 for |x| ≥ L and
|u|2 dx ≤ C∇u2 (Poincar´ ´e’s inequality), Ω∩BL

˜ := E(t) + u(t)2 if k is large. When we easily see that X(t) is equivalent to E(t) N Ω = R , that is, we consider the Cauchy problem, we should use the inequality  

2 2 2 |u| dx ≤ C ∇u + |u| dx |x|≥L

BL

instead of the above Poincare’s ´ inequality. Thus, we conclude from (4.2.23) that
∞

k ε1 ∞ a(x)|ut |2 dx ds + E(s) ds ≤ X(0). (4.2.25) 8 0 4 0 Ω Returning to the inequality (4.2.23) and using the estimate (4.2.25) we obtain
∞

∞   sup0≤t 0. For later use we first consider the equation with a forcing term f ∈ L2 ([0, T ]; L2(Ω)),   N  ∂ ∂ aij (x, t) utt − u = f (x, t) ∂xi ∂xj i,j=1

in

Ω × [0, T ].

(5.1.3)

We derive the following differential inequality: Proposition 5.1. Let u ∈ C 1 ([0, T ]; L2 (Ω))∩C([0, T ]; H01 (Ω)) be a solution of (5.1.3) and let ω be an open set in Ω such that ω ⊃ Γ(x0 ) for some x0 ∈ RN , where Γ(x0 ) is the part of the boundary ∂Ω defined in the introduction. Then there exists δ0 > 0 such that if |∇aij | ≤ δ0 , it holds
T   ∇u(t)2 + ut (t)2 dt 0  
T (5.1.4)
T
 2  2 2 |ut | + |u| dx ds + f (t) dt ≤ C E(0) + E(T ) + 0

ω

0

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with some C > 0 independent of u, where we set ⎞ ⎛
 N 1 ∂u ∂u E(t) := ⎝ut (t)2 + aij (x, t) dx⎠ . 2 ∂xi ∂xj Ω i,j=1 Then, by use of the above inequality we can prove the following theorem on the unique continuation: Theorem 5.1. Let ω be an open set in Ω such that ω ⊃ Γ(x0 ). Then, there exist δ1 > 0 and T0 > 0 such that if T > T0 and &

∂ (5.1.5) sup |∇aij (x, t)| + aij (x, t) ≤ δ1 , ∂t i,j;(x,t)∈QT a solution u ∈ C 1 ([0, T ]; L2(Ω))∩C([0, T ]; H01 (Ω)) of (5.1.1) with u ≡ 0 on ω×[0, T ] vanishes identically on QT := Ω × [0, T ]. Although Theorem 5.1 seems to be interesting, for our later use we need the following theorem: Theorem 5.2. Let u ∈ X2 (T ) := C 2 ([0, T ]; L2 (Ω)) ∩ C 1 ([0, T ]; H01 (Ω)) ∩ C([0, T ]; H 2 ∩ H01 (Ω)) be a solution of the problem (5.1.1) with ut (x, t) ≡ 0 on ω × [0, T ], where ω is an open set in Ω satisfying ω ⊃ Γ(x0 ). Then, there exist constants δ2 > 0 and T1 > 0 such that if T > T1 and &

∂ ∂ sup (5.1.6) |∇aij (x, t)| + ∇ aij (x, t) + aij (x, t) ≤ δ2 , ∂t ∂t i,j,(x,t)∈QT then u(x, t) ≡ 0 on Ω × [0, T ]. Next, we consider the wave equation with variable coefficients in an exterior domain Ω with a dissipation:   N  ∂ ∂u utt − (5.1.7) aij (x, t) + a(x)ut = f (x, t) in Ω × (0, ∞), ∂xi ∂xj i,j=1 u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0.

(5.1.8)

We assume N 

aij (x, t)ξi ξj ≥ k0 |ξ|2 ,

ξ ∈ RN ,

i,j=1

with some k0 > 0. We also assume that aij belong to C 1 (Ω × [0, T ]), that Ω is an 1,2 exterior domain with boundary from C 1 , and f ∈ Wloc ([0, ∞); L2 (Ω)). Proposition 5.2. Under the assumption (5.1.6) on aij from Theorem 5.2 with QT replaced by Q∞ there exists a constant T0 > 0 such that if T > T0 , the solutions

Decay and Global Existence

235

u(t) ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) of the problem (5.1.7)–(5.1.8) satisfy
t+T

t+T X(t + T ) − X(t) + ε0 E(s) ds + k a(x)|ut |2 dx ds t t Ω (5.1.9)
t+T
≤C (|∇u| + |u| + k|ut |) |f | dx ds, t ≥ 0, Ω

t

with some ε0 > 0 and an arbitrarily large k > 0, where we set
1 a(x)|u|2 dx + kE(t). X(t) = (ut , (φ(x − x0 ) + c0 h) · ∇u)+((α + 2c1 η)ut , u)+ 2 Ω (5.1.10) We see easily that X(t) is equivalent to E(t) + u(t)2 for a large k > 0 since 


|u|2 dx ≤ C ΩL

Ω2L \ΩL

|u|2 dx +

|∇u|2 dx Ω2L

with some C > 0. Note that the above Proposition 5.2 is a generalization of (4.2.23). It is not difficult to derive some energy decay properties from (5.1.9) if f = 0. Indeed, this will be used to derive various types of energy decays for quasilinear wave equations in Section 8. 5.2. Proof of Proposition 5.1 We use a similar multiplier method employed in the previous sections. We first consider the solutions u ∈ X2 . The identities (A) to (C) from Section 2 become


 d 1 2 ˜ E(t) + a(x)|ut | dx = − aij,t uxi uxj dx + f ut dx, (A) dt 2 Ω i,j Ω Ω N
 i,j=1

0

T


Ω

T
ηaij uxi uxj dx dt = (ut , ηu) − 0




T







N  Ω i,j=1

aij uxj ηxi u dx dt

˜ (B)

  η|ut |2 + f ηu dx dt,

+ 0

0

T

Ω

 
T
 ∂ ∂ u h · ∇u dx dt aij − ∂xj 0 Ω i,j ∂xi

1 T =− ((h · ∇)aij + (∇ · h)aij ) uxi uxj dx dt 2 i,j 0 Ω 2


T
∂u 1 T ∂ − aij νi νj (ν · h) dS dt + aij uxj uxk hk dx dt. 2 i,j 0 ∂ Ω ∂n ∂xi Ω i,j 0 ˜ (C)

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Mitsuhiro Nakao

˜ that Now, taking h = x − x0 we have from (C)
T

T
  2   N |ut | − aij uxi uxj dx dt + aij uxi uxj dx dt 2 0 Ω 0 Ω i,j i,j 2

T ∂u 1 T = −(ut , h · ∇u) + aij νi νj (ν · h) dS dt 2 ∂n 0 ∂Ω i,j 0


T 1 T − (h · ∇aij )uxi uxj dx dt + (f, h · ∇u) dx dt. 2 i,j 0 Ω 0

(5.2.1)

˜ with η(x) = 1 and using the assumption on aij we have Combining (5.2.1) and (B)
T
 2   |ut | + aij uxi uxj dx dt 0

Ω

i,j



≤ C ut (T )(∇u(T ) + u(T )) + ut (0)(∇u(0) + u(0))





T

+ sup |∇aij | x,t,i,j

+

0




⎞  1/2 1/2 
2
T T ∂u 2 2 aij νi νj dS dt⎠ + f (t) dt ∇u , ∂ν 0 0 Γ(x0 )

and hence
T
0

Ω

 2   aij uxi uxj dx dt |ut | + i,j



≤C

aij uxi uxj dx dt




T

0

i,j

Ω i,j




+


i,j

0

T




T

f (t)2 dt + sup |∇aij | x,t,i,j 0 ⎞ 2 ∂u aij νi νj dS dt⎠ . ∂ν

E(T ) + E(0) +
Γ(x0 )

T

E(t)dt

(5.2.2)

0

To estimate the boundary integral in (5.2.2) we make the usual device. That is, by choosing h(x) as in (3.1.9) and further choosing η(x) as in (3.1.10) we can derive the estimates 2 
T
∂u aij νi νj dS dt ∂ν Γ(x0 ) i,j 0 
T

≤C

E(0) + E(T ) + sup |∇aij | x,t,i,j




T





|ut |2 dx dt +

+ 0

ω

E(t) dt

T

f (t)2 dt + 0

(5.2.3)

0


i,j

0

T


ω

⎞ aij uxi uxj dx dt⎠

Decay and Global Existence

237

and 
i,j

≤C

T 0




ω

aij uxi uxj dx dt


T




E(0) + E(T ) + 0

 2  |u| + |ut |2 dx dt +






T

(5.2.4)

f (t) dt . 2

0

ω

The estimate (5.1.4) follows from (5.2.2), (5.2.3) and (5.2.4). Since every term appearing in (5.2.4) is meaningful for finite energy solutions this is in fact valid for such a solution.
5.3. Proof of Theorems 5.1 and 5.2 Let u(t) be a solution of (5.2.1) with u(x, t) ≡ 0 on ω × [0, T ]. We take δ1 ≤ δ0 . Then, from Proposition 5.1 we have
T E(t)dt ≤ C1 (E(0) + E(T )). (5.3.1) 0

Let E(t∗ ) = min0≤t≤T E(t), 0 ≤ t∗ ≤ T . Then, by the identity

 1 t E(t) = E(t∗ ) + aij,s (x, s)uxi uxj dx ds, 2 t∗ Ω i,j

˜ (A)

we see


∗

E(t) ≤ E(t ) + C2 sup aij,t L∞ (QT ) i,j




∗

0

T


 Ω i,j

aij uxi uxj dx dt

(5.3.2)

T

≤ E(t ) + C2 δ1

E(s) ds 0

for all t, 0 ≤ t ≤ T . Hence, if 2C1 C2 δ1 < 1/2 we have from (5.3.1) and (5.3.2)
T (T E(t∗ ) ≤) E(t)dt ≤ C3 E(t∗ ) (5.3.3) 0

for a certain constant C3 > 0. Taking T > T0 := C3 we obtain E(t∗ ) = 0, i.e., ut (t∗ ) = u(t∗ ) ≡ 0. Thus, we conclude from the uniqueness of solutions to the initial-boundary value problem that u(x, t) ≡ 0 on Ω × [0, T ].
Remark 5.1. Modifying the proof of Theorem 5.1 we easily see the inequality  


T T  2  2 2 |ut | + |u| dx dt + f (t) dt E(0) ≤ C 0

ω

0

for solutions u(t) of (5.1.3), which may be more convenient for applications.

238

Mitsuhiro Nakao

Let us proceed to the proof of Theorem 5.2. Differentiating the equation with respect to t we have   ∂  ∂ut   ∂  ∂u aij = aij,t . (5.3.4) uttt − ∂xi ∂xj ∂xi ∂xj i,j i,j Applying Proposition 5.1 with u replaced by ut to (5.3.4) we have ⎛ ⎞   2
T
T
 ∂ ∂u dx dt⎠ , E1 (t)dt ≤ C ⎝E1 (0) + E1 (T ) + aij,t ∂xj 0 0 Ω i,j ∂xi (5.3.5) where E1 (t) is defined by E(t) with u replaced by ut . Here, we see   2
T
 ∂ ∂u dx dt aij,t ∂xj 0 Ω i,j ∂xi 2 2 2 
T
 ∂ u ∂ 2 2 ≤C |∇aij,t | |∇u| + aij dx dt ∂t ∂xi ∂xj 0 Ω
T
T
T ≤ Cδ2 ∆u2 dt ≤ Cδ2 utt 2 dt ≤ Cδ2 E1 (t) dt, 0

0

(5.3.6)

0

where in the last step we have used the equation (5.1.1) and an elliptic regularity result for bounded domains. It follows from (5.3.5) and (5.3.6) that for small δ2 > 0,
T E1 (t)dt ≤ C(E1 (0) + E1 (T )). (5.3.7) 0 ∗

Setting again E1 (t ) = min0≤t≤T E1 (t) we have from the equation (5.3.4) that    
t
1 ∂ ∂ut ∂ut ∂u ∗ aij,t E1 (t) = E1 (t ) + + aij,t utt dx dt 2 ∂xi ∂xj ∂xi ∂xj ∗ Ω i,j t   
T ≤ E1 (t∗ ) + C δ2 + δ2 E1 (s) ds, 0 ≤ t ≤ T, 0

(5.3.8) and hence, assuming δ2 < 1, we have from (5.3.7) and (5.3.8),  
T
T √ ∗ ∗ (T E1 (t ) ≤) E1 (t)dt ≤ C E1 (t ) + ε2 E1 (t)dt 0

(5.3.9)

0

for some C > 0. Thus, if we take δ2 small enough and large T > 0 we obtain E1 (t∗ ) = 0, which implies ut (x, t) ≡ 0 on Ω × [0, T ]. Returning to the equation (5.1.1) we have   ∂  ∂u − aij (x, t) = 0 on Ω × [0, T ] ∂xi ∂xj i,j and we conclude, by the boundary condition, u(x, t) ≡ 0 on Ω × [0, T ].




Decay and Global Existence

239

5.4. Proof of Proposition 5.2 Here we consider the problem (5.1.7) in the exterior domain Ω. We combine the ˜ (B) ˜ and (C) ˜ which are true even for the case of exterior domains identities (A), (in fact we use these identities in forms of differential equalities). Taking h = φ(r)(x − x0 ) as in the proof of Theorem 4.1 we can show the following inequality (see (4.2.4)):  
d 1 a(x)|u|2 dx + kE(t) (ut , φ(x − x0 ) · ∇u) + α(ut , u) + dt 2 Ω    0 0 
(N φ+φ r) (x −x )(x −x φ(x−x0 )·∇aij i j i j)  +φ +φ+α α aij + uxi uxj dx − 2 r 2 i,j Ω 
 N φ + φ r − α + ka(x) |ut |2 dx + 2 Ω 2

∂u k 1 ≤ aij,t uxi xj dx + aij νi νj φν · (x − x0 ) dS 2 2 i,j Γ(x0 ) ∂ν Ω

i,j |a(x)ut φ(x − x0 ) · ∇u| dx. + (φ(x − x0 ) · ∇u + u + kut ) f dx + Ω

Ω

(5.4.1) Here, we see that     (xi −x0i )(xj −x0j ) φ(x−x0 )·∇aij (N φ+φ r)  +φ +φ+α aij − u xi u xj − 2 r 2 i,j   ˜1  |)ε δ (N φ + φ r) (L + |x 0 0 + φ r + φ + α − ≥ − aij uxi uxj , 2 2 i,j (5.4.2) where δ˜1 = k0−1

 i,j

∇aij 2∞

1/2 .

Also, assuming (L + |x0 |)δ˜1 < 1/4, we can choose α > 0 and k − and

1 such that



ε0 N φ + φ r (L + |x0 |)ε0 l0 − + φ r + φ + α ≥ 2 2 8 k ε0 N φ + φ r − α + a(x) ≥ . 2 2 8

Finally, note that 

 1 1 ¯1 ≤ δ a u u dx aij uxi uxj dx ij,t x x i j 2 Ω i,j i,j 2 Ω * with δ¯1 = k0−1 ( i,j aij,t 2∞ )1/2 . We assume δ¯1 is sufficiently small so that k δ¯1 ≤ ε0 /4.

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Mitsuhiro Nakao

Then we obtain from (5.4.1) and (5.4.2) that  
1 d a(x)|u|2 dx + kE(t) (ut , φ(x − x0 ) · ∇u) + α(ut , u) + dt 2 Ω
ε0 k 2 + E(t) + a(x)|ut | dx 8 2 Ω 2

∂u dS + C (|∇u| + |u| + k|ut |)|f | dx ≤C Γ(x0 ) ∂ν Ω

(5.4.3)

with a constant C > 0. To control the first term of the right-hand side of (5.4.3) we use the usual argument obtaining (5.2.3) and (5.2.4). Then we have: 2


∂u   d 2 2 dS ≤ C |∇u| dx + C + |u | |f ||∇u|dx − c0 (ut , h · ∇u) t ∂ν dt Γ(x0 ) ω ∩Ω ω (5.4.4) and


 2  d 2 2 |∇u| dx + c1 (ut , ηu) + c1 a(x)η|u| dx ≤ C |ut | + |u|2 dx (5.4.5) dt ω ∩Ω Ω ω for some c1 > 0, C > 0. It follows from (5.4.3)–(5.4.5) that
dX(t) ε0 k + E(t) + a(x)|ut |2 dx dt 
8 2 Ω 

(5.4.6) |u|2 dx + (|∇u| + |u| + k|ut |)|f | dx ≤C Ω

ω

with some C > 0 and a large k > 0, where we set


1 X(t) = (ut , (φ(x−x0 )+c0 h)·∇u) + ((α+2c1 η)ut , u) + a(x)|u|2 dx + kE(t). 2 Ω We summarize the above arguments in the following statement:

Proposition 5.3. For any large k, say k > N + 1, if   1/2   1/2 2 2 ∇aij ∞ + aij,t ∞ sup 0≤t 0 there exists a constant Cε > 0 such that
t+T
t+T

t+T
  a(x)|ut |2 + |f |2 dx ds+ε |u|2 dx ds ≤ Cε E(t)dt (5.4.8) t

ω

for any t > 0.

t

Ω

t

Decay and Global Existence

241

Proof. The proof is given in a quite similar way as the proof of Proposition 4.1. But, we need a unique continuation property as stated in Theorem 5.2. We note that on the limit function v appearing in the proof we may assume that v ∈ C 2 ([0, T ]; L2loc(Ω)), ∇v ∈ C([0, T ]; L2 (Ω)), D2 u ∈ C([0, T ]; L2 (Ω)). Because, we can use a mollifier ρ(t) with respect to t to regularize v. Details are left to the readers as an exercise.
Without loss of generality we may assume δ2 ≤ δ1 . Then, by combining Propositions 5.2 and 5.3 we arrive at the desired inequality stated in Proposition 5.1. Finally, we note that by a standard density argument, the inequality is valid for finite energy solutions u(t).


6. Lp estimates for the wave equation in exterior domains 6.1. Problem and result Let us consider again the linear wave equation with a localized dissipation: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(6.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, (6.1.2) where a = a(x) is a continuous function on Ω satisfying Hyp. A. For each (u0 , u1 ) ∈ H01 (Ω) × L2 (Ω) there exists a unique solution u of problem (6.1.1)–(6.1.2) in the class C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)). On the basis of the local energy decay proved in Section 3 we shall prove Lp estimates of the solutions under additional regularity conditions. These results will be used in the next section for the proof of global existence of solutions to semilinear wave equations. We denote the norm of W m,p (Ω) or of W m,p (RN ) by || · ||m,p . We begin with the Cauchy problem in the whole space, utt − ∆u = 0 in RN × [0, ∞), u(x, 0) = u0 (x), The following estimate is known:

ut (x, 0) = u1 (x).

(6.1.3) (6.1.4)

Theorem 6.1. Let N ≥ 2. If u0 , u1 ∈ C0∞ (RN ), the solution u(t) of the Cauchy problem (6.1.3)–(6.1.4) satisfies the estimates u(t)p ≤ Ct−b (u0 1,p∗ + u1 p∗ ) and

(6.1.5)

u(t)∞ ≤ Ct−d (u0 M,1 + u1 M−1,1 ) (6.1.6) for any p, 2 < p ≤ p0 , where C denotes a constant independent of u and we set  N 2(N + 1) 2N p M= , p∗ = , + 1, p0 = 2 N −1 (N + 1)p + 2 b = (N − 1)(1/2 − 1/p) and

d = (N − 1)/2.

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The proof of Theorem 6.1 is not trivial. For the proof see Brenner [5], Pecher [54] and Mochizuki and Motai [32] etc., where more general results are proved. For general more precise estimates concerning the wave equation in the whole space see, e.g., Georgiev [9]. Remark 6.1. By a density argument we see that the above estimates are valid for more general solutions with initial data for which the right-hand sides are finite. In particular, the compactness of the supports of u0 , u1 is not necessary. By combining Theorem 3.1 from Section 3 and Theorem 6.1 with a ‘cut-off’ method as in Shibata-Tsutsumi [63] we can prove: Theorem 6.2. Let N ≥ 2 and m be a nonnegative integer. Let a ∈ C 2M+m (Ω) and in addition to Hyp. A we assume supp a(·) is compact. Concerning initial data we assume that u0 ∈ H01 (Ω) ∩ H 2M+m (Ω) ∩ W 2M+m,1 (Ω), u1 ∈ H 2M+m−1 (Ω) ∩ W 2M+m−1,1 (Ω) and that these data satisfy the compatibility condition of (M + m − 1)-th order. Then there exists a unique solution u(t) of the problem (6.1.1)– (6.1.2) in C([0, ∞); H 2M+m (Ω) ∩ H01 (Ω)) ∩ C 1 ([0, ∞); H 2M+m−1 (Ω) ∩ H01 (Ω)) and it satisfies
(6.1.7) u(t)m,p ≤ C I˜M +m (1 + t)−b for 2 < p ≤ p0 , and
u(t)m,∞ ≤ C I˜2M+m (1 + t)−d , (6.1.8) where we set  (u0 k+1,i + u1 k,i ), I˜k = i=1,2

⎧ ⎪ ⎨b  b = b−δ ⎪ ⎩ b−1−δ

if N is odd and N ≥ 5, if N = 3, if N is even,

and

⎧ ⎪ ⎪(N − 1)/2 if N is odd and N ≥ 5, ⎪ ⎨1 − δ if N = 3, d = ⎪−δ if N is even and N ≥ 4, ⎪ ⎪ ⎩ −1/2 − δ if N = 2. We note that δ > 0 in the above can be chosen arbitrarily small. By Shibata and Tsutsumi [63] we know that when V is convex and a(x) ≡ 0, then ER (t) ≤ CE(0)(1+t)−2(N −1) for solutions with initial data with compact support. Using this we have the following: Corollary 6.1. Let V be convex and let N ≥ 4 be even. We assume that a(x) ≡ 0. Then, the estimates (6.1.7) and (6.1.8) are valid with b = b and d = d. Remark 6.2. sides of (6.1.7) and (6.1.8) can be re*mIt is clear that the*left-hand m placed by k=0 Dtk u(t)p and k=0 Dtk u(t)∞ , respectively, where Dt denotes the partial differentiation with respect to t.

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243

Remark 6.3. When N is even the estimates (6.1.7) and (6.1.8) do not imply any decay property but show some growth property. These estimates, however, are never trivial since the standard energy inequality E(t) ≤ E(0) gives only
t    u(t)2 ≤ u0  + ut (s) ds ≤ u0  + E(0) (1 + t). 0

Remark 6.4. By the energy inequality and Sobolev’s embedding theorem we easily see that for 2 ≤ q ≤ ∞, u(t)m,q ≤ C(u0 M+m,2 + u1 M+m−1,2 ),

0 ≤ t ≤ 1.

This fact will be used to estimate the solutions near t = 0. Remark 6.5. See Racke [56], where Lp –Lq estimates are discussed for thermoelastic systems. 6.2. Proof of Theorem 6.2 We follow [44]. First, we take a smooth function φ(x) such that  1 if |x| ≥ L + 2, φ(x) = 0 if |x| ≤ L + 1, ˜(x, t) of the where we assume supp a(·) ⊂ BL . Let us consider the solution u problem u˜tt − ∆˜ u + a(x)˜ ut = 0 in Ω × [0, ∞), (6.2.1) ˜|∂Ω = 0. (6.2.2) u˜(x, 0) = φ(x)u0 (x), u˜t (x, 0) = φ(x)u1 (x) and u We can expect that u˜(x, t) behaves like the original solution u(x, t) if |x| is large. To estimate u ˜(x, t) we introduce the solution v of the Cauchy problem: vtt − ∆v = 0 in RN × [0, ∞), v(x, 0) = φ(x)u0 (x), vt (x, 0) = φ(x)u1 (x). We further take a smooth function ρ(x) such that  1 if |x| ≥ L + 1, ρ(x) = 0 if |x| ≤ L,

(6.2.3) (6.2.4)

and consider ρv, which satisfies (ρv)tt − ∆(ρv) = −∇ρ · ∇v − ∆ρv in RN × [0, ∞)

(6.2.5)

and (ρv)(x, 0) = φ(x)u0 (x), (ρv)t (x, 0) = φu1 (x). Note that ρφ = φ. We set w = u˜ − ρv. Then we see that wtt − ∆w + awt = ∇ρ · ∇v + ∆ρv =: g and w(x, 0) = 0,

wt (x, 0) = 0 and w|∂Ω = 0.

(6.2.6)

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Since v is a solution of the Cauchy problem in RN we know by Theorem 6.1 that for any nonnegative integer m, ||ρvm,p ≤ Ct−b (u0 m+1,p∗ + u1 m,p∗ )

(6.2.7)

for 2 < p ≤ 2(N + 1)/(N − 1) =: p0 , and ρvm,∞ ≤ C(1 + t)−d (u0 m+M,1 + u1 m+M−1,1 + u0 m+M,2 + u1 m+M−1,2 ), (6.2.8) where we recall b = (N − 1)(1/2 − 1/p) and d = (N − 1)/2. We denote the solutions of the linear equation (6.1.1)–(6.1.2) by S(t; (u0 , u1 )). Then, by variation of constants we see from (6.2.6)
t w(t) = S(t − s; (0, g)) ds. 0

Thus, we have by Theorem 6.1 that for any integer m ≥ 1,
t S(t − s; (0, g))H m (ΩL+2 ) ds w(t)H m (ΩL+2 ) ≤ 0
t ≤C Γ(t − s)gH m−1 (BL+2 \BL+1 ) ds
0 t ≤C Γ(t − s)vH m (BL+2 \BL+1 ) ds, 0

where we set

 Γ(t) =

e−λt (1 + t)(−1+δ)/2

if N is odd , if N is even.

(6.2.9)

Needless to say, Theorem 6.1 (Corollary 6.1) implies that if (u0 , u1 ) ∈ H m+1 (Ω) × H m (Ω) have compact support and satisfy the compatibility condition of the m-th order, then the solution u(t) satisfies Dm+1 u(t)L2 (Ω∩BR ) ≤ CR (u0 H m+1 + u1 H m )Γ(t),

R

1,

where D = (∇, Dt ). We have used this fact just above. (When V is convex, N ≥ 4 is an even number and a(x) ≡ 0 we take Γ(t) = (1 + t)−(N −1) . Hereafter, we say V is convex for such a case.) Hence, noting b ≤ (N − 1)/(N + 1) < 1,
t Γ(t − s)v(s)m,p(BL+2 \BL+1 ) ds w(t)H m (ΩL+2 ) ≤ C (6.2.10) 0
t ≤C Γ(t − s)s−b (v(0)m+1,p∗ + vt (0)m,p∗ ) ds 0 ˆ

≤ C(1 + t)−b (u0 m+1,p∗ + u1 m,p∗ ), where

 ˆb =

b b − 1/2 − δ/2

if N is odd, if N is even.

Decay and Global Existence When V is convex we can take ˆb = b. Similarly, we have
t Γ(t − s)v(s)m,∞(BL+2 \BL+1 ) ds w(t)H m (ΩL+2 ) ≤ C 0
t  ≤C Γ(t − s)(1 + s)−d (v(0)m+M,i + vt (0)m+M−1,i ) ds 0 ˆ

≤ C(1 + t)−d



245

(6.2.11)

i=1,2

(v(0)m+M,i + vt (0)m+M−1,i ),

i=1,2

where

⎧ ⎪ ⎨d ˆ d = 1/2 − δ/2 ⎪ ⎩ −δ/2

if N is odd, if N is even and N ≥ 4, if N = 2.

When V is convex we can take dˆ = d. In particular, replacing m by M + m in (6.2.11) and using Sobolev’s embedding lemma we have ˆ  w(t)m,∞(ΩL+2 ) ≤ C(1 + t)−d (u0 2M+m,i + u1 2M+m−1,i ). (6.2.12) i=1,2

To estimate w(t) outside of the domain ΩL+2 we take a smooth function µ(x) such that  1 if |x| ≥ L + 2, µ(x) = 0 if |x| ≤ L + 1, and set µw = w. ˜ Then, w ˜tt − ∆w ˜ = −∇µ · ∇w − ∆µw =: h in RN × [0, ∞)

(6.2.13)

with w(x, ˜ 0) = 0, w ˜t (x, 0) = 0. Denoting by S0 (t; (u0 , u1 )) the solution of the free wave equation in RN with initial data (u0 , u1 ) we see
t w(t) ˜ = S0 (t − s; (0, h)) ds 0

and hence, by Theorem 6.1 and (6.2.11),
t w(t) ˜ (t − s)−b h(s)m,p∗ (BL+2 \BL+1 ) ds m,p ≤ C 0
t ≤C (t − s)−b w(s)m+1,2(ΩL+2 ) ds
0 t ˆ  ≤C (t − s)−b (1 + s)−d (u0 m+M+1,i + u1 m+M,i ) ds 0 −˜ b

≤ C(1 + t)



i=1,2

(u0 m+M+1,i + u1 m+M,i )

i=1,2

(6.2.14)

246 with

Mitsuhiro Nakao ⎧ ⎪ ⎪b ⎪ ⎨ ˜b = b − δ ⎪ b − (1 + δ)/2 ⎪ ⎪ ⎩ b − 1 − δ/2

if if if if

N N N N

is odd and N > 3, = 3, is even and N ≥ 4, = 2,

˜b = b if V is convex. Similarly,
t w(t) ˜ (1 + t − s)−d w(s)m+M,2(BL+2 ) ds m,∞ ≤ C 0
t ˆ  (t − s)−d (1 + s)−d (u0 m+2M,i + u1 2M−1,i ) ds ≤C 0 −d˜

≤ C(1 + t)



i=1,2

(u0 m+2M,i + u1 m+2M−1,i ),

i=1,2

(6.2.15)

⎧ d ⎪ ⎪ ⎪ ⎨1 − δ/2 d˜ = ⎪ (1 − δ)/2 ⎪ ⎪ ⎩ −(1 + δ)/2

if if if if

N N N N

is odd and N ≥ 5, = 3, is even and N ≥ 4, = 2.

The estimates (6.2.10) and (6.2.14) yield, in particular, ˜  (u0 m+M+1,i + u1 m+M,i ), w(t)m,p ≤ C(1 + t)−b

(6.2.16)

i=1,2

for 2 < p ≤ p0 . The estimates (6.2.12) and (6.2.15) yield ˜  (u0 m+2M,i + u1 m+2M−1,i ). w(t)m,∞ ≤ C(1 + t)−d

(6.2.17)

i=1,2

It follows from (6.2.7), (6.2.8), (6.2.16), and (6.2.17) that ˜  (u0 m+M+1,i + u1 m+M,i ), ˜ u(t)m,p ≤ C(1 + t)−b

(6.2.18)

i=1,2

and ˜

˜ u(t)m,∞ ≤ C(1 + t)−d



(u0 m+2M,i + u1 m+2M−1,i ).

(6.2.19)

i=1,2

Next, we have to estimate u ˆ = u−u ˜. For this we repeat a similar argument used in the derivation of the estimates for u˜(t). In fact, this time the argument is simpler, but the decay property for even and higher-dimensional domains is broken in this step. From (6.2.1) we see uˆtt − ∆ˆ u + a(x)ˆ ut = 0 in Ω × [0, ∞),

(6.2.20)

and u ˆ(x, 0) = (1 − φ(x))u0 (x),

u ˆt (x, 0) = (1 − φ(x))u1 (x) and u ˆ|∂Ω = 0.

Decay and Global Existence Taking a smooth function ρˆ such that  0 if ρˆ(x) = 1 if

247

|x| ≥ L + 4, |x| ≤ L + 3,

we have from Section 3 ρˆu ˆ(t)m,p ≤ C∇ˆ u(t)m,2(ΩL+4 ) ≤ CΓ(t)(u0 m+1,2 + u1 m,2 )

(6.2.21)

ρˆuˆ(t)m,∞ ≤ Cˆ u(t)m+M,2 ≤ CΓ(t)(u0 m+M,2 + u1 m+M−1,2 ).

(6.2.22)

and

Further, setting w ˆ = uˆ − ρˆuˆ, we have w ˆtt − ∆wˆ = −∇ˆ ρ · ∇ˆ u − ∆ˆ ρu ˆ =: gˆ in Ω × [0, ∞)

(6.2.23)

and w(x, ˆ 0) = 0,

wˆt (x, 0) = 0

and w| ˆ ∂Ω = 0.

This problem is very similar to (6.2.6), but, we note, the properties of g and gˆ are different. We see in this case,
t w(t) ˆ Γ(t − s)ˆ u(s)H m (BL+4 \BL+3 ) ds (6.2.24) H m (ΩL+4 ) ≤ C
0 t ≤C Γ(t − s)Γ(s)(u0 m,2 + u1 m−1,2 ) ds 0

ˆ ≤ Γ(t)(u 0 m,2 + u1 m−1,2 ) with

 ˆ = Γ(t)

e−λt (1 + t)δ

if N is odd, if N is even.

ˆ = Γ(t) = (1+t)−(N −1) . In particular, replacing When V is convex we can take Γ(t) m by m + M , we have ˆ w(t) ˆ m,∞(ΩL+4 ) ≤ C Γ(t)(u0 m+M,2 + u1 m+M−1,2 ).

(6.2.25)

¯ = µw, ˆ where µ Finally, to estimate w(t) ˆ in the complement to ΩL+2 , we set w(t) is the function appearing in (6.2.13). Then, ˆ in RN × [0, ∞) w ¯tt − ∆w ¯ = −∇µ · ∇w ˆ − ∆µwˆ =: h with w(x, ¯ 0) = 0,

w ¯t (x, 0) = 0.

(6.2.26)

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Mitsuhiro Nakao

As in (6.2.14) and (6.2.15), we see by (6.2.24),
t ˆ w(t) ¯ (t − s)−b h(s) m,p ≤ C m,2(BL+4 \BL+3 ) ds 0
t ≤C (t − s)−b w(s) ˆ m+1,2(ΩL+2 ) ds 0
t ˆ ≤C (t − s)−b Γ(s)(u 0 m+1,2 + u1 m,2 ) ds 0

(6.2.27)





≤ C(1 + t)−b (u0 m+1,2 + u1 m,2 ) with

 

b = and

b b−1−δ




t

w(t) ¯ m,∞ ≤ C

if N is odd, if N is even,

ˆ (1 + t − s)−d h(s) m+M−1,2 ds

0



≤ C(1 + t)−d (u0 m+M,2 + u1 m+M−1,2 ) with

 

d =

d −δ

(6.2.28)

if N is odd, if N is even.

Thus, we obtain



−b (u0 m+1,2 + u1 m,2 ) w(t) ˆ m,p ≤ C(1 + t)

(6.2.29)

and



−d (u0 m+M,2 + u1 m+M−1,2 ). w(t) ˆ m,∞ ≤ C(1 + t)

(6.2.30)

Further, it follows from (6.2.21) and (6.2.29) that ˆ(t)m,p + w(t) ˆ ˆ u(t)m,p ≤ ρˆu m,p



≤ C(1 + t)−b (u0 m+1,2 + u1 m,2 )

(6.2.31)

and from (6.2.22) and (6.2.30) that



ˆ u(t)m,∞ ≤ C(1 + t)−d (u0 m+M,2 + u1 m+M−1,2 ).

(6.2.32)

Summarizing the above estimates (6.2.18), (6.2.19), (6.2.31), and (6.2.32) we ob˜ d }. tain the desired results (6.1.7) and (6.1.8) with b = min{˜b, b } and d = min{d,


Decay and Global Existence

249

7. Semilinear wave equations 7.1. Problem and result In this section we consider semilinear wave equations in exterior domains: utt − ∆u + a(x)ut = f (u) in Ω × [0, ∞),

(7.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(7.1.2)

where f (u) is a nonlinear term like f (u) = |u|α u. We shall consider this problem ˜ = Hyp. A+ Hyp. A on the under two types of assumptions Hyp. A and Hyp. A dissipation a(x)ut . Recall that under Hyp. A we know the Lp decay of solutions for ˜ we know the total energy the linear equation (Theorem 6.2), while under Hyp. A decay of solutions (Theorems 4.1 and 4.2). We apply these results to the semilinear problem (7.1.1)–(7.1.2). The latter case is easier to treat and we begin with this case. The arguments here are taken from [43] and [44]. We make the following assumptions on f (u). Hyp. B. The function f (u) is continuously differentiable on R and satisfies |f (j) (u)| ≤ kj |u|α+1−j , j = 0, 1, for u ∈ R with some kj > 0 and α > 0. ˜ The function f (u) is 2M − 1 times continuously differentiable on R and Hyp. B. satisfies |f (j) (u)| ≤ kj |u|α+1−j , j = 0, 1, 2, · · · , 2M − 1, for u ∈ R with some kj > 0 and α ≥ 2M − 2. Let us state our existence theorems. Theorem 7.1. Let 1 ≤ N ≤ 4 and Hyp. B be satisfied with 4/N < α ≤ 2/(N − 2) ˜ there exists δ > 0 such that if (4/N < α < ∞ if N = 1, 2). Then, under Hyp. A 1 2 (u0 , u1 ) ∈ H0 (Ω) × L (Ω) and I0 := u0 H1 + u1  ≤ δ, the initial-boundary value problem (7.1.1)–(7.1.2) admits a unique solution u(t) in the space C([0, ∞); H01 (Ω))∩C 1 ([0, ∞); L2 (Ω)), satisfying the decay or boundedness estimates (7.1.3) E(t) ≤ C0 I02 (1 + t)−1 and u(t) ≤ CII0 < ∞. Remark 7.1. When N = 1 the problem is reduced to the case Ω = (0, ∞) or (−∞, 0) and we may assume Γ(x0 ) = φ. If we seek for more regular solutions the condition on α in Theorem 7.1 can be relaxed. We illustrate this for the so-called H 2 solutions.

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Theorem 7.2. Let 3 ≤ N ≤ 6 and 2/(N − 2) < α ≤ 2/(N − 4)+ . We assume, in addition to the conditions of Hyp. B with this α, that f ∈ C 2 (R) and |f  (u)| ≤ k1 |u|α−1 . Then, there exists δ1 > 0 such that if (u0 , u1 ) ∈ H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) and I1 := ||u0 ||H 2 + ||u1 ||H 1 < δ1 , the problem admits a unique solution u(t) in X2 := C 2 ([0, ∞); L2 (Ω)) ∩ C 1 ([0, ∞); H01 (Ω)) ∩ C([0, ∞); H 2 (Ω) ∩ H01 (Ω)), satisfying u(t) ≤ C1 I1 < ∞

(7.1.4)

and u(t)X2 := ut (t) + ∇u(t) + utt (t) + ∇ut (t) + ∆u(t) ≤ C1 I1 (1 + t)−1 with some C1 > 0.

(7.1.5)

Remark 7.2. If f (u) is more smooth and further (u0 , u1 ) belongs to H m+1 (Ω) × H m (Ω), m ≥ 2, and satisfies the required compatibility conditions, then we can prove that the solutions in Theorems 7.1, 7.2 are more smooth (cf. [48]). Hyp. C. There exists p, 2 < p ≤ p0 , such that α ≥ p − 1 and α > (2M − 1)(1 − (1 − θ)(p − 2)/p), where

 θ=

1 1 1 + − N p 2(2M − 1)



2M − 1 1 1 + − N p 2

−1 .

√ Remark √ 7.3. When N = 3 and N = 4 Hyp. C is reduced to α > 2 3 − 1 and α > 15 − 2, respectively. Theorem 7.3. We make the assumptions Hyp. A, Hyp. B and Hyp. C. Let N ≥ 3 be odd. Then, there exists ε1 > 0 such that if I˜2M ≤ ε1 , the problem (7.1.1)–(7.1.2) i 2M−i admits a unique solution u(·) ∈ XM := ∩2M−1 (Ω) ∩ H01 (Ω)) ∩ i=0 C ([0, ∞); H 2M 2 C ([0, ∞); L (Ω)) satisfying the estimates
u(t)p ≤ C I˜2M (1 + t)−b

and


u(t)∞ ≤ C I˜2M (1 + t)−d , where d = d := (N − 1)/2 and b = b := (N − 1)(1/2 − 1/p) if N ≥ 5 and b = b − δ and d = d − δ if N = 3. Here we recall I˜2M = ||u0 ||2M,2 + ||u1 ||2M−1,2 .

Corollary 7.1. When N is even and N ≥ 4 we assume that V is convex and a(x) ≡ 0. Then, the same statements as in Theorem 7.3 hold with d = d and b = b.

Decay and Global Existence 7.2. Proof of Theorems 7.1 and 7.2 Setting U (t) = (u(t), ut (t))T , the problem (7.1.1)–(7.1.2) is rewritten as
t U (t) = S(t)U U0 + S(t − s)F (s) ds,

251

(7.2.1)

0

where U0 = (u0 , u1 )T ∈ H01 (Ω)×L2 (Ω), F (s) = (0, f (u(s)))T , and S(t) denotes the semi-group in H01 (Ω) × L2 (Ω) associated with the linear problem. It is well-known that under Hyp. B with 0 < α ≤ 2/(N − 2)+ , the problem (7.1.1)–(7.1.2) has a unique solution u ∈ C([0, T ); H01 (Ω)) ∩ C 1 ([0, T ); L2 (Ω)) for some T > 0, and to show the global existence in Theorem 7.1 it suffices to derive the a priori estimate E(t) ≤ C(II0 )(1 + t)−1 in the interval of existence. Let K > 0 and assume that the following estimates hold in an interval [0, T ), T > 0, ut (t) + ∇u(t) ≤ KII0 (1 + t)−1/2

(7.2.2)

and u(t) ≤ KII0 . (7.2.3) Note that (7.2.2) and (7.2.3) are valid on some interval if we take a large K > 0. By Theorem 4.1 we know S(t)U U0 E ≤ CII0 (1 + t)−1/2 , where (u(t), ut (t))E = ∇u(t) + ut (t). Thus, by (7.2.1), we have
t −1/2 U (t)E ≤ CII0 (1 + t) +C (1 + t − s)−1/2 f (u(s))ds.

(7.2.4)

(7.2.5)

0

Here, by Hyp. B and Gagliardo-Nirenberg inequality, (α+1)(1−θ) f (u(s)) ≤ k0 uα+1 ∇u(α+1)θ 2(α+1) ≤ k0 u

(7.2.6)

with θ = N α/2(α + 1). Thus, as long as (7.2.2) and (7.2.3) hold, we have f (u(s)) ≤ k0 K α+1 I0α+1 (1 + s)−N α/4

(7.2.7)

and hence, U (t)E ≤ C0 (II0 + k0 K α+1 I0α+1 )(1 + t)−1/2 (7.2.8) for some C0 > 0 independent of K, I0 , where we have used the assumption N α/4 > 1. Now, we make an assumption on I0 : Q0 (II0 , K) := C0 (1 + k0 K α+1 I0α ) < K.

(7.2.9)

Then we obtain U (t)E ≤ Q0 (II0 , K)II0 (1 + t)−1/2 < KII0 (1 + t)−1/2 .

(7.2.10)

Next, we check the boundedness of u(t). Since u(t) ≤ CII0 for solutions of the linear equation we have from (7.2.1) that
t u(s)α+1 I0 + k0 K α+1 I0α+1 ) u(t) ≤ CII0 + Ck0 2(α+1) ds ≤ C0 (I 0

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Mitsuhiro Nakao

for some C0 , which can be identified with the one in (7.2.8). Thus, under the assumption (7.2.9), we have u(t) ≤ Q0 (K, I0 )II0 < KII0 .

(7.2.11)

The estimates (7.2.10) and (7.2.11) show that under the assumption (7.2.9), the (local) solution u(t) exists in fact globally in time and these estimates hold for all t ≥ 0. Needless to say, (7.2.9) is valid if we take K > C0 and I0 < δ for a small δ > 0. The proof of Theorem 7.1 is finished.
Next we shall show the proof of Theorem 7.2. The existence of H 2 local solutions are rather standard. Indeed, all of the estimates required for the existence of H 2 local solutions follow more or less easily from similar estimations below. We note that the assumption |f  (u)| ≤ k0 |u|α−1 is used in this step. For the proof of Theorem 7.2 it will be sufficient to derive the following a priori estimates: u(t) ≤ KI1

(7.2.12)

and ut (t) + ∇u(t) + utt (t) + ∇ut (t) + ∆u(t) ≤ KI1 (1 + t)−1/2

(7.2.13)

for some K > 0, where I1 = u0 H 2 + u1 H 1 . We may assume that the (local) solutions u belong to C 2 ([0, T ); L2(Ω)) ∩ C 1 ([0, T ); H01 (Ω)) ∩ C([0, T ); H 2 (Ω)) and satisfy the estimates (7.2.12) and (7.2.13) for some interval [0, T ), T > 0. Then, we start again from (7.2.1). But, instead of (7.2.6) and (7.2.7), we use the estimates: ˜ (α+1)(1−θ)

˜

f (u(s) ≤ k0 u2N/(N −2) D2 u(α+1)θ ˜

˜

≤ Ck0 ∇u(α+1)(1−θ) (∇u + ∆u)(α+1)θ

with θ˜ = (α(N − 2) + 2)/2(α + 1), where we have used the assumption &

2 2 ,1 < α ≤ . max N −2 (N − 4)+ Thus, we have u(t)E ≤ CII0 (1 + t)−1/2 + Ck0 K α+1 I1α+1 ≤ C1 (1 + k0 K α+1 I1α )I1 (1 + t)−1/2




t

(1 + t − s)−1/2 (1 + s)−(α+1)/2 ds

0

(7.2.14) with some C1 , where uE = ut  + ∇u. Similarly, u(t) ≤ C(I1 + Ck0 K α+1 I1α )I1 .

(7.2.15)

Next, differentiating the equation we have uttt − ∆ut + a(x)ut = f  (u)ut in Ω × [0, ∞),

(7.2.16)

ut (x, 0) = u1 (x), utt (x, 0) = u2 (x) := ∆u0 (x) − a(x)u1 (x) and ut |∂Ω = 0. (7.2.17)

Decay and Global Existence Therefore, −1/2

ut (t)E ≤ CI1 (1 + t)


+C

t

253

(1 + t − s)−1/2 f  (u(s))ut (s)ds.

(7.2.18)

0

Here, we easily see f  (u(s))ut (s) ≤ k0





(N −2)/2N 
1/N
|ut |2N/(N −2) dx |u|N α dx

Ω

≤ Ck0 ∇ut ∇u

ˆ α(1−θ)

Ω αθˆ

(∇u + ∆u)

with θˆ = (α(N − 2) − 2)/2α. Thus, we obtain ut (t)E ≤ C1 (1 + k0 K α+1 I1α )I1 (1 + t)−1/2

(7.2.19)

for some C1 > 0. Finally, from the equation itself, we see ∆u(t) ≤ C1 (1 + k0 K α+1 I1α )I1 (1 + t)−1/2 .

(7.2.20)

Now, we take a large K such that K > C1 and make the assumption on I1 : Q1 (I1 , K) := C1 (1 + k0 K α+1 I1α ) < K,

(7.2.21)

which is valid if I1 ≤ δ1 for some small δ1 > 0. Then we obtain u(t) ≤ Q1 (I1 , K)I1 < KI1

(7.2.22)

and ut (t) + ∇u(t) + utt (t) + ∇ut (t) + ∆u(t) ≤ Q1 (I1 , K)I1 (1 + t)−1/2 < KI1 (1 + t)−1/2 .

(7.2.23)

2

We conclude that under the assumption (7.2.21) the local H solutions exist globally in time and the estimates (7.2.22) and (7.2.23) hold for all t ≥ 0. The proof of Theorem 7.2 is now complete.
7.3. Proof of Theorem 7.3 In this section we assume Hyp. A and Hyp. C on a(x) and f (u), respectively, and under these assumptions we prove the existence of global solutions. The following well-known result concerning the local existence is the starting point (cf. Kato [17]). Proposition 7.1. We set M = [N/2]+1. Let (u0 , u1 ) belong to H 2M (Ω)×H 2M−1 (Ω) and satisfy the compatibility condition of the (2M − 1)-order. Then, under Hyp. B, there exist T > 0 and a unique solution u(t) of the problem (7.1.1)–(7.1.2) in the i 2M−i (Ω)∩H H01 (Ω))∩C 2M ([0, T ); L2 (Ω)). Further, class X(T ) := ∩2M−1 i=0 C ([0, T ); H *2M−1 Dtk u(t)H 2M −k + this solution can be continued in t as long as the norm k=0 *2M k k=1 Dt u(t)2 is bounded. In order to prove Theorem 7.3 we set for 2 < p ≤ p0 , V (T ) := {u ∈ X(T )| uV (T ) < ∞},

254

Mitsuhiro Nakao

where uV (T ) :=

' sup 0≤t 0 independent of T > 0 for assumed solutions u(t) ∈ V (T ) of (7.3.1). Note that the boundedness of u(t)2 for any finite interval [0, T ] follows 0t from the inequality ||u(t)|| ≤ ||u0 || + 0 ||ut (s)||ds. Proposition 7.2. Let (u0 , u1 ) belong to H 2M(Ω)∩W 2M,1(Ω)×H 2M−1(Ω)∩W 2M,1(Ω) and satisfy the compatibility condition of the 2M − 1-th order. Let u(t) be the local solution of (7.1.2) with u(·)V (T ) ≤ K < ∞ with some K > 0. Then, for p satisfying Hyp. C, we have ' (

sup (1 + t)d u(t)∞ + (1 + t)b (u(t)p + Du(t)p ) ≤ C(I˜2M + K α+1 ), 0≤t 0,

Decay and Global Existence

255

under the assumptions that α ≥ p − 1 and α > p − pb /d . Note that the latter condition is equivalent to α > p − pb/d = 2 because we can take δ > 0 arbitrarily small. The second term is treated as follows:


2M−1 

 f (j) (u)(Du)ν1 · · · (D2M−1 u)ν2M −1 dx

Ω j=1 ν∈S j

⎛

≤C⎝

⎞ 2M−1 2M−1  
1 |u|α D2M−1 u dx + |u|α+1−j |Di u|νi dx⎠




⎛

Ω

j=2 ν∈Sj

≤ C ⎝uα 2α Du2M−2,2 +

2M−1 



Ω

i=1

α+1−j ||u||∞

j=2 ν∈Sj

2M−1 1

⎞

||Di u||νpii νi ⎠ ,

i=1

2M−1 satisfying where we should choose {pi }i=1

1 ≤ pi ≤ ∞,

2M−1 

1/pi = 1 and pi νi ≤ 2N/(N − 4M + 2i)+ .

i=1

Note that r := max {r, 0} and the last inequality on pi νi assures H 2M−i ⊂ Lpi νi . Such a choice of {pi } is certainly possible because +

2M−1  i=1

2M−1  iνi N − 4M 1 1 νi (N − 4M + 2i)+ ≤ + = − < 1. 2N 2N N 2 N i=1

Now, we easily see that α−p/2

uα 2α Du2M−2,2 ≤ Ku∞


p/2

up







≤ CK α+1 (1 + t)−(α−p/2)d −b p/2 ≤ CK α+1 (1 + t)−d −ε ,

ε > 0,

provided that α > p/2 − b p/2d , which is equivalent to α > p/2 − bp/2d. Further, by Gagliardo-Nirenberg inequality, we have i i Di upi νi ≤ CDu1−θ Duθ2M−1,2 p

with

 θi =

i−1 1 1 + − N p pi νi



2M − 1 1 1 + − N p 2

−1 .

Hence, α+1−j ||u||∞

2M−1 1 i=1

||D

i

u||νpii νi

≤ CK

α+1

−(α+1−j)d
−

(1 + t)

2M P−1 i=1

(1−θi )νi b


.

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Mitsuhiro Nakao

But, we see that 2M−1 



 −1 2M−1  2M − 1 1 1 2M − 1 1 1 − + + − νi N 2 N N p 2 i=1   −1 2M−1   iνi 2M − 1 1 1 1 + − + + − N p 2 N pi i=1   −1 2M − 1 1 1 2M − 1 1 1 = − + + − j N 2 N N p 2  −1 2M − 1 1 1 −(2M − 1 − N )N −1 + − N p 2 =: ηj .

(1 − θi )νi =

i=1

Therefore, α+1−j ||u||∞

2M−1 1







||Di u||νpii νi ≤ CK α+1 (1 + t)−(α+1−j)d −ηj b .

i=1

By an easy calculation we can show that min

2≤j≤2M−1

=

min

{(α + 1 − j)d + ηj b } 

2≤j≤2M−1



(α + 1)d − (2M − 1 − N )b N −1

+d (p − 4M )N −1 p−1



2M − 1 1 1 + − N p 2

2M − 1 1 1 + − N p 2 -

−1

−1

j

= (α + 1 − 2M + 1)d + η2M−1 b ˜  = (α + 2 − 2M )d + (2M − 1)(1 − θ)b with



 −1 1 1 2M − 1 1 1 − + − , p 2M − 1 N p 2 where we have used the fact p ≤ p0 ≤ 4M . Summarizing we obtain θ˜ =




f (u(t))2M−1,1 ≤ CK α+1 (1 + t)−d −ε ,

ε > 0,

provided that α ≥ p − 1,

˜ α > 2 and α > (2M − 1)(1 − (1 − θ)b/d).

Similarly, we can show that



f (u(s))2M−1,2 ≤ C +



uα+1 2(α+1)

2M−1 



j=2 ν∈Sj

|u| |D 2α

+ Ω

α+1−j u∞

2M−1 1 i=1

2M−1

1/2 u| dx ⎞ 2

Di uν2pi i νi ⎠ .

(7.3.5)

Decay and Global Existence

257

Thus, replacing p by p/2 and repeating a quite similar argument getting (7.3.5), we have
(7.3.6) f (u(t))2M−1,2 ≤ CK α+1 (1 + t)−d −ε , ε > 0, provided that α ≥ p/2 − 1, where

α > p/2 − pb/2d and α > (2M − 1)(1 − (1 − θ)b/d), 

θ=

1 1 − p 2(2M − 1)



2M − 1 1 1 + − N p 2

−1 .

The second condition is equivalent to α > 1. We note that 0 < θ˜ < θ < 1. Thus, under Hyp. C, we obtain (7.3.5) and (7.3.6), hence the estimate for ||u(t)||∞ . Next, we consider the estimate for u(t)p and Dup , D = (∇, Dt ). We again use the formula (7.3.1) to get first
t
 −b
u(t)p ≤ CI1,M (1 + t) +C (1 + t − s)−b f (u)M,i ds. s

Also, −b


Du(t)p ≤ CI1,M+1 (1 + t)

i=1,2


+C




(1 + t − s)−b

Ω



f (u)M+1,i ds.

i=1,2

Since 2M − 1 ≥ M + 1 we can apply the estimates (7.3.5) and (7.3.6) to get the desired estimates for f (u)M+1,i and, consequently,


u(t)W 1,p + ut (t)p ≤ C(I˜2M + K α+1 )(1 + t)−b . The proof of Proposition 7.2 is complete.

(7.3.7)


Finally, we shall show: Proposition 7.3. Assume that u0 2M,2 + u1 2M−1,2 ≤ 1. Then, under Hyp. C, the local solution u(t) with u(t)V (T ) ≤ K satisfies 2M−1  k=0

Dtk ∇u(t)2M−k−1 +

2M 

Dtk u(t) ≤ C(u0 2M,2 + u1 2M−1,2 + K α+1 )

k=1

with a constant C > 0 independent of K, t and (u0 , u1 ). Proof. Let 1 ≤ l ≤ 2M − 1 be an integer. Differentiating the equation l times with respect to t we have Utt − ∆U + a(x)U Ut = Dtl f (u), (7.3.8) l where U = Dt u. Since α ≥ 1, it is standard to show that U Ut (0)2 + ∇U (0)2 ≤ C(IIM ) < ∞, where IM := u0 H 2M + u1 H 2M −1 . In particular, if IM ≤ 1, we can prove that U Ut (0)2 + ∇U (0)2 ≤ CIIM

(7.3.9)

258

Mitsuhiro Nakao

with some constant C > 0 independent of IM . Now, multiplying the equation by Ut and taking account the boundary condition U |∂Ω = 0, we have
t
Ut (0)22 + ∇U (0)22 + 2 |Dtl f (u)||U Ut |dx ds U Ut (t)22 + ∇U (t)22 ≤ U 0 Ω
t 2 ≤ CIIM + C sup U Ut (s)2 Dtl f (u)2 ds. 0≤s≤t

0

(7.3.10) Here, by the same argument deriving the estimate (7.3.6) we see


Dtl f (u) ≤ CK α+1 (1 + t)−d −ε , ε > 0.

(7.3.11)

Since d + ε > 1 (note that d + ε = 1 − δ + ε > 1 for sufficiently small δ > 0 if N = 3) we have from (7.3.10) and (7.3.11) that Dtl+1 u(t)2 + ∇Dtl u(t) ≤ C(IIM + K α+1 ) < ∞.

(7.3.12)

Further, returning to the equation −∆Dtl−1 u(t) = −Dtl+1 u(t) − a(x)Dtl u(t) + Dtl−1 f (u) and using the estimate Dtl−1 f (u)2 ≤ CK α+1 we see by elliptic regularity theory that Dtl−1 ∇uH 1 ≤ C(IIM + K α+1 ).

(7.3.13)

Next, differentiating the equation l − 1 times with respect to space variables and using the inequality uH m+2 ≤ C(∆uH m + uH m+1 ) we obtain 2M−1 

Dtk ∇u(t)H 2M −1−k +

k=1

2M 

Dtk u(t)2 ≤ C(IIM + K α+1 )

k=1




for some constant C > 0.

Completion of the proof of Theorem 7.3. Let u(t) be the local solution guaranteed by Proposition 7.1. This solution belongs to V (T ) for a small T > 0. From Propositions 7.2, 7.3 we conclude under Hyp. C that if I˜2M ≤ 1 and u(·)V (T ) ≤ K, then u(·)V (T ) ≤ C(I˜2M + K α+1 ). Therefore, if we choose K > 0 such that C(I˜2M + K α+1 ) < K, (7.3.14) we have u(·)V (T ) < K and hence, u(t) is continued to exist in X(T˜ ), T˜ > T , beyond T . This extended solution still satisfies the same estimate on [0, T˜ ], i.e., u(·) ˜ ≤ C(I˜2M + K α+1 ) < K. V (T )

Decay and Global Existence

259

Thus, we can conclude that the solution u(t) exists globally on [0, ∞) and satisfies the estimate u(·)V (T ) < K for all T > 0. Such a choice of K as in (7.3.14) is of course possible if I˜2M is sufficiently small. The proof of Theorem 7.3 is now complete.


8. Quasilinear wave equations 8.1. Problem and result Following [45], we consider in this section the initial-boundary value problem for the quasilinear wave equation: utt − div{σ(|∇u|2 )∇u} + a(x)ut = 0 in Ω × [0, ∞),

(8.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(8.1.2)

N where Ω is an exterior domain in the N dimensional Euclidean √ space R with a smooth boundary ∂Ω and σ(v) is a function like σ(v) = 1/ 1 + v. Concerning the dissipation a(x)ut we make two types of assumptions specified later, which are intended to make the effect of this term as weak as possible. As in the previous ˜ on the dissipation section, under two types of assumptions Hyp. A and Hyp. A a(x)ut , we will prove theorems of global existence of the problem (8.1.1)–(8.1.2). Concerning σ(·) we make the following assumptions.

Hyp. D. The function σ(·) is a differentiable function on R+ = [0, ∞) and satisfies the conditions: σ(v 2 ) ≥ k0 > 0 and σ(v 2 ) − 2|σ  (v 2 )|v 2 ≥ k0 > 0, if |v| ≤ L, where L > 0 is an arbitrarily fixed constant and k0 := k0 (L) is a positive constant (we may assume σ(0) = 1 for simplicity). The following result concerning local in time solutions is standard (cf. Kato [17]). We use the notation Im,q := ||u0 ||m,q + ||u1 ||m−1,q and Im := Im,2 . Proposition 8.1. Let m > M := [N/2] + 1 be an integer and assume that σ(·) ∈ C m+1 ([0, ∞)), a(·) ∈ C m+1 (Ω) and ∂Ω is of class C m+1 . Let the initial data (u0 , u1 ) ∈ H m+1 (Ω) × H m (Ω) satisfy the compatibility condition of order m associated with the problem (8.1.1)–(8.1.2). Then, there exists T = T (IIm ) > 0 such that the problem admits a unique solution u(t) on [0, T ) belonging to T Xm :=

m 2 k=0

C k ([0, T ); H m+1−k (Ω) ∩ H01 (Ω))

2

C m+1 ([0, T ); L2 (Ω)).

260

Mitsuhiro Nakao

∞ We set Xm = Xm . From Proposition 8.1 it suffices for the existence of global solutions in Xm to derive the a priori estimate

sup 0≤t 0, where u(t) is an assumed local solution on [0, T ). In what follows we assume that ∂Ω is sufficiently smooth, i.e., ∂Ω is of class C m+1 . Our first result in this section reads as follows: Theorem 8.1. Let N ≥ 1 be any integer and assume that σ(·) ∈ C m+1 (R+ ) and ˜ and Hyp. D, there a(·) ∈ C m+1 (Ω) with m > [N/2] + 1. Then, under Hyp. A exists δ > 0 such that if (u0 , u1 ) ∈ H m+1 (Ω) × H m (Ω) satisfy the compatibility condition of order m and the smallness condition Im := u0 H m+1 + u1 H m < δ, the problem (8.1.1)–(8.1.2) admits a unique solution u(t) in the class Xm . Further, the following estimates hold: 2 Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k ≤ CIIm (1 + t)−k−1

for 0 ≤ k ≤ m

and 2 ∇u(t)2H m−k ≤ CIIm (1 + t)−1

for 0 ≤ k ≤ m.

Remark 8.1. When Ω is a bounded domain we can consider a more delicate situation, where a(x) is localized and further degenerates on (N − 1)-dimensional submanifolds in Ω (see [42]). The results obtained by a second approach are stated separately in the cases N ≥ 4 and N = 3. Theorem 8.2. Let N ≥ 4. When N is even we assume that V is convex. Assume that σ(·) and a(·) are of class C 3M . We assume that (u0 , u1 ) belong to H 3M+1 (Ω)∩ W 2M+1,1 (Ω) × H 3M (Ω) ∩ W 2M,1 (Ω) and satisfy the compatibility conditions of the order 3M associated with the quasilinear problem (8.1.1)–(8.1.2) and also with the linear problem with σ ≡ 1. Further, we assume that a(·) satisfies Hyp. A and supp a(·) is compact. Then, under Hyp. D, there exists δ > 0 such that if I˜3M := u0 H 3M +1 + u0 W 2M +1,1 + u1 H 3M + u1 W 2M,1 ≤ δ, there exists a unique solution u(t) in the class Y3M :=

3M 2 k=0

C k ([0, ∞); H 3M+1−k (Ω) ∩ H01 )(Ω) 2

2

C 3M+1 ([0, ∞); L2 (Ω))

W k,∞ ([0, ∞); W M+1−k,∞ (Ω)),

satisfying 3M  k=0

Dtk ∇u(t)H 3M −k ≤ C I˜3M < ∞

Decay and Global Existence and

M 

261

Dtk ∇u(t)W M −k,∞ ≤ C I˜3M (1 + t)−d

k=0

with d = (N − 1)/2. More interesting is the case N = 3, where the situation is also more delicate. Theorem 8.3. Let N = 3. Assume that σ(·) and a(·) are of class C 4M+2 . We assume that (u0 , u1 ) belong to H 4M+3 (Ω) ∩ W 4M+2,q (Ω) × H 4M+2 (Ω) ∩ W 4M+1,q (Ω) and satisfy the compatibility conditions of order 4M + 2 associated with the quasilinear problem (6.1.1)–(6.1.2) and also with the linear problem with σ ≡ 1. We assume that a(·) satisfies Hyp. A and supp a(·) is compact. Then, under Hyp. D, there exists δ such that if I˜4M+2 := u0 H 4M +3 + u0 W 4M +2,q + u1 H 4M +2 + u1 W 4M +1,q ≤ δ, there exists a unique solution u(t) in the class Y4M+3 :=

4M+2 2

C k ([0, ∞); H 4M+3−k (Ω) ∩ H01 (Ω))

2

C 4M+2 ([0, ∞); L2 (Ω)),

k=0

satisfying 4M+3 

Dtk ∇u(t)H 4M +3−k ≤ C I˜4M+2 < ∞

k=0

and

M+1 

Dtk ∇u(t)W M +1−k,p ≤ C I˜4M+2 (1 + t)−d(p)

k=0

with d(p) = (p − 2)(1 − ε)/p, 0 < ε " 1, where we should take 6 ≤ p < ∞ and q = p/(p − 1). Remark 8.2. Concerning the regularity of the initial data, in Shibata-Tsutsumi [63] the data (u0 , u1 ) are required to belong to H 20 (Ω) × H 19 (Ω) if N = 3, while here we impose (u0 , u1 ) ∈ H 11 (Ω) × H 10 (Ω). 8.2. Energy decay for the quasilinear equation ˜ and Hyp. D. Let u(t) be a local solution on [0, T˜), 0 < T˜ ≤ ∞ We assume Hyp. A of the problem (8.1.1)–(8.1.2) in Proposition 8.1. In this section, on the basis of the arguments from Section 5 we first derive the L2 -boundedness and decay estimates for E(t). Next, we also derive the decay of the energy for U = ut . Proposition 8.2. There exists δ2 > 0 such that if sup (Dx2 Dt u(t)∞ + D2 u(t)∞ + Du(t)∞ ) ≤ δ2 ,

D = (∇, Dt ),

(8.2.1)

0≤t 2(1 + ν)/(1 − ν). Then, for each K > 0 there exists an open neighborhood SK of (0, 0) in H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) such that if (u0 , u1 ) ∈ SK , the problem (9.1.6), (9.1.7) admits a 2 unique solution u(t) in Xloc , satisfying utt (t) + ∇ut (t) + ∆u(t) ≤ K and u(t)2 ≤ C(K, I1 )(1 + t)ν and E(t) ≤ C(K, I1 )(1 + t)ν−1 with some constant C(K, I1 ) > 0. The conclusion is valid also for the Cauchy problem in Ω = RN , if N = 3. For the case q = 0 we can prove an existence theorem for the initial data (u0 , u1 ) ∈ H01 (Ω) × L2 (Ω). Theorem 9.3. Let 1 ≤ N ≤ 3. We assume Hyp. E with q = 0 and Hyp. F is valid with α such that 4/(N − 2ν) < α ≤ 2/(N − 2)+ , 0 ≤ ν < N/2. Then there exists δ > 0 such that if I0 < δ, the problem (9.1.6), (9.1.7) admits a unique solution u(t) ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) satisfying u(t)2 ≤ C(II0 , δ)(1 + t)ν and E(t) ≤ C(II0 , δ)(1 + t)ν−1 with some C(II0 , δ) > 0. We note that ν = |p|/(p + 2) in this situation. When N = 3, the assertion is valid also for Ω = RN . 9.2. A basic inequality Let us consider the problem with a forcing term utt − ∆u + ρ(x, ut ) = f (x, t)

in Ω × (0, ∞),

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(9.2.1) (9.2.2)

where we assume f ∈ 1,2 For a moment we assume that f ∈ Wloc ([0, ∞); L2 (Ω)) and u(t) is a solution 2 in Xloc of the problem (9.2.1), (9.2.2). L2loc ([0, ∞); L2 (Ω)).

284

Mitsuhiro Nakao

For such solutions we have familiar identities corresponding to (A), (B), and (C). Our basic inequality reads as follows: Proposition 9.1. There exists T0 > 0 such that if T > T0 and k > 0 is sufficiently large, then
t+T

t+T X(t + T ) − X(t) + k ρ(x, ut )ut dx ds + ε1 E(s) ds t t  (9.2.3)
Ω

t+T 
|ut |2 dx + |ρ(x, ut )|2 dx + (|f ||u| + |f |2 )dx ds ≤C t

ΩR

ω

Ω

with some C > 0, and ε1 > 0, where X(t) is a certain quantity equivalent to E(t) + u(t)2 . The proof is given by the almost same argument deriving (4.2.23) (see also (5.4.6)) and we give an outline of it. First, we obtain for any k > 0, 


d α ut φ(r)(x − x0 ) · ∇u dx + α ut u dx + a(x)|u|2 dx + kE(t) dt 2 ΩCR Ω Ω 

 N φ(r) + rφ (r) − α |ut |2 dx +k ρ(x, ut )ut dx + 2 Ω Ω 
 N φ(r) + rφ (r) + φ(r) |∇u|2 dx + α− 2
Ω    + φ (r)r−1 |(x − x0 ) · ∇u(t)|2 + φ(r)|∇u(t)|2 dx Ω

2 ∂u ν · φ(r)(x − x0 ) dS + ρ(x, ut )φ(r)(x − x0 ) · ∇u dx − ∂ν Ω ∂ Ω

+α ρ(x, ut )u dx = f (kut + αu + φ(r)(x − x0 ) · ∇u) dx, ΩR

Ω

(9.2.4)

where we have used the fact that due to the ‘half-linearity’:

α d ρ(x, ut )u dx = a(x)u2 dx. α C 2 dt ΩC Ω R R Now we take the function φ(r) defined in Section 4.2. We choose k ≥ N + 1 and an appropriate α > 0. Then, I1 (x) :=

k  C N φ(r) + rφ (r) − α + χ BR a(x) ≥ ε1 > 0 2 2

and I2 (x) := α −

N φ(r) + rφ (r) + φ(r) + φ (r)r > ε1 > 0 2

Decay and Global Existence

285

with some ε1 > 0. Noting ρ(x, ut )ut ≥ ε0 |ut |2 if x ∈ ΩC L and using Young’s inequality, we obtain 


α d 2 ut φ(r)(x − x0 ) · ∇u dx + α ut u dx + a(x)|u| dx + kE(t) dt 2 ΩCR Ω Ω
ε1 k ρ(x, ut )ut dx + E(t) + 4 Ω 2 2
∂u (L + |x0 |)ε0 dS ≤ 2 Γ(x0 ) ∂ν  


  2 2 2 2 |f ||u| + k |f | dx (|ρ(x, ut )| + |u| ) dx + +C ΩL

Ω

ΩL

ω

(9.2.5) for a large k. The boundary integral can be treated as usual (see Section 4) and we obtain 


d ut (φ(r)(x − x0 ) − C0 h(x)) · ∇u dx + (α + η 2 )ut u dx dt Ω Ω 

1 k ε1 + αa(x)|u|2 dx + kE(t) + ρ(x, ut )ut dx + E(t) 2 ΩCR 4 Ω 2 



 2    2 2 2 2 ≤C |u| + ρ(x, ut ) dx + |ut | dx + |f ||u| + k |f | dx . Ω

(9.2.6) The first term of the right-hand side is the most delicate one to control. But, we can show the following lemma which is proved by use of unique continuation as in Proposition 4.1. Lemma 9.1. There exists T0 > 0, independent of u, such that if T > T0 , then the inequality
t+T
|u|2 dx ds t ΩL 

t+T 

t+T   2 2 2 |ρ(x, ut )| + |f | dx + |ut | dx ds + ε E(s) ds ≤ Cε (T ) Ω

t

C ∪ω BL

t

holds for any fixed number ε > 0. Now we set X(t) =


ut (φ(r)(x − x0 ) − C0 h(x)) · ∇u dx + Ω
1 αa(x)|u|2 dx + kE(t). + 2 ΩCR


(α + η 2 )ut u dx

(9.2.7)

Ω

Then we arrive at the basic inequality (9.2.3), where we have replaced the notations k/8 and ε1 /4 by k and ε1 , respectively. It remains to show that X(t) is equivalent to E(t) + ||u(t)||2 . But, we have:

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Lemma 9.2. For a large k > 0 there exist constants C1 > 0 and C2 > 0 such that for any t ≥ 0, C1 (E(t) + ||u(t)||2 ) ≤ X(t) ≤ C2 (E(t) + ||u(t)||2 ).

(9.2.8)

Proof. Since the second inequality of (9.2.8) holds trivially, it is sufficient to show the first inequality. We easily see,
ut (φ(r)(x − x0 ) − C0 h(x)) · ∇u dx −CE(t) ≤ Ω

with some constant C > 0. For any ε > 0 we have


(α + η 2 )ut u dx ≥ −ε |u|2 dx − C(ε) |ut |2 dx, Ω

Ω

Ω

and by the assumption a(x) ≥ ε0 > 0 in

1 αε0 2 αa(x)|u| dx ≥ |u|2 dx. 2 BRC 2 ΩCL ΩC L,

Finally, we note that Poincar´ ´e’s inequality  


2 2 2 |u| dx ≤ C |∇u| dx ≤ C |∇u| dx ΩL

ΩL

Ω

holds due to the boundary condition u|∂Ω = 0. When Ω = RN this should be replaced by the inequality 


2 2 2 |u| dx ≤ C |∇u| dx + |u| dx . BL

Ω2L

B2L \BL

Combining these we obtain the first inequality.




9.3. Proof of Theorem 9.1 For convenience we set Ω1R (t) = {x ∈ ΩR | |ut (x, t)| ≤ 1} and Ω2R (t) = ΩR \ Ω1R . 2 of the problem (9.1.1)–(9.1.2). Then, by the Let u(t) be a solution from Xloc Proposition 9.1 with f ≡ 0 it holds
t+T

t+T X(t + T ) + k ρ(x, ut )ut dx ds + ε1 E(s) ds t  Ω t (9.3.1)  0 t+T 0 0 2 2 ≤ X(t) + C t ω |ut | dx + ΩR |ρ(x, ut )| dx ds

for a large k > 0 and some ε1 > 0. To consider the integrals of the right-hand side of (9.3.1) we first note that by the assumption on ρ(x, v) and the energy identity, 
∞ 

p+2 q+2 E(t) + a(x)|ut | dx + a(x)|ut | dx dt ≤ CE(0) < ∞, 0 ≤ t. 0

Ω1R

Ω2R

(9.3.2)

Decay and Global Existence We set


t



t
|ut | dx ds =: I1 (t) and

|ρ(x, ut )|2 dx ds =: I2 (t).

2

0

287

0

ω

ΩR

We shall estimate I1 (t) and I2 (t) separately in the following four cases: Case (1): 0 ≤ p < ∞ and 0 ≤ q ≤ 2/(N − 2). In this case, by Holder’s ¨ inequality and (9.3.2), we have 2 

 p+2 

t

I1 (t) ≤ C 0
t

p  p+2

t

Ω1R

a(x)|ut |p+2 dx ds

dx ds 0


+C a(x)|ut |q+2 dx ds 2   0 ΩR ≤ C E(0)2/(p+2) (1 + t)p/(p+2) + E(0) .

Ω1R

To treat I2 (t) we assume for a moment N ≥ 3. By use of Sobolev’s inequality,
t
2 I2 (t) ≤ k1 a(x)|ut |2 dx ds 0 ΩR
t

t
2 2 2(p+1) 2 +k1 a(x) |ut | dx ds + k1 a(x)2 |ut |2(q+1) dx ds 0

Ω1R

0

Ω2


Rt
 ≤ C E(0)2/(p+2) (1 + t)p/(p+2) + E(0) + C 

0 2(q+1)(1−θ)/(q+2)

Ω1R

a(x)|ut |p+2 dx ds


t
2(q+1)θ +C a(x)|ut |q+2 dx ||ut ||2N/(N −2)(ΩL ) ds 2 ΩR   0 2/(p+2) (1 + t)p/(p+2) + E(0) ≤ C E(0) 2(q+1)(1−θ)/(q+2) 
t
a(x)|ut |q+2 dx ds +C(K02 + E(0))(q+1)θ 0

Ω2R


t  2θ(q+1)−q q+2 · ds  0  ≤ C E(0)2/(p+2) (1 + t)p/(p+2) + E(0) +C(K02 + E(0))(q+1)θ E(0)

2(q+1)(1−θ) q+2

q(N −2)

(1 + t) 4−(N −2)q

with θ = N q/(q + 1)(4 − (N − 2)q), where we have used 0 ≤ q ≤ 2/(N − 2)+ . Thus we obtain I1 (t) + I2 (t) ≤ CQ0 (t + 1)ν

(9.3.3)

with Q0 , ν defined in Theorem 9.1. We can show similar estimates for N = 1, 2 with some trivial modification.

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Mitsuhiro Nakao

Case (2): −1 < p ≤ 0 and 0 ≤ q ≤ 2/(N − 2). In this case we get
t

t
I1 (t) ≤ a(x)|ut |p+2 dx ds + Ω1R

0

and

0


t


a(x)|ut |q+2 dx ds ≤ CE(0),

Ω2R


t


I2 (t) ≤ C

a(x)|ut | dx ds + C 2

0

0

ΩR

+C(K02 + E(0))(q+1)θ E(0)

Ω1R

2(q+1)(1−θ) q+2

a(x)|ut |2(p+1) dx ds q(N −2)

(1 + t) 4−(N −2)q ,

N ≥ 3.

Here, we see that
t
Ω1R

0

a(x)|ut |2(p+1) dx ds


t


≤C

2(p+1)/(p+2) a(x)|ut |

p+2

0

Ω1R

dx ds

(1 + t)−p/(p+2) .

Hence we have again I1 (t) + I2 (t) ≤ CQ0 (1 + t)ν . Some modifications give the results for the cases N = 1, 2.

(9.3.4)

Case (3): 0 ≤ p < ∞ and −1 < q ≤ 0. Let N ≥ 3. Then we have, by use of H¨ ¨older’s inequality,
t
a(x)|ut |2 dx ds 0

≤

Ω2R

C(K02

+ E(0))

−N q 4−(N −2)q

a(x)|ut |

q+2

0

and

Ω2R


t
0

Ω2R

 4−(N4−2)q


t
dx ds

−(N −2)+

(1 + t) 4−(N −2)q ,


t
a(x)2 |ut |2(q+1) dx ds ≤ C 0

Ω2R

a(x)|ut |q+2 dx ds ≤ CE(0).

So we obtain the estimate as (9.3.4) with Q0 , ν as in Theorem 9.1. Case (4): −1 < p ≤ 0 and −1 < q ≤ 0. Utilizing the arguments in the cases (2), (3) above, we can conclude the same type of an inequality as (9.3.4). Now we are ready to complete the proof of Theorem 9.1. First we note that if 0 ≤ t ≤ T , then by (9.3.1) with f = 0 and the same estimations for I1 (t), I2 (t) X(t) ≤ X(0) + CQ0 ,

0 ≤ t ≤ T,

and also X(t) ≤ X(nT ) + CQ0 ,

nT < t < (n + 1)T.

Decay and Global Existence

289

Taking these into account we can show by (9.3.1) and the estimates for I1 (t)+II2 (t) that (9.3.4) X(t) ≤ X(0) + CQ0 (1 + t)ν . In particular we see that ||u(t)||2 ≤ C(Q0 + u0 2 )(1 + t)ν . The differential inequality d/dt((1+t)E(t)) = E(t)+(1+t)d/dtE(t) ≤ E(t) implies
t (1 + t)E(t) ≤ E(0) + E(s)ds. 0

But, from (9.3.1) and (9.3.4) we see that
t E(s)ds ≤ CX(0) + CQ0 (1 + t)ν 0

for any t > 0. Hence, we obtain E(t) ≤ CX(0)(1 + t)−1 + CQ0 (1 + t)ν−1 ≤ C(Q0 + u0 2 )(1 + t)ν−1 .


The proof of Theorem 9.1 is complete. 9.4. Proof of Theorems 9.2 and 9.3

Let (u0 , u1 ) ∈ H2 (Ω) ∩ H01 (Ω) × H01 (Ω) and let u(t) be the local (in time) solution, say, in X2 (T˜), T˜ > 0. For the proof of Theorem 9.2 we assume for a moment u(t)2 ≤ δ02 (1 + t)ν ,

E(t) ≤ δ02 (1 + t)ν−1

(9.4.1)

and ∇ut (t) + ∆u(t) ≤ K

(9.4.2)

for 0 ≤ t < T˜ . Under the assumptions (9.4.1), (9.4.2) we shall derive the a priori estimates   ˜ 0 + u0 2 )(1 + t)ν , ˜ 0 + u0 2 (1 + t)ν−1 (9.4.3) E(t) ≤ C Q u(t)2 ≤ C(Q and ˜1 ∇ut (t) + ∆u(t) ≤ Q

(9.4.4)

˜ 0 will be defined by Q0 with K0 replaced by K and for 0 ≤ t < T˜ , where Q ˜ ˜ I0 , I1 , K) = 0. By a ˜ Q1 = Q1 (II0 , I1 , K) will be a quantity such that limI0 ,I1 →0 Q(I ˜ 0 +u0 2 ) < δ 2 standard argument, the estimates (9.4.3), (9.4.4) assure that if C(Q 0 ˜ 1 < K, then the local solution u(t) exists in fact on the infinite interval [0, ∞) and Q and satisfies the estimates on [0, ∞). These conditions will be satisfied if I0 and I1 are small. Then, we define SK by the set of initial data satisfying all of these conditions. Without loss of generality we may assume T˜ > T0 (otherwise, the

290

Mitsuhiro Nakao

estimates can be proved much more easily). We can use the basic inequality in Proposition 9.1 with f = f (x, u) to get
t+T
t+T
ρ(x, ut )ut dx ds + ε1 E(s) ds X(t + T ) − X(t) + k t t Ω 
 


t+T |ut |2 dx + |ρ(x, ut )|2 dx ds ≤C (9.4.5) t ω ΩR 

t+T

(|f (x, u)||u| + |f (x, u)|2 ) dx ds .

+ t

Ω

Here, when 0 < α ≤ 4/(N − 2)+ we have by the Gagliardo-Nirenberg inequality,
t+T

t+T
|f (x, u)||u| dx ds ≤ C |u|α+2 dx ds (9.4.6) t t Ω Ω
t+T ≤C u(α+2)(1−θ0 ) ∇u(α+2)θ0 ds t
t+T α ≤ Cδ0 (1 + t)2˜µ E(s)ds t

˜ = (α + 2)(1 − θ0 )ν + ((α + 2)θ0 − 2)(ν − 1). with θ0 = αN/2(α + 2) and µ We note that (α + 2)θ0 = αN/2 ≥ 2 by the assumption on α. Further, we confirm that µ ˜ ≤ 0 is equivalent to θ0 ≥ (να + 2)/(α + 2), that is, α ≥ 4/(N − 2ν) which is also fulfilled under our assumption on α. Thus, we have
t+T

t+T α |f (x, u)||u| dx ds ≤ Cδ0 E(s) ds. t

Ω

t

Similarly, if 0 < α ≤ 2/(N − 2)+ , then
t+T

|f (x, u)|2 dx ds ≤ Cδ02α t

Ω

t+T

E(s) ds.

t

Thus, when 0 < α ≤ 2/(N − 2)+ we make the assumption ε1 C(δ0α + δ02α ) ≤ . 4

(9.4.7)

When N ≥ 3 and 4/(N − 2)+ < α ≤ 2/(N − 4)+ we have, instead of (9.4.6),
t+T

t+T
|f (x, u)||u| dx ds ≤ C |u|α+2 dx ds t t Ω Ω
t+T (α+2)(1−θ ) u2N/(N −2) 1 D2 u(α+2)θ1 ds ≤C (9.4.8)
t t+T ≤C ∇u(α+2))(1−θ1 ) (∇u + ∆u)(α+2)θ1 ds t
t+T (α+2)(1−θ1 )−2 ≤ Cδ0 (K + δ0 )(α+2)θ1 E(s) ds t

Decay and Global Existence

291

with θ1 = (α(N −2)−4)/2(α+2), where we have used the fact (α+2)(1−θ1 )−2 ≥ 0. Similarly, if 2/(N − 2)+ < α ≤ 2/(N − 4)+ , then
t+T
t+T
2(α+1)(1−θ˜1 )−2 2 2(α+1)θ˜1 |f (x, u)| dx ds ≤ Cδ0 (K + δ0 ) E(s) ds Ω

t

t

with θ˜1 = (α(N − 2) − 2)/2(α + 1). When 2/(N − 2)+ < α ≤ 2/(N − 4)+ we assume  ε  ˜ 1 2(α+1)(1−θ˜1 )−2 (9.4.9) (K + δ0 )2(α+1)θ1 < . C δ0α + δ0 4 Under the assumption (9.4.7) or (9.4.9) we obtain from (9.4.5)
t+T

t+T 1 X(t + T ) − X(t) + k ρ(x, ut )ut dx ds + ε1 E(s) ds 2 t t Ω 
t+T 

2 2 ≤C |ut | dx + |ρ(x, ut )| dx ds. t

ΩL

ω

This inequality is essentially the same as (9.3.1) and the proof of Theorem 9.1 yields the same estimates (9.1.9) for u(t) and E(t) with K0 replaced by K. Further, the desired inequality   ˜ 0 + u0 2 < δ0 C Q will be satisfied if I0 is small. In what follows we assume 1 ≤ N ≤ 3. We shall derive the estimate for the second derivatives of u. Differentiating the equation we have (9.4.11) uttt − ∆ut + ρv (x, ut )utt − βu (x, u)ut = 0. From this we have
d (utt (t)2 + ∇ut (t)2 ) ≤ C |u|α |ut ||utt | dx dt Ω 
1/2
≤C |u|2α |ut |2 dx utt .

(9.4.12)

Ω

Here, 



1/2 α/2 |u| |ut | dx ≤ Cuα ∇uα/2 ut  if N = 1, ∞ ut  ≤ Cu 2α

Ω

2

1/2 |u|2α |ut |2 dx ≤ Cuα 2aα ut 2b ,



Ω

≤ Cu

α(1−θ2 )

∇u

a−1 + b−1 = 1, αθ2

ut 

1−θ˜2

a

∇ut 

with θ2 = 1 − 1/aα, θ˜2 = 1 − 1/b = 1/a " 1, and 1/2 

|u|2α |ut |2 dx ≤ Cuα ∞ ut  Ω

≤ C∇uα/2 (∆u + ∇u)α/2 ut  if N = 3.

θ˜2

1, if N = 2

292

Mitsuhiro Nakao

Thus,

⎧ 2 (α+1)/2 ˜ ⎪ (1 + t)−µ ⎨K(Q0 + u0  ) α 1+ε 2 (α+1−ε)/2 ˜ 0 + u0  ) C |u| |ut ||utt | dx ≤ K (Q (1 + t)−µ ⎪ α+2 Ω ⎩ ˜ 0 + u0 2 ) 4 (1 + t)−µ (K + I0 )1+α/2 (Q


where µ is defined by ⎧ 1 ⎪ ⎨ 2 (−(α + 1)ν + α/2 + 1) µ = 12 (−(α + 1)ν + α + 1 − ε(2 − ν)) ⎪1 ⎩ 2 (−(α/2 + 1)ν + α/2 + 1)

if N = 1, if N = 2, ε " 1, if N = 3,

if N = 1, if N = 2, if N = 3.

We note that the condition µ > 1 is equivalent to α > 2(1 + ν)/(1 − 2ν) if N = 1, α > (1 + ν)/(1 − ν) if N = 2, α > 2(1 + ν)/(1 − ν) if N = 3, and these conditions are satisfied under our assumptions on α. Now, we obtain from (9.4.12) that if N = 3, then utt (t)2 + ∇ut (t)2

 (α+2)/4 ˜ 0 + u0 2 ≤ utt (0)2 + ∇ut (0)2 + C(µ − 1)−1 (K + I0 )1+α/2 Q . (9.4.13)

Similar results hold also for N = 1, 2. Thus, utt (t)2 + ∇ut (t)2

 γ ˜ 0 + u0 2 ≤ utt (0)2 + ∇ut (0)2 + C(K, I0 )(µ − 1)−1 Q

(9.4.14)

˜ 0 (E(0), K) is the constant Q0 with K0 ˜0 = Q with some γ > 0. Recall that Q ˜ 0 (0, K) = 0. We easily replaced by K defined in the Theorem 9.1 and satisfies Q see utt (0) ≤ ∆u0  + ρ(x, u1 ) + f (x, u0 ) −

−

≤ C(I1 + I0 + u1 (q+1)(1−θ2 ) ∇u1 (q+1)θ2 + u1 1+p + u1 1+q ), (9.4.15) with the notation r− := min{r, 0} and a certain 0 ≤ θ2 ≤ 1, where we have used Gagliardo-Nirenberg inequality and the assumption (9.4.15). From (9.4.14) and (9.4.15) we have utt (t) + ∇ut (t)   (q+1)(1−θ2 ) (q+1)θ2 (1+p− ) (1+q− ) I1 + I0 + I0 ≤ C I1 + I0 + I0  γ ˜ 0 + u0 2 =: Q1 (II0 , I1 , K). +C(K, I0 )(µ − 1)−1/2 Q

(9.4.16)

Decay and Global Existence

293

Similarly as in (9.4.15),   − − ρ(x, ut ) ≤ C I0 + ut (q+1)(1−θ2 ) ∇ut (q+1)θ2 + ut 1+p + ut 1+q   − − (q+1)(1−θ2 ) (q+1)θ2 ˆ 1 (II0 , I1 , K). =: Q Q1 + I01+p + I01+q ≤ C I0 + I0 (9.4.17) We also note that β(x, u) ≤ C(u0 2 +u1 2 ) under our assumption (see (9.4.22) below). Thus, ∆u(t) ≤ utt (t) + ρ(x, t) + f (x, u) ˆ 1 (II0 , I1 , K)) =: Q ˜ 1 (II0 , I1 , K). ≤ C(Q1 (II0 , I1 , K) + Q

(9.4.18)

We conclude that ˜ 1 (II0 , I1 , K), ∇ut (t) + ∆u(t) ≤ Q (9.4.19) ˜ I0 , I1 , K). Since ˜ I0 , I1 , K) for C Q(I where for simplicity we use the same notation Q(I ˜ limI0 →0,I1 →0 Q1 (II0 , I1 , K) = 0 we see that if I0 , I1 are small, then ˜ 1 (II0 , I1 , K) < K Q which is the desired second condition on the initial data. The proof of Theorem 9.2 is now complete.
Finally, we give the proof of Theorem 9.3. This is in fact essentially included in the first part of the proof of Theorem 9.2. Let q = 0 and let u(t) be a local in time solution in C 1 ([0, T˜ ); L2 (Ω)) ∩ C([0, T˜); H01 (Ω)), T˜ > 0. Assume for a moment u(t)2 ≤ δ02 (1 + t)ν ,

E(t) ≤ δ02 (1 + t)ν−1 , 0 < t < T˜.

(9.4.20)

Then as it is already shown, under the assumption (9.4.7), the estimates (9.4.3) ˜ 0 and ν are given by hold, where Q0 = Q   ˜ 0 = C E(0) + E(0)min{2/(p+2),1} and ν = |p|/(p + 2). (9.4.21) Q Thus, in addition to (9.4.7), we make the assumption ˜ 0 + u0 ) < δ02 . C(Q

(9.4.22)

Then we can conclude that the local solution u(t) is in fact global in time and the estimates (9.4.3) hold on [0, ∞).


10. Some open problems We give here some open problems related to the topics discussed in this article. 1. Consider utt − u + ρ(x, ut ) = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

294

Mitsuhiro Nakao

where Ω is an exterior domain. When ρ(x, ut ) = a(x)|ut |r ut or ρ(x, ut ) = a(x)(ut + |ut |r ut ), derive some decay rate of local energy, where a(x) ≥ 0 is a localized function on ω. 2. Consider utt − u + a(x)ut = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, where a(x) is localized near ω and near infinity. Under appropriate assumptions on (u0 , u1 ), say, (u0 , u1 ) ∈ H01 (Ω) ∩ Lp (Ω) × L2 (Ω) ∩ Lp (Ω), 1 ≤ p < 2, derive some sharper decay rate of total energy like E(t) ≤ C(1 + t)−1−γ , γ > 0. For the case a(x) ≡ 1 see Ono [53]. 3. Consider utt − u + a(x)ut = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, where V ≡ R \ Ω consists of some convex bodies as in Ikawa [12], [13] and a(x) is localized near infinity. By utilizing the result in [13], derive a decay rate of the total energy under the assumption on a(x) including the case, where a(x) = 0 near the boundary ∂Ω and a(x) ≥ ε0 > 0 near infinity. N

4. Consider utt − u + ρ(x, ut ) = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, where ρ(x, ut ) is a half-linear localized dissipation treated in the final section. Derive a sharper decay estimate for E(t) under additional assumptions on the data (u0 , u1 ). Further, derive a decay estimate for the second-order derivatives D2 u(t), D = (∇, Dt ), and apply it to semilinear wave equations with source term f (u). 5. Consider the periodicity problem utt − u + ρ(x, ut ) = f (x, t) in Ω × [0, ∞), u(x, t + T ) = u(x, t) and u|∂Ω = 0, where f (t, x) is a force term with period T in time. Discuss the existence and stability of periodic solutions. For related results in the case of bounded domains see [46]. 6. Discuss the global existence of small data solutions to the problem utt − u + a(x)ut = |ut |p−1 ut in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, ˜ under the hypothesis Hyp. A or under the hypothesis Hyp. A.

Decay and Global Existence

295

7. Consider the equation with Neumann boundary condition: utt − u + a(x)ut = 0 in Ω × [0, ∞), ∂ u|∂Ω = 0, ∂n where a(x) is localized near ω. Discuss the local energy decay for this problem. For the problem with Neumann boundary condition the standard multiplier technique does not work well and we must employ another method, e.g. microlocal analysis. For such a technique see Bardos, Lebeau and Rauch [3]. Generally speaking, it is a very challenging problem how to combine multiplier technique (energy method) with microlocal analysis for the study of the wave equation with some Neumann type boundary conditions. A problem concerning a Neumann type boundary dissipation in a bounded domain is treated in Lasiecka and Triggiani [24], where energy method and microlocal analysis are well combined. See also [2] for an exterior problem. See Shibata and Zheng [64] for general fully nonlinear hyperbolic equations with a Neumann type dissipation in a bounded domain, where spectral analysis for the linearized stationary problem plays an essential role. u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and

Acknowledgement. The author would like to thank Professor M. Reissig for his kind invitation to this contribution and for many helpful suggestions during the preparation of the manuscript.

References [1] L. Aloui and M. Khenissi, Stabilization for the wave equation on exterior domains, Carleman Estimates and Applications to Uniqueness and Control Theory, F. Colombini and C. Zuily ed., Birkhauser ¨ (2001), 1–13. [2] J. J. Bae and M. Nakao, Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains, Dis. Cont. Dyn. Syst., to appear. [3] G. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), 1024–1065 [4] C. O. Bloom and N. D. Kazarinoff, Local energy decay for a class of the non starshaped bodies, Arch. Ration. Mech. Anal. 55 (1975), 73–85. [5] P. Brenner, On Lp − L
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[46] M. Nakao, Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation, J. Differential Equations 190 (2003), 81–107. [47] M. Nakao and Il Hyo Jung, Energy decay for the wave equation with a half-linear dissipation in exterior domains, Differential Integral Equations 16 (2003), 927–948. [48] M. Nakao and K. Ono, Global existence to the Cauchy problem for the semilinear dissipative wave equations, Math. Z. 214 (1993), 325–342. [49] M. Nakao and K. Ono, Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation, Funkcial. Ekvac. 38 (1995), 417–431. [50] T. Narazaki, Lp − Lq estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan 56 (2004), 587–626 [51] K. Nishihara, Lp − Lq estimates of solutions to the damped wave equation in 3dimensional space and their application, Math. Z. 244 (2003), 631–649. [52] K. Ono, The time decay to the Cauchy problem for semilinear dissipative wave equations, Adv. Math. Sci. Appl. 9 (1999), 243–262. [53] K. Ono, Decay estimates for dissipative wave equations in exterior domains, Math. Anal. Appl. 286 (2003), 540–562.

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[66] D. Tataru, The Xθs spaces and unique continuation for solutions to the semilinear wave equation, Comm. Partial Differential Equations 21 (1996), 841–887. [67] L. R. Tcheugou´ ´e T´ ´ebou, Stabilization of the wave equation with localized nonlinear damping, J. Differential Equations 145 (1998), 502–524. [68] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2000), 464–489. [69] T. Yamazaki, Global solvability for quasilinear hyperbolic equation of Kirchhoff type in exterior domains of dimension larger than three, Math. Methods Appl. Sci., to appear. [70] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations 15 (1990), 205–235. [71] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl. 70 (1991), 513–529. Mitsuhiro Nakao Graduate School of Mathematics Kyushu University Ropponmatsu Fukuoka 810-8560, Japan e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 159, 301–385 c 2005 Birkhauser
¨ Verlag Basel/Switzerland

Global Existence in the Cauchy Problem for Nonlinear Wave Equations with Variable Speed of Propagation Karen Yagdjian Abstract. We consider nonlinear wave equations with variable coefficients. Special attention is devoted to the parametric resonance phenomena. Mathematics Subject Classification (2000). Primary 35L70; Secondary 58J45. Keywords. Nonlinear wave equation, global existence, variable coefficients, parametric resonance.

1. Introduction This paper is aimed to give a collection of some results for the existence and nonexistence of global solutions to the Cauchy problem for hyperbolic equations with variable coefficients. Developed in the last quarter of the last century the theory of Fourier Integral Operators allows to construct locally in time a parametrix and fundamental solution to the Cauchy problem for the linear strictly hyperbolic operators with coefficients depending both on time and on the spatial variables. Moreover, some specific classes of those operators are used for the operators with characteristics of variable multiplicity. In [37], [38], [39], [40] one can find the theory developed for two specific classes of the operators with the basic types of the behavior of the characteristics. The main ingredients of that theory are: the splitting of the cotangent space into so-called “pseudo-differential” ff and “hyperbolic zones”, the investigation of the complete symbol of the operator in both zones, Levi conditions (conditions on the lower order terms of the operator), developing calculus of special classes of symbols, construction of exponents of pseudo-differential operators, and the investigation of the corresponding Hamiltonian vector fields.

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It turns out that the basic concepts and technique of the above-mentioned theory developed for equations with multiple characteristics work out in the investigation of the existence of a global solution to the Cauchy problem for some classes of second-order hyperbolic operators with variable coefficients. In particular, if the coefficients of the equation depend on time alone, then the phase function exists globally that allows to get global in time results, like Lp − Lq decay estimates, for second-order linear equations. The papers [28], [29], give several results based on this approach. Moreover these basic concepts and techniques provide us with a key to study global existence results for nonlinear equations. In this paper we describe only few of them, which can be exposited in transparent way highlighting basic links between operators with multiple characteristics and the problem of the existence of global solutions for the strictly hyperbolic equations. We pay special attention to equations with oscillating coefficients. In the theory of linear weakly hyperbolic equations too fast oscillations break down the local well-posedness of the Cauchy problem. Corresponding phenomena in the unbounded in time problem for semi-linear wave equations lead to the blow-up of the solution. In [43] it is detected that in bottom of these two phenomena lies the well-known “parametric resonance” of Physics. To bear in mind the extraordinary importance of the parametric resonance for many branches of the applied sciences we give here a detailed proof of Borg’s theorem, and track down the mathematical consequences of the parametric resonance for nonlinear wave equations. For the sake of a self-contained representation we first describe in Section 2 and Section 3 well-known results for the wave equation and then generalize them to operators with variable coefficients. To find out the influence of variable coefficients on the global existence in the Cauchy problem for the wave equation we choose the model equation ∇x u|2 ) = 0 utt − a2 (t, x)∆u + f (u)(u2t − a2 (t, x)|∇ that in the case of constant function a(t, x) = const > 0 is the wave map type equation studied in [26]. The wave map equations appear in physics and are important also for other applications. A good introduction on wave map equations with comprehensive references can be ffound in [33]. In Section 4 we consider an equation with periodic, non-constant, smooth, and positive coefficient a = a(t). The proof of the nonexistence of the global in time solution for that equation (see Theorem 4.1) is given in Section 5. The condition on the function f = f (u) that is assumed through the paper, is discussed in Theorem 4.2. After studying the periodic case we turn in Section 6 to the equation with a coefficient stabilizing to a periodic one. Namely we consider the model equation with a(t) = exp(tα )b(t), t ∈ [1, ∞), for α < 0. Here b = b(t) is defined on R, a periodic, non-constant, smooth, and positive function. We study the global solvability of the Cauchy problem with data prescribed on t = 1. In Section 7 we prove the main result of Section 6, Theorem 6.1, that shows that oscillations, which approach for large time the pure periodic behavior (α < −1), in general

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break the global existence. The global existence for small data for the equation with fast oscillating coefficients, that is for the case α ∈ [1/2, ∞), is proved in Section 11. In Section 8 we give an example of the influence of the decay of the coefficient a(t, x) on the global existence of the solution of the model equation with a(t) = t−2 exp(−t−1 ). Namely, for arbitrary small initial data we construct blowing up solution. That proves Theorem 8.1, the main result of that section. It is easy to see that this equation has a permanently restricted domain of influence. In Section 9 we consider the model equation with increasing coefficient a(t, x) = tl , l > 0. According to Theorem 9.1 for the Cauchy problem for this equation small data solutions exist globally. In Section 10 we show that the approach of the previous sections works out also for the model equation with the exponentially growing coefficient a(t, x) = exp(t). Namely, due to Theorem 10.1 small data solutions exist globally. We try to make the proofs elementary, so that they can be read without special background in the theory of hyperbolic equations. We intentionally make sections independent from each other and hope that it makes them more convenient to use for lectures. This also explains some reiterations in the formulas. In fact this paper basically is a fragment of the lecture notes given at the University of Tsukuba from April 1998 till March 2000. I am very grateful to the editors of this volume published by Birkhauser, ¨ Prof. Michael Reissig and Prof. Bert-Wolfgang Schulze, not only for the invitation to write a contribution, but also for the allowing to keep this way of writing.

2. Counterexamples to the global existence Example 1 (Nirenberg’s Example [21]). The following one is an example of an equation without global classical solution to the Cauchy problem with large initial data in R3 : utt − ∆u = |∇ ∇x u|2 − |ut |2 , u(0, x) = 0,

ut (0, x) = ϕ1 (x) ,

x ∈ R3 ,

where further conditions on ϕ1 are given below. If u = u(t, x) is a solution to this Cauchy problem, then the function v = v(t, x) defined by v(t, x) = eu(t,x) solves the following Cauchy problem for the wave equation: vtt − ∆v = 0 , v(0, x) = 1, vt (0, x) = ϕ1 (x) ,

x ∈ R3 .

The solution v is unique and there is an explicit representation for v:
t v(t, x) = 1 + ϕ1 (x + ty)ds . 4π |y|=1

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Karen Yagdjian

If v > 0, then one obtains



t u(t, x) = log 1 + 4π




|y |=1

ϕ1 (x + ty)ds

.

For every given positive t0 and point x0 ∈ R3 one can find a function ϕ1 (x) ∈ C0∞ (R3 ) such that the solution u(t, x) develops singularity not later than at time t0 at the point x0 . Indeed, ϕ1 has only to satisfy the following relation:
t0 ϕ1 (x0 + t0 y)ds = −1 . 4π |y|=1 On the other hand it is easily seen that if ϕ1 (z) = O(|z|−1 ) as |z| → ∞, ∇ϕ1 1 < 4π, ϕ1 ∞ < 1, then u is defined globally, that is for all x ∈ R3 and all t ∈ R. To prove the last statement we note that for “large time”, t > 1, there is some decay of the varying part v(t, x) − 1 of the solution v = v(t, x) to the wave equation. Namely, the L∞ -norm of that part satisfies the following estimates:

∞ t d |v(t, x) − 1| = ϕ1 (x + τ y)dτ ds 4π |y|=1 t dτ

∞ t = y · (∇ϕ1 )(x + τ y)dτ ds 4π |y|=1 t

z  t ∞ ds = · (∇ϕ1 )(x + z) 2 dτ 4π t τ |z |=τ τ
∞
t 1 ≤ dτ |(∇ϕ1 )(x + z)|ds 2 4π t τ |z |=τ
1 = |∇ϕ1 (x + z)|dz 4πt |z|≥t 1 1 ∇ϕ1 1 ≤ < 1 . ≤ 4πt t Analogously for t < −1. For a small time t, |t| ≤ 1, that difference vanishes at t = 0, and we have t
ϕ1 (x + ty)ds ≤ |t|ϕ1 ∞ < 1 . |v(t, x) − 1| = 4π |y|=1 Therefore, v(t, x) is positive for all (t, x) ∈ R × R3 and u = u(t, x) is defined globally in R × R3 . From now on through this paper “blow-up” means the nonexistence of a global in time classical solution u ∈ C 2 ([0, ∞) × Rn ) to the Cauchy problem under consideration.

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Example 2. Assume that u = u(x, t) is a smooth solution of utt − ∆u = |u|p on [0, T ] × Rn with 1 < p < (n + 1)/(n − 1), and ut (x, 0) = g(x).

u(x, 0) = f (x) ,

Suppose that supp f , supp g ⊂ { x ∈ Rn ; |x| ≤ R }. By the domain of dependence property, supp u(t, ·) ⊂ { x ∈ Rn ; |x| ≤ R + t }. By integrating the equation with respect to spatial variables we obtain

(utt (t, x) − ∆u(t, x)) dx = |u(t, x)|p dx . Rn

Rn

On the other hand the divergence theorem gives
∆u(t, x) dx = 0 , Rn

while d2 dt2 Hence, d2 dt2





u(t, x) dx = Rn

Rn

∂ 2 u(t, x) dx . ∂t2





u(t, x) dx = Rn

Rn

Let

|u(t, x)|p dx .


u(t, x) dx ,

F (t) = Rn

then


F¨ (t) = Rn

|u(t, x)|p dx .

Using the compact support of u(t, ·) and Hlder’s inequality we get with 1/p+1/q = 1, p/q = p − 1, τn the volume of the unit ball in Rn ,
p
p u(t, x) dx n u(t, x) dx = R |x|≤R+t 
 
p/q ≤ 



|x|≤R+t

= Rn

=

|u(t, x)|p dx

1 dx |x|≤R+t

 |u(t, x)|p dx τn (R + t)np/q

F¨ (t)ττn (R + t)n(p−1) .

Thus we have obtained the following differential inequality: F¨ (t) ≥ τn−1 (R + t)−n(p−1) |F (t)|p

(2.1)

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Karen Yagdjian

for all t in the interval [0, T ]. In particular, F¨ (t) ≥ 0, and so F (t) ≥ F˙ (0)t + F (0). Now

˙ F (0) = ut (0, x) dx = g(x) dx ≡ Cg . Rn

Rn

If Cg > 0, then F (t) ≥ (pos. const.)t ,

t

large .

(2.2)

The next lemma shows that any function satisfying (2.1) and (2.2) cannot remain finite if 1 < p < (n + 1)/(n − 1). Hence, T < ∞. This is a special case of Kato’s theorem [20]. Lemma 2.1. Suppose F (t) ∈ C 2 ([a, b)), and for a ≤ t < b, F (t) ≥ C0 (k + t)r , F¨ (t) ≥ C1 (k + t)−q F (t)p , where C0 , C1 , and k are the positive numbers. If p > 1, r ≥ 1, and (p− 1)r > q − 2, then b must be finite. Proof. By the hypotheses of the lemma we get F¨ (t) ≥ C1 (k + t)−q F (t)p ≥ C1 (k + t)−q C0p (k + t)pr ≥ C(k + t)pr−q on [a, b). Upon integration, one has
F˙ (t) − F˙ (a) ≥ C

t

(k + s)pr−q ds . a

But pr − q ≥ −1 and so the last inequality implies that unless b is finite, F˙ (t) must be positive for t sufficiently large. Thus, one may assume that there exists an a0 such that a < a0 < b and F˙ (t) > 0 for all t ∈ [a0 , b) . By interpolating with θ ∈ (0, 1) between the assumed inequalities, one has F¨ (t)

≥ C1 (k + t)−q F (t)θp+p(1−θ) ≥ C(k + t)−q F (t)θp (k + t)rp(1−θ) = C(k + t)rp(1−θ)−q F (t)θp .

Let α = θp and β = q − rp(1 − θ). Since (p − 1)r > q − 2 and r ≥ 1, pr − q > r − 2 ≥ −1 , Then one can choose q−2 1 1) F (a0 )(1−α)/2 − F (t)(1−α)/2
C(α − 1) t ≥ (k + s)−β/2 ds 2 a0   1 C(α − 1) (k + t)1−β/2 − (k + a0 )1−β/2 = 2 1 − β/2 for all t ∈ [a0 , b). Since 1 − β/2 > 0, it is clear that t cannot be arbitrary large. Indeed, the left-hand side is bounded by F (a0 )(1−α)/2 while the right-hand side tends to ∞ as t → ∞.


308

Karen Yagdjian

It is easily seen that this lemma can be proved also by Bihari’s inequality [5]. Thus there is the blow-up of solution u ∈ C 2 ([0, T ) × Rn ) to the Cauchy problem considered in Example 2 if 1 < p < (n + 1)/(n − 1), that is T < ∞. The next example shows that this result can be extended to some operators with variable coefficients. Example 2a. Assume that u = u(x, t) is a smooth solution of utt − a2 (t)∆u = m(t)|u|p on [0, T ] × Rn , u(x, 0) = f (x) , ut (x, 0) = g(x) on Rn , where for the speed of propagation a = a(t) and for the function m = m(t) the following conditions are fulfilled with some positive constants C and k:
t ∞ max a(τ ) ds , 0 < a0 ≤ a(t) , a(t), m(t) ∈ C (R+ ) , A(t) := 0 −c

m(t) ≥ C(k + t)

n(p−1)

(k + A(t))

for large t

τ ≤s

with

c < p +1.

Suppose that supp f , supp g ⊂ { x ∈ R ; |x| ≤ R }. By the domain of dependence property, supp u(t, ·) ⊂ { x ∈ Rn ; |x| ≤ R + A(t) }. By integrating the equation with respect to spatial variables we obtain

  2 utt (t, x) − a (t)∆u(t, x) dx = m(t)|u(t, x)|p dx . n

Rn

Rn

Similarly to the consideration of the previous example we obtain

d2 u(t, x) dx = m(t)|u(t, x)|p dx . dt2 Rn n R Let
F (t) = u(t, x) dx ,


then F¨ (t) =

Rn

m(t)|u(t, x)|p dx . Rn

Using the compact support of u(t, ·) and Hlder’s inequality we get with 1/p+1/q = 1, p/q = p − 1, τn the volume of the unit ball in Rn ,
p
p u(t, x) dx n u(t, x) dx = R |x|≤R+A(t) 
 
p/q ≤ 

= =

|x|≤R+A(t)

|u(t, x)|p dx

1 dx |x|≤R+A(t)

 1 (R + A(t))np/q m(t)|u(t, x)|p dx τn m(t) n R 1 ¨ (R + A(t))np/q . F (t)ττn m(t)

Global Existence for Nonlinear Wave Equations

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Thus

F¨ (t) ≥ τn−1 m(t)(R + A(t))−n(p−1) |F (t)|p for all t in the interval [0, T ]. In particular, F¨ (t) ≥ 0, and so F (t) ≥ F˙ (0)t + F (0). Now

F˙ (0) = ut (0, x) dx = g(x) dx ≡ Cg . Rn

Rn

If Cg > 0, then F (t) ≥ (pos. const.)t , t large . The Lemma 2.1 with r = 1 and c = q satisfying (p − 1)r > q − 2, shows that the function F (t) cannot remain finite. Hence, T < ∞.
If we consider the case with a(t) = tl and m(t) = tm for large t, then the assumptions on these functions imply p < [n(l + 1) + 1 + m]/[n(l + 1) − 1]. Similar conclusions one can obtain for a(t) = exp(t), m(t) = tm1 exp(m2 t). Open Problem: Prove local and global existence results for the case with a(t) = tl , m(t) = tm . Then the expected sufficient condition for the global existence is p > [n(l + 1) + 1 + m]/[n(l + 1) − 1]. Conjecture: Assume that a(t) is a non-vanishing, non-constant, positive and periodic function. Then for every n, s, p > 1, and positive number ε there are initial data f (x), g(x) ∈ C0∞ (Rn ) with small Sobolev norms f (s+1) + g(s) ≤ ε such that the solution u(t, x) to the problem utt − a2 (t)∆u = |u|p on [0, ∞] × Rn , u(x, 0) = f (x) , ut (x, 0) = g(x) on Rn , blows up in finite time. Example 3. For x ∈ R3 , n = 3, in Example 2 we have 1 < p < 2. By improving √ the lower bound (2.2) one can obtain the following sharper result: if 1 < p < 1 + 2, then the solution blows up in finite time. Assume that u = u(x, t) is a smooth solution of utt − ∆u = |u|p on [0, T ] × R3 with 1 < p, and u(x, 0) = f (x) ,

ut (x, 0) = g(x).

Suppose that supp f , supp g ⊂ { x ∈ R ; |x| ≤ R } =: K. Then, let u = u0 (x, t) be a solution to the Cauchy problem for the linear wave equation 3

u0tt − ∆u0 = 0 ,

u0 (x, 0) = f (x) ,

u0t (x, 0) = g(x).

By Kirchhoff’s formula the function u = u0 (x, t) can be represented as follows:

1 u0 (x1 , x2 , x3 , t) = f (y1 , y2 , y3 ) dσ 4πt St (x)

∂ 1 + g(y1 , y2 , y3 ) dσ , ∂t 4πt St (x)

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where St (x) := {y ∈ R3 ; |x−y| = t}. It follows also the strong Huyghens’ principle, which states supp u0 ⊂ { (x, t) ; St (x) ∩ K = ∅ } . Hence, supp u0 ⊂ { (x, t) ; t − R ≤ |x| ≤ t + R } . Since u0 solves the linear wave equation, one obtains upon integration, that
∂2 u0 (x, t) dx = 0 for all t ≥ 0 . ∂t2 R3


It follows

R3

u0 (x, t) dx = Cg t + Cf ,




where Cg =


g(x) dx ,

Cf =

R3

f (x) dx . R3

Now we regard u(x, t) as a solution to the linear wave equation with the right-hand side ϕ(x, t) = |u|p : utt − ∆u = ϕ . If vϕ (t, x) solves this equation and takes vanishing initial data, vϕ (0, x) = ∂t vϕ (0, x) = 0, then


1 ϕ(y1 , y2 , y3 , t − |x − y|) dy1 dy2 dy3 vϕ (t, x) = 4π |x − y| Bt (x)


|u|p (y1 , y2 , y3 , t − |x − y|) 1 dy1 dy2 dy3 ≥ 0 , = 4π |x − y| Bt (x) where Bt (x) := {y ∈ Rn ; |x−y| ≤ t}. On the other hand this leads to the following estimate: u(x, t) = u0 (x, t) + vϕ (x, t) ≥ u0 (x, t) . Thus




Cg t + Cf


0

=

t−R≤|x|≤t+R


≤ ≤

u0 (x, t) dx

u (x, t) dx = R3

u(x, t) dx t−R≤|x|≤t+R



(p−1)/p

(vol {t − R ≤ |x| ≤ t + R}) 



≤

C(t + R)2(p−1)/p 0

R3

1/p |u(x, t)| dx p

R3

1/p |u(x, t)|p dx .

Assuming Cg > 0 we get for F (t) = R3 u(x, t) dx
F¨ (t) = |u(x, t)|p dx ≥ C(t + R)p−p2(p−1)/p = C(t + R)2−p R3

for large t .

Global Existence for Nonlinear Wave Equations Integrating, one has (1 < p < 3)  1  (t + R)3−p − (a0 + R)3−p for F˙ (t) − F˙ (a0 ) ≥ C 3−p Then with a new constant C we get F˙ (t) ≥ C(t + R)3−p for large t .

311

t ∈ [a0 , ∞) .

Integrating once more we derive F (t) ≥ C(t + R)4−p

for large t .

On the other hand, similarly to Example 2 we obtain the differential inequality (2.1), F¨ (t) ≥ τn−1 (R + t)3−3p |F (t)|p . It is very suggestive to use immediately Lemma 2.1. To do so we set r = 4−p ≥ 1,

q = 3p − 3 ,

then condition (p − 1)r > q − 2 is satisfied due to the inequality (p − 1)(4 − p) > 3(p − 1) − 2

⇐⇒

−(p − 1)2 + 2 > 0 ,

and this completes the consideration.

√




Remark 2.2. According to [31] for n = 3 at the borderline value p = 1 + 2 small data solutions must blow up in finite time. On the other hand for n = 3 and √ p > 1 + 2 small data solutions exist globally [19]. 0 In the all previous examples to get blow-up we assumed Rn g(x) dx > 0. Next example provides blow-up without this assumption. We assume that n = 3, small data, and consider the real-valued solution. Theorem 2.3 (F. John [19]). For the semilinear wave equation
u = u2 in three space dimensions, x ∈ R3 , all smooth data with compact support, ϕ0 , ϕ1 ∈ C0∞ (R3 ) produce singularities in finite time. Proof. Let us assume that the data are supported in the ball of radius R, and are not both identically zero. We let v = u2 and denote by I(r, t) and J(r, t) the spherical means of u and v about the origin respectively:
1 I(r, t) = u(y, t) dσy , 4πr2 Sr

1 1 J(r, t) = v(y, t) dσy = u2 (y, t) dσy , 4πr2 Sr 4πr2 Sr where Sr := {y ∈ R3 ; |y| = r}. For the function u = u(x, t) one can also define the spherical means of u about the point x:
1 I(x, r, t) = u(x + y, t) dσy , 4πr2 Sr and then an operator Ωr

: u(x, t)
−→ rI(x, r, t) .

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One can prove that for all h ∈ C(R3 )  2  ∂ ∂2 ∂2 ∂2 ∆x Ωr h = + + h = Ωr h . Ω r ∂ x21 ∂ x22 ∂ x23 ∂ r2 Therefore, we get for the function Ωr u  2  2   ∂ ∂ ∂2 − ∆ u = −
Ωr u = Ω Ωr u . x r ∂ t2 ∂ t2 ∂ r2 Since the commutator [−∆, Ωr ] = 0 vanishes we have
Ωr u = Ωr
u + [
, Ωr ]u = Ωr u2 + [−∆, Ωr ]u = Ωr v . Consequently,     rI(x, r, t) tt − rI(x, r, t) rr = rJ(x, r, t)

for all t ≥ 0 ,

r ≥ 0,

x ∈ R3 .

We need this equation only at x = 0, so simplifying the notations we have     rI(r, t) tt − rI(r, t) rr = rJ(r, t) . One can continue the functions I and J for negative r by setting I(r, t) := I(−r, t) ,

J(r, t) := J(−r, t)

for all r ≤ 0 .

One still has the equation for these values of r, too. Our aim is to prove that if we assume that a smooth solution exists up to the time 6R, we can derive arbitrary large lower bounds on I for 0 < r < t − 6R, and thus we reach a contradiction. Step 1. Preliminaries. The Cauchy-Schwarz inequality gives  2
1 I 2 (r, t) = u(y, t) dσ y 4πr2 Sr   

1 1 2 ≤ 1 dσ u(y, t) dσ , y y 4πr2 Sr 4πr2 Sr  
1 2 u(y, t) dσ for all r, t . = y 4πr2 Sr Hence, I 2 (r, t) ≤ J(r, t) . If we denote


 1 = φ0 (x + y, t) dσy , 4πr Sr
  1 = φ1 (x + y, t) dσy , 4πr Sr

 rI(r, 0) = Φ0 (r) rIIt (r, 0) = Φ1 (r)

Global Existence for Nonlinear Wave Equations

313

then due to d’Alembert’s formula and to Duhamel’s principle
r+t 1 1 {Φ0 (r + t) + Φ0 (r − t)} + Φ1 (ρ) dρ I(r, t) = 2r 2r r−t
t 
r+(t−τ )  1 + ρJ(ρ, τ ) dρ dτ . 2r 0 r −(t−τ ) The function ρJ(ρ, τ ) is odd, −ρJ(−ρ, τ ) = −ρJ(ρ, τ ). The function rI(r, t) takes initial data Φ0 (r) and Φ1 (r) which vanish for |r| > R. Therefore, if we choose a point (r, t) such that |t + r| > R and |t − r| > R, then
t 
r+(t−τ )  1 ρJ(ρ, τ ) dρ dτ . I(r, t) = 2r 0 r −(t−τ ) Since ρJ(ρ, τ ) is odd, the part of the integral, corresponding to the region symmetric with respect to the τ axis, is zero. One can therefore restrict the integration to the region ' ( Z(r, t) := (ρ, τ ) ; τ > 0 and t − r < τ + ρ < t + r and τ − ρ < t − r , that is 1 I(r, t) = 2r



for |t + r| > R and |t − r| > R . (2.3)

ρJ(ρ, τ ) dρ dτ Z (r,t)

Letting r go to zero, we find
t ρJ(t − ρ, ρ) dρ I(0, t) =

for t > R .

0

Step 2. I(0, t) is not identically zero for R < t < 3R. Indeed, otherwise we would have J(τ, ρ) ≡ 0 for τ + ρ ∈ (R, 3R), and therefore

1 J(τ, ρ) = u2 (τ, ρy) dS ≡ 0 4π |y |=1 implies u(τ, ρy) = 0

∀τ + ρ ∈ (R, 3R) , y ∈ S 2 .

Letting τ → R, we find in particular that u(R, ρy) = 0 ,

ut (R, ρy) = 0

for all ρ ∈ (0, 2R) , y ∈ S 2 ,

that is u(R, x) = 0 ,

ut (R, x) = 0

for all |x| ≤ 2R .

If u ∈ C ((0, ∞) × R ), then one can regard the equation
u = u2 as a linear equation
u−a(t, x)u = 0 with a smooth coefficient a(t, x) = u(t, x) ∈ C 2 ([0, ∞)× R3 ). Then the finite propagation speed forces 2

3

u(0, x) = 0 ,

ut (0, x) = 0 for all |x| ≤ R ,

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Karen Yagdjian

and the solution would then vanish identically. The obtained contradiction implies that I(0, t) is not identically zero for R < t < 3R, and consequently, there is a point (0, t0 ), R < t0 < 3R, near which I does not vanish, I(0, t0 ) > 0 ,

R < t0 < 3R .

Moreover, I(r, t) does not vanish on the half-line {(r, t) ; t = t0 + r, r > 0 }, i.e., I(r, t) > 0

for all t = t0 + r ,

r > 0.

Indeed, if I(r , t ) = 0 for some t = t0 + r , r > 0, then from the representation
t

r
+(t
−τ )  1 I(r , t ) =  ρJ(ρ, τ ) dρ dτ 2r 0 r
−(t
−τ ) with the integration restricted to the trapezoid ( ' Z(r , t ) := (ρ, τ ) ; τ > 0 and t − r < τ + ρ < t + r and τ − ρ < t − r follows that J(ρ, τ ) ≡ 0 in Z(r , t ). The estimate I 2 ≤ J then implies I(ρ, τ ) ≡ 0 in Z(r , t ). But the points (0, t0 ) and (r , t ) are the vertexes of Z(r , t ), so that I(0, t0 ) must vanish. With this conclusion we turn from the theory of partial differential equations to some integral inequalities. Step 3. Two-variables integral inequality. Thus we have from (2.3) and I 2 ≤ J the following two-variables integral inequality:

1 ρI 2 (ρ, τ ) dρ dτ for |t + r| > R and |t − r| > R . I(r, t) ≥ 2r Z (r,t) Lemma 2.4. The continuous function I(r, t) which is defined in {r ≥ 0, t ≥ 0} and which satisfies I(r, t) > 0

for all

t = t0 + r ,

cannot solve the inequality

1 ρI 2 (ρ, τ ) dρ dτ I(r, t) ≥ 2r Z (r,t)

for all

t0 ∈ (R, 3R) ,

t+r > R, t−r > R,

where the domain of integration Z(r, t) is ' ( Z(r, t) := (ρ, τ ) ; τ > 0 and t − r < τ + ρ < t + r, and τ − ρ < t − r . Namely, there exists a constant γ such that I(0, γ) = ∞. Proof. Consider the set ' ( T = (ρ, τ ) ; 4R < τ + ρ < 6R, R < τ − ρ < 3R .

Global Existence for Nonlinear Wave Equations

315

The half-line {(r, t) ; t = t0 + r, r > 0 } (with t0 ∈ (R, 3R)) intersects T . On that half-line both J and I are positive, therefore the following number is positive:

1 ρI 2 (ρ, τ ) dρ dτ > 0 . c := 2 T Let S denote the half-strip ' ( S = (ρ, τ ) ; 6R < τ + ρ , 3R < τ − ρ < 4R . Then T ⊂ Z(r, t) for (r, t) ∈ S. Consequently,



1 1 c 2 I(r, t) ≥ ρI (ρ, τ ) dρ dτ ≥ ρI 2 (ρ, τ ) dρ dτ = . 2r 2r r Z (r,t) T Thus

c for all (r, t) ∈ S . r In the last part of the proof we assume that 0 < r < t − 6R. Let S(r, t) denote the rectangle ' ( S(r, t) := Z(r, t) ∩ S = (ρ, τ ) ; t − r < τ + ρ < t + r , 3R < τ − ρ < 4R . I(r, t) ≥

Then I(r, t) ≥

1 2r



ρJ(ρ, τ ) dρ dτ ≥ S (r,t)

c2 2r





ρ−1 dρ dτ .

S (r,t)

If (ρ, τ ) ∈ S(r, t), then τ + ρ < t + r and ρ < τ − 3R imply 2ρ ≤ t + r − 3R and ρ−1 ≥ 2(t + r − 3R)−1 . Hence, if we set R ≥ 1, then



c2 c2 I(r, t) ≥ ρ−1 dρ dτ ≥ (t + r − 3R)−1 dρ 2r r S (r,t) S (r,t)

c2 = 1 dρ dτ r(t + r − 3R) S (r,t) c2 (t + r) − (t − r) 4R − 3R c2 2rR √ √ · · · = r(t + r − 3R) r(t + r − 3R) 2 2 2 c2 c2 ≥ ≥ for all 0 < r < t − 6R . t + r − 3R t+r Now we introduce the rectangle ' ( T (r, t) := (ρ, τ ) ; t − r < τ + ρ < t + r , 6R < τ − ρ < t − r , =

contained in Z(r, t). We have



1 1 ρI 2 (ρ, τ ) dρ dτ ≥ ρI 2 (ρ, τ ) dρ dτ I(r, t) ≥ 2r 2r Z (r,t) T (r,t) for all 0 < r < t − 6R. The proof of Lemma 2.4 will be completed if we prove the following statements:

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Karen Yagdjian

Lemma 2.5. The continuous function I(r, t) defined on {r ≥ 0, t ≥ 0} cannot solve the inequalities

1 ρI 2 (ρ, τ ) dρ dτ for all 0 < r < t − 6R , I(r, t) ≥ 2r T (r,t) c2 t+r

≥

I(r, t)

for all

0 < r < t − 6R ,

simultaneously. Namely, there exists a constant γ such that I(0, γ) = ∞. Proof. We find I(r, t) ≥

1 2r





c4 ρI (ρ, τ ) dρ dτ ≥ 2r T (r,t)





2

ρ T (r,t)

1 dρ dτ . (τ + ρ)2

Setting α = τ + ρ and β = τ − ρ and noting that in T (r, t) ρ≥

1 (t − r − β) , 2

β ≤ t−r,

while α ∈ [t − r, t + r] and β ∈ [6R, t − r], we write

ρ T (r,t)

1 dρ dτ (τ + ρ)2




= =

t−r

dα


≥




t+r

= t−r t+r

t−r




dβ ρ 6R

1 dα 2 4α




1 1 2 (τ + ρ) 2

t−r

(t − r − β) dβ 6R


t−r  1 dα (t − r − β) dβ 2 t−r α 6R  (t − r − 6R)2  2r . 4 (t + r)(t − r) 2

1 4 1

t+r

Finally, for t ≥ 6R + r we obtain I(r, t) ≥

2r (t − r − 6R)2 c4 (t − r − 6R)2 c4 1 = . 2r 4 (t + r)(t − r) 2 8(t + r) (t − r)

Let us suppose for (r, t), t ≥ 6R + r, r > 0, that I satisfies the inequality I(r, t) ≥

C (t − r − 6R)k , t+r (t − r)q

with numbers C, k, q, C > 0,

k ≥ 0,

q ≥ 0,

Global Existence for Nonlinear Wave Equations

317

as well as the first one of the lemma. Then acting similar to the previous calculations we have

1 ρI 2 (ρ, τ ) dρ dτ I(r, t) ≥ 2r T (r,t)

(τ − ρ − 6R)2k 1 C2 ρ dρ dτ ≥ 2 2r (τ − ρ)2q T (r,t) (ρ + τ )

t−r C 2 t+r 1 (τ − ρ − 6R)2k 1 ≥ dα dβ ρ , 2r t−r 2 (ρ + τ )2 (τ − ρ)2q 6R and consequently, I(r, t)







t+r

≥

C2 2r

≥

 C2 8r(t − r)2q

t−r

dα t−r

1 (β − 6R)2k 1 (t − r − β) 2 4 α (t − r)2q
t−r   1 2k dα (t − r − β)(β − 6R) dβ . α2 6R

dβ


6R t+r

t−r

The last integrals can be evaluated explicitly: 
t+r 1 
t−r  C2 dα (t − r − β)(β − 6R)2k dβ 2q 2 8r(t − r) t−r α 6R  (t − r − 6R)2k+2   2 2r C = 8r(t − r)2q (t + r)(t − r) (2k + 1)(2k + 2) 2 2k+2 C (t − r − 6R) = . 4(2k + 1)(2k + 2)(t + r)(t − r)2q+1 Hence, I(r, t)

≥

(t − r − 6R)2k+2 C2 . 4(2k + 2)2 (t + r) (t − r)2q+1

Thus we have done one step that resulted in a substitution C (t − r − 6R)k t+r (t − r)q

→

(t − r − 6R)2k+2 C2 2 4(2k + 2) (t + r) (t − r)2q+1

in the estimate from below. Let us define the sequences {C Cj }j≥0 , {kj }j≥0 , {qqj }j≥0 by C0 = c2 , Cj +1

k0 = q0 = 0, Cj2 = , kj+1 = 2kj + 2 , 4(2kj + 2)2

qj +1 = 2qqj + 1 .

Then a recursive argument gives I(r, t) ≥ Cj (t + r)−1 (t − r − 6R)kj (t − r)−qj for 0 < r < t − 6R and all j = 0, 1, . . . .

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Karen Yagdjian

It is easy to solve the recursion formulas and to find kj = 2j+1 − 2 , qj = 2j − 1 , j     Cj = exp 2j log C0 − 21−i log(2i+2 − 4) . i=1

Indeed, Cj

Cj2−1 Cj2−1 Cj2−1 Cj2−1 = = = 4(2kj−1 + 2)2 4(2(2j − 2) + 2)2 16(2j − 1)2 (2j+2 − 4)2

=

implies log Cj = 2 log Cj −1 − log (2j+2 − 4)2 = 2 log Cj −1 − 2 log (2j+2 − 4) (one step). If we make one more step, then we obtain ' ( log Cj = 2 2 log Cj −2 − 2 log (2j+1 − 4) − 2 log (2j+2 − 4) =

2 · 2 log Cj −2 − 2 · 2 log (2j+1 − 4) − 2 log (2j+2 − 4) (two steps).

In j steps we arrive at log Cj = 2 · 2 · 2 log Cj −3 −2 · 2 · 2 log (2j −4)−2 · 2 log (2j+1 −4)−2 log (2j+2 −4) ··· j  21+j−i log (2i+2 − 4) (j-steps) = 2j log C0 − i=1



= 2j log C0 −

j  i=1

 21−i log(2i+2 − 4) .

Global Existence for Nonlinear Wave Equations

319

Thus I(r, t)

j     ≥ exp 2j log C0 − 21−i log(2i+2 − 4) i=1 −1

×(t + r) (t − r − 6R)2 −2 (t − r)−2 +1 j     = exp 2j log C0 − 21−i log(2i+2 − 4) j+1

j

i=1 j+1 j t−r (t − r − 6R)2 (t − r)−2 × (t + r)(t − r − 6R)2 j     j ≥ exp 2 log C0 − 21−i log(2i+2 − 4)

i=1

   t−r 1  j 2 j exp 2 × exp 2 log(t − r − 6R) log (t + r)(t − r − 6R)2 (t − r)   t−r exp 2j Wj (t − r) ≥ (t + r)(t − r − 6R)2 for 0 < r < t − 6R and j = 0, 1, . . . , where (t − r − 6R)2  1−i − log C0 + log 2 log(2i+2 − 4) t−r i=1 j

Wj (t − r) :=

∞

≥

log C0 + log

(t − r − 6R)2  1−i − 2 log(2i+2 − 4) =: W∞ (t − r) . t−r i=1

Obviously, there exists a constant γ such that W∞ (t − r) > 0 for t − r ≥ γ. Hence, for j → ∞ we find that I(r, t) = ∞ for t − r ≥ γ ,

r ≥ 0.


Lemma 2.5 is proved.

Completion of the proof of Theorem 2.3. From Lemma 2.5 in particular it follows u(0, γ) = I(0, γ) = ∞ . The solution has a life span T ≤ γ, and the theorem is proved.




According to the next theorem there is a blow-up for a system that is similar to the equation from Example 2. Theorem 2.6 (R. Agemi, Y. Kurokawa, H. Takamura [1], D. Del Santo [9], D. Del Santo, V. Georgiev, and E. Mitidieri [10], D. Del Santo and E. Mitidieri [11]). Let f1 , f2 , g1 , g2 ∈ C0∞ (R3 ) with ∅ = supp f1 ∪ supp f2 ∪ supp g1 ∪ supp g2 ⊂ {|x| ≤ R} .

320

Karen Yagdjian

Let T ∈ (0, ∞) and let (u, v), where u, v ∈ C 2 (R3 × [0, T )), be the solution of the Cauchy problem for the system

utt − ∆u = |v|p , vtt − ∆v = |u|q , with initial data ut (x, 0) = g1 (x) , u(x, 0) = f1 (x) , v(x, 0) = f2 (x) , vt (x, 0) = g2 (x) . Suppose that p, q > 1 and

& p + 2 + q −1 q + 2 + p−1 , max ≥ 1. pq − 1 pq − 1 Then T < ∞. On the other hand due to the next theorem small data solutions exist globally in time. Theorem 2.7 (R. Agemi, Y. Kurokawa, H. Takamura [1], D. Del Santo [9], D. Del Santo, V. Georgiev, and E. Mitidieri [10]). Let f1 , f2 , g1 , g2 ∈ C0∞ (R3 ) with supp f1 ∪ supp f2 ∪ supp g1 ∪ supp g2 ⊂ {|x| ≤ R} . Suppose that p, q ≥ 2 and



max

p + 2 + q −1 q + 2 + p−1 , pq − 1 pq − 1

& < 1.

Then there exists ε0 > 0 such that for each ε ∈ (0, ε0 ) the Cauchy problem

utt − ∆u = |v|p , vtt − ∆v = |u|q ,

u(x, 0) = εf1 (x) , ut (x, 0) = εg1 (x) , v(x, 0) = εff2 (x) , vt (x, 0) = εg2 (x) , has a unique global classical solution (u, v), where u, v ∈ C 2 (R3 × [0, ∞)).

3. Blow-up for the problem with large potential energy of nonlinearity Consider the wave equation utt − ∆u + f (u) = 0 0u in R × R . We define F (u) = 0 f (s) ds, and the “complete energy” &


 1 2 2 u + |∇x u| + F (u) dx . E(t) = 2 t Rn n

Global Existence for Nonlinear Wave Equations

321

Theorem 3.1 (H. Levine [22]). Assume that f is smooth, f (0) = 0, uf (u) ≤ (2 + ε)F (u) for some positive ε, and that u is a smooth solution with E(0) < 0. Further, assume that the data have compact support, u(0, x) = u0 (x) ∈ C0∞ (Rn ) ,

ut (0, x) = u1 (x) ∈ C0∞ (Rn ) .

Then u must develop a singularity in finite time. Proof. For the energy of the classical solution we have
.  / d E(t) = ut utt + ∇x u · ∇x ut + ut f (u) dx dt n
R .  / = ut utt − ∆u + f (u) dx = 0 Rn

and E(t) is therefore constant, E(t) ≡ E(0). Multiplying the equation by u and integrating we find
  uutt + |∇x u|2 + uf (u) dx = 0 . Rn

Adding (with α = ε/4) &


 1 2 ut + |∇x u|2 + F (u) dx = −(2 + ε)E(0) −(2 + ε) 2 Rn to both sides, we find,

  uutt + |∇x u|2 + uf (u) dx − (2 + ε) Rn



Rn

that is,
Rn

&  1 2 2 u + |∇x u| + F (u) dx 2 t = −(2 + ε)E(0),

  −(1 + 2α)u2t − 2α|∇x u|2 + uutt + uf (u) − (2 + 4α)F (u) dx = −(2 + 4α)E(0) .

Hence, after using the assumption on f , (2 + 4α)E(0)
  = (1 + 2α)u2t + 2α|∇x u|2 − uutt − uf (u) + (2 + 4α)F (u) dx n
R   (1 + 2α)u2t − uutt dx . ≥ Rn

If we define now the positive function I(t) =

1 1 β(t + τ )2 + 2 2


u2 dx > 0 , Rn

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Karen Yagdjian

where β and τ are positive constants, then

d d2 I(t) = β(t + τ ) + uut dx , I(t) = β + (u2t + uutt ) dx . dt dt2 Rn Rn Using E(0) < 0 we choose β such that 2E(0) + β < 0 , and then choose τ such that






I (0) = βτ + Rn

u0 u1 dx > 0 .

Our goal is to prove that I(t) cannot remain bounded. We write
I  (t) = β + (u2t + uutt ) dx n R

' ( (1 + 2α)u2t − uutt dx u2t dx − ≥ β + 2(1 + α) n Rn
R ≥ β + 2(1 + α) u2t dx − (2 + 4α)E(0) Rn
= 2(1 + α)β + 2(1 + α) u2t dx − (2 + 4α)E(0) − (1 + 2α)β n R
  2 = 2(1 + α) β + ut dx − (1 + 2α)(β + 2E(0)) . Rn

Thus


 I  (t) > 2(1 + α) β + Rn

 u2t dx .

This implies I(t)I  (t) − (1 + α)(I  (t))2

' ('  ( 2 2 > β(t + τ ) + u dx (1 + α) β + u2t dx Rn Rn
(2 ' uut dx −(1 + α) β(t + τ ) + Rn 

   = (1 + α) β(t + τ )2 + u2 dx β + u2t dx  − β(t + τ ) +


Rn

Rn

2 uut dx

Rn

.

The right-hand side is√nonnegative. Indeed, √ using the Cauchy-Schwarz inequality on the product of (u, β(t + τ )) and (ut , β) in L2 (Rn ) × R we obtain


  2    β β(t + τ ) + uut dx ≤ β(t + τ )2 + u2 dx β + u2t dx . Rn

Rn

Rn

Global Existence for Nonlinear Wave Equations

323

Hence, I(t)I  (t) − (1 + α)(I  (t))2 after multiplication with −αI

−α−2

(I

> 0

implies −α 

) (t) < 0 .

On the other hand the parameters have been chosen so that   I −α (0) > 0 , I −α (0) < 0 . It follows that the function J(t) := I −α (t) must vanish in finite time. Indeed, 
t
t
s    J(t) = J(0) + J (s)ds = J(0) + J (τ )dτ ds ≤ J(0) + J  (0)t J (0) + 0

0

0

implies that it happens no later than at tls = −J(0)/J  (0), hence limt→tls I(t) = ∞. The theorem is proved.
Thus according to this theorem one can say, if the absolute value of the negative initial potential energy of the nonlinearity is larger than the initial total energy of the linear problem, then the problem cannot have global solutions. Next we take two examples for the function f (u) and discuss the condition on E(0). 1) If we take f (u) = −|u|p−1 , then to verify the main condition we write for positive u
u up (−|s|p−1 )ds = −(2 + ε) , uf (u) = −u|u|p−1 = −up ≤ (2 + ε)F (u) = (2 + ε) p 0 which leads to 0 < ε ≤ p −2. For negative u we obtain uf (u) = −u|u|p−1 = |u|p


≤


=

(2 + ε)

u

(−|s|p−1 )ds

(2 + ε)F (u) = (2 + ε) 0 |u|

|s|p−1 ds = (2 + ε)

0

|u|p p

which leads to 0 < ε, ε ≥ p−2. Finally we have to assume ε = p − 2 > 0, that is, p > 2. 2) If we take f (u) = −up−1 , then to verify the main condition we write
u up p (−sp−1 )ds = −(2 + ε) . uf (u) = −u ≤ (2 + ε)F (u) = (2 + ε) p 0 This holds for negative and positive u simultaneously only for p = 2 + ε, and we get the condition p > 2.

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3) Consider now the assumption E(0) < 0: &


 1 1 2 2 p u1 (x) + |∇x u0 (x)| − |u0 (x)| dx < 0 , E(0) = 2 p Rn where we have assumed u0 (x) ≥ 0. If we replace u0 (x) and u1 (x) by εu0 (x) and εu1 (x), respectively, then for ε > 0 small it will never be satisfied because of p > 2. Thus we have an example with blow-up for large data. According to Theorem 6.5.2 [17] the Cauchy problem with the initial data εu0 (x) ∈ C0∞ (Rn ) and εu1 (x) ∈ C0∞ (Rn ) has a C ∞ solution for all t ≥ 0 if n ≥ 4 and ε is sufficiently small, that is, the existence of global small data solutions. It can be similarly proved the nonexistence of global solutions for the Cauchy problem0 for the equation utt − a(t)∆u + f (u) = 0, where a (t) ≤ 0 and uf (u) ≤ u (2 + ε) 0 f (s)ds.

4. Parametric resonance and wave map type equations In this section we consider global existence and blow-up of the solutions to the problem for the scalar nonlinear wave equation:

∇x u|2 ) = 0 , utt − a2 (t, x)∆u + f (u)(u2t − a2 (t, x)|∇ (4.1) ut (0, x) = u1 (x) , u(0, x) = u0 (x) , where u = u(t, x) is a real-valued unknown function, f (u) is a real-valued continuous function, a(t, x) ≥ 0, while ∆ is the Laplace operator in Rn . The equation in (4.1) is a model and a special case of an equation for wave maps. Let M be a Riemannian manifold with metric (gij ) = g. A wave map u : Rn+1 −→ M , is by definition a solution of the Euler-Lagrange equations associated with the functional
∂α u, ∂ α ug(u) dσ . u −→ J(u) := Rn

    ∂x1 )α1 , . . . , (−∂ ∂x1 )αn , Here ∂α = ∂tα0 , ∂xα11 , . . . , ∂xα1n , ∂ α = ∂tα0 , (−∂ ·, ·g(u) is the product in the metric g(u), and the usual Einstein summation convention is in force, while dσ denotes the volume measure on Rn+1 with respect to the standard metric. In local coordinates, u is seen to satisfy the equation
ul + Γlbc (u)∂α ub ∂ α uc = 0 ,

l = 1, 2, . . . , m ,

where Γlbc (u) refer to the Riemann-Christoffel symbols associated with the metric g. The scalar problem (4.1) is similar to the wave map problem, having the same form (4.1) in the case that the target M is one dimensional. The constant coefficient case, a(t, x) ≡ 1, is studied in [26]. There are given necessary and sufficient conditions on f = f (u) for which the Cauchy problem

Global Existence for Nonlinear Wave Equations

325

(4.1) has a global smooth solution for any smooth initial data. The conditions are written as 
 

∞
s
0
s exp f (r)dr ds = ∞ and exp f (r)dr ds = ∞. (4.2) 0

0

−∞

0

For instance, the function f (u) = cuk with c ∈ R and nonnegative integer k, satisfies condition (4.2) if and only if c ≥ 0 and k is odd. When n = 3, f = f (u) and a = a(t, x) are constant functions, say f (u) ≡ 1 and a(t, x) ≡ 1, equation (4.1) becomes Nirenberg’s example, which is globally solvable for small data only. Moreover, when a(t, x) ≡ 1 and f = f (u) is smooth enough, it still has a global solution for small data [26] independently of condition (4.2). First we will prove that the condition a(t, x) ≡ 1 is crucial for the global solvability of problem (4.1), namely even for smooth, positive, periodic, non-constant function a = a(t, x) the problem is not globally solvable even for arbitrary small data. Secondly we will show that condition (4.2) is equivalent to the global solvability of the problem (4.1) for arbitrary (in particularly large) initial data. The proof of the second statement is short and almost identical to the one suggested in [26]. It will be given at the end of this section, while the proof of the first one based on the properties of the Hill’s equation and takes a lot of efforts. Parametric resonance breaks down the small data solution We give an example of the influence of the behavior of a time-dependent coefficient a = a(t) from (4.1), and in particular its oscillating behavior, on the global existence of solutions to nonlinear hyperbolic equations. Namely for arbitrary small initial data we will construct blowing up solutions. To this end we consider in R × Rn the equation utt − a2 (t, x)∆u + f (u)(u2t − a2 (t, x)|∇ ∇x u|2 ) = 0 ,

(4.3)

where u = u(t, x) is a real-valued unknown function, f = f (u) is a real-valued smooth function. We restrict ourselves to the case of a(t, x) = b(t): n    (uxj )2 = 0 . utt − b2 (t)∆u + f (u) (ut )2 − b2 (t)

(4.4)

j=1

The next theorem shows that if the function b = b(t) differs from the case to be a constant, for instance, oscillates, the situation with the global existence for small data changes dramatically. For the equation (4.4) with f (u) ≡ 1 it was detected in [41], [43]. In the next theorem ϕ(s) denotes the norm of the function ϕ = ϕ(x) from the Sobolev space H s (Rn ).

326

Karen Yagdjian

Theorem 4.1. Let b = b(t) be a defined on R, a periodic, non-constant, smooth, and positive function. Suppose that (4.2) does not hold. Then for every n, s, and for every positive δ there are data u0 ∈ C0∞ (Rn ) and u1 ∈ C0∞ (Rn ) such that u0 (s+1) + u1 (s) ≤ δ,

(4.5)

but a solution u ∈ C (R+ × R ) to the problem with data 2

n

u(0, x) = u0 (x),

ut (0, x) = u1 (x) ,

x ∈ Rn ,

(4.6)

does not exist. For a simplest example of the equation (4.4) one can take f (u) = uk , k even, while b(t) = 1 + ε sin(2πt), with a number ε ∈ (−1/2, 1/2), ε = 0. The oscillations are responsible for the blow-up of the solutions. Indeed, if, for instance for the simplified equation with f (u) ≡ 1, we switch off them, that is if we set ε = 0, then for n ≥ 3 the problem has small data global in time solution. (See, e.g., [17], [27].) The next theorem explains why in Theorem 4.1 we have to suppose that (4.2) does not hold. Indeed, if it is satisfied, then there is a global solution even for large data. Large Data Global Solution Theorem 4.2. Let f ∈ C ∞ (R) and let a(t, x) ∈ C ∞ (R+ × Rn ) be a uniformly positive bounded function. Then (4.3), (4.6) has a global classical solution u ∈ C ∞ (R1+n ) for any u0 and u1 ∈ C ∞ (Rn ) if and only if the function f satisfies (4.2). Proof of Theorem 4.2. In fact equation (4.3) is transformed into the linear wave equation by 

u
s v = G(u) := exp f (r)dr ds . (4.7) 0

0

Since G ∈ C 2 (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C 2 (a, b) , where we denote a := lim G(u) , u→−∞

b := lim G(u) . u→∞

First we show that (4.2) is a sufficient condition for global existence. Assume (4.2) and let ϕ, ψ ∈ C ∞ (Rn ). Denote by W (ϕ, ψ) the solution v of the Cauchy problem for the linear wave equation:

vtt − a2 (t, x)∆v = 0 , v(0, x) = ϕ(x) , vt (0, x) = ψ(x) . Set v = W (G(ϕ), G (ψ)). We claim that u = H(v) is a classical solution of (4.3). Indeed, G (u) = f (u)G (u) , G (H(v))H  (v) = 1 ,

Global Existence for Nonlinear Wave Equations

327

imply H  (u) = −f (H(v))H  (v)2 , while ∂α u = H  (v)∂α v, ∂α2 u = H  (v)(∂α v)2 + H  (v)∂α2 v, Thus, we obtain utt − a2 (t, x)∆u

for α = 0, 1, . . . , n.

=

H  (v)(vtt − a2 (t, x)∆v) + H  (v)((∂ ∂t v)2 − a2 (t, x)|∇ ∇x v|2 )

=

∇x u|2 ) , −f (u)((∂ ∂t u)2 − a2 (t, x)|∇

(4.8)

so that u is a global smooth solution of (4.3). We choose 


u0 (x)
s ϕ(x) := exp f (r)dr ds , ψ(x) := u1 (x) exp 0

0



u0 (x)

f (r)dr

.

0

Now since t = W (0, 1) the function u(t, x) = H(t) = H(W (0, 1)) is a classical solution of (4.3) with (ϕ, ψ) = (0, 1) for a < t < b and satisfies lim u(t, x) = −∞ and

t→a+0

lim u(t, x) = −∞

t→b−0

for any x ∈ Rn . Hence, (4.2) is also a necessary condition for which (4.3) has a global smooth solution for any smooth initial data.


5. Proof of Theorem 4.1: Parametric resonance We follow [8] to prove Borgs’s theorem. 5.1. Some properties of the Hill’s equation Consider the auxiliary ordinary differential equation with a periodic coefficient wtt + λb2 (t)w = 0 .

(5.1)

It can be written also as a system of differential equations for the vector-valued function x(t) = t (wt , w):   d 0 −λb2 (t) x(t) = A(t)x(t) , where A(t) := . 1 0 dt Let the matrix-valued function Xλ (t, t0 ), depending on λ, be the solution of the Cauchy problem   d 1 0 . (5.2) X = A(t)X , X(t0 , t0 ) = 0 1 dt Thus, Xλ (t, t0 ) gives the fundamental solution to the equation (5.1). This matrix is called matrizant. In what follows we often omit subindex λ of Xλ (t, t0 ). There is an explicit representation formula for the matrizant:
t1
tk−1 ∞
t  X(t, t0 ) = I + A(t1 ) dt1 A(t2 ) dt2 . . . A(tk ) dtk . k=1

t0

t0

t0

328

Karen Yagdjian

The Liouville formula







t

W (t) = W (t0 ) exp

S(τ )dτ

,

t0

where W (t) := det X(t, t0 ) ,

S(t) :=

2 

Akk (t)

k=1

with S(t) ≡ 0 guarantees the existence of the inverse matrix Xλ (t, t0 )−1 . This allows to prove that 1. The matrizant of any system satisfies the identities X(t, t0 ) = X(t, t1 )X(t1 , t0 ) ,

X(t, t0 )−1 = X(t0 , t)

for all t0 , t1 , t ∈ R .

Indeed, both sides of the first equality satisfy the equation and coincide at t = t1 . By the uniqueness of the solution to the Cauchy problem both sides coincide identically. 2. If A(t) is a 1-periodic matrix, then X(t + t0 , t0 ) is independent of t0 ∈ N. In particular X(t + 1, 1) = X(n + 1, n) =

X(t, 0) for all t ∈ R , X(1, 0) for all n ∈ N .

Indeed, the matrix Y (t) := X(t+t0 , t0 ), t0 ∈ N, solves the equation dY /dt = A(t)Y and takes initial data Y (0) = X(t0 , t0 ) = I. So does the matrix X(t, 0). Uniqueness in the Cauchy problem implies X(t + t0 , t0 ) = X(t, 0) is independent of t0 ∈ N. Problem. Let A(t) be a 1-periodic matrix. Is X(t + t0 , t0 ) independent of t0 when t0 runs over R? Consider now the matrix C(t, t0 ) := (X(t, t0 )) 1, t0 ), then d C(t, t0 ) = dt = = =

−1

X(t + 1, t0 ) = X(t0 , t)X(t +

d (X(t, t0 ))−1 X(t + 1, t0 ) dt   d −1 −1 d (X(t, t0 )) X(t + 1, t0 ) X(t + 1, t0 ) + (X(t, t0 )) dt dt   d (X(t, t0 ))−1 X(t + 1, t0 ) + (X(t, t0 ))−1 A(t + 1)X(t + 1, t0 ) dt   d −1 −1 (X(t, t0 )) X(t + 1, t0 ) + (X(t, t0 )) A(t)X(t + 1, t0 ) . dt

d −1 Further, the identity (X(t, t0 )) X(t, t0 ) ≡ I leads to (X(t, t0 )−1 X(t, t0 )) = 0, dt so that   d −1 −1 d (X(t, t0 )) X(t, t0 ) = 0 X(t, t0 ) + (X(t, t0 )) dt dt

Global Existence for Nonlinear Wave Equations implies



d −1 (X(t, t0 )) dt

Thus, d C(t, t0 ) = dt = =



 =

− (X(t, t0 ))

−1

=

− (X(t, t0 ))

−1

=

− (X(t, t0 ))

−1



329

 d −1 X(t, t0 ) (X(t, t0 )) dt

A(t)X(t, t0 ) (X(t, t0 ))

−1

A(t) .

 d −1 (X(t, t0 )) X(t + 1, t0 ) + (X(t, t0 ))−1 A(t)X(t + 1, t0 ) dt

− (X(t, t0 )) 0.

−1

A(t)X(t + 1, t0 ) + (X(t, t0 ))

−1

A(t)X(t + 1, t0 )

That is, the matrix C(t, t0 ) is independent of t. We set t = t0 in C(t0 ) = (X(t, t0 ))−1 X(t + 1, t0 ) and get C(t0 ) = X(t0 + 1, t0 ) ,

X(t + 1, t0 ) := X(t, t0 )C(t0 ) for all t, t0 ∈ R.

We will write just C for the matrix C(0). Hence, C = X(1, 0) , X(t + 1, 0) := X(t, 0)C for all t ∈ R , X(t + n, 0) := X(t, 0)C n for all t ∈ R , n ∈ Z . 3. For the matrix X(1, 0) we will use the notation   b11 b12 Xλ (1, 0) = . b21 b22 This matrix is called a monodromy matrix and its eigenvalues are called multipliers of system (5.2) (see [23],[44]). Thus, the monodromy matrix is the value at t = 1 (the “end”of the period) of the fundamental matrix X(t, 0) defined by the initial condition X(0, 0) = I (i.e., matrizant), and the multipliers are the roots of the equation det [X(1, 0) − µI] = 0 .

(5.3)

4. The matrizant of any 1-periodic system satisfies the identities X(t + 1, t0 + 1) = X(t, t0 )

for all t0 , t ∈ R .

Indeed, X(t + 1, t0 + 1) = =

X(t + 1, t0 )X(t0 , t0 + 1) = X(t, t0 )C(t0 )X(t0 + 1, t0 )−1 X(t, t0 )C(t0 )C −1 (t0 ) = X(t, t0 ) .

5. Let a be an eigenvector of the monodromy matrix relative to some multiplier µ, X(1, 0)a = µa .

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Karen Yagdjian

The vector a is in general complex, even when A(t) is a real-valued matrix. Consider the solution x = x(t) of the system dx/dt = A(t)x with initial condition x(0) = a , then it is expressed in terms of the matrizant by x(t) = X(t, 0)a . Using the properties X(t, t0 ) = X(t, t1 )X(t1 , t0 ) and X(t + 1, t0 + 1) = X(t, t0 ) we have x(t + 1) = = =

X(t + 1, 0)a = X(t + 1, 1)X(1, 0)a X(t, 0)X(1, 0)a = X(t, 0)µa µX(t, 0)a = µx(t) .

Thus, for each multiplier there is a solution x = x(t) of the system satisfying x(t + 1) = µx(t)

for all t ∈ R .

(5.4)

One can derive the characteristic equation (5.3) by determining a nontrivial solution x(t) of the system with the property (5.4), where µ is some number. Indeed, the vector functions x(t+ 1) and µx(t) satisfy the system. If they coincide at t = 0, x(1) = µx(0) ,

(5.5)

then due to the uniqueness it follows that (5.4) is true. Thus for the solutions property (5.5) is equivalent to (5.4). It is thus sufficient to find a solution with the property x(1) = µx(0). Since x(1) = X(1, 0)x(0), it follows that this property is equivalent to [X(1) − µI] x(0) = 0 , so that µ is a root of the characteristic equation (i.e., a multiplier) and x(0) is an eigenvector of the monodromy matrix. 6. Theorem 5.1 (Lyapunov-Floquet). In system (5.2) let A(t) be a continuous matrix which is periodic of period P , A(t+P ) = A(t). Then the matrizant X(t, 0) of (5.2) has a representation of the form X(t, 0) = Z(t)etB ,

where

Z(t + P ) = Z(t)

is P -periodic .

The matrix B, generally speaking, is complex-valued. If in system (5.2) the matrix A(t) is real-valued with period P , then the (real-valued) matrizant X(t, 0) of system (5.2), the system regarded as a periodic one with period 2P , has a representation of the form X(t, 0) = Z(t)etB1 ,

where

Z(t + 2P ) = Z(t)

is 2P -periodic ,

while B1 is a real-valued matrix. Proof. We have proved that X(t+P, 0) = X(t, 0)C. For every C with the property det C = 0 there is a (non-unique) matrix B such that C = eP B ,

Global Existence for Nonlinear Wave Equations

331

i.e., C has logarithm P B. Hence, X(t + P, 0) = X(t, 0)eP B . Define Z(t) by

Z(t) = X(t, 0)e−tB ,

then =

  X(t + P, 0)e−(t+P )B = X(t + P, 0)e−P B e−tB     X(t, 0)Ce−P B e−tB = X(t, 0)eP B e−P B e−tB

=

X(t, 0)e−tB = Z(t, 0) .

Z(t + P, 0) =

Thus, it holds Z(t + P, 0) = Z(t, 0) as claimed. We shall now assume that A(t) is a real matrix, X(t, t0 ) is a matrizant, and C is a monodromy matrix, so that X(t + P, 0) = X(t, 0)C and C is real. We have X(t + 2P, 0) = X(t, 0)C 2 . One can easily prove that there exists a real matrix B1 such that e2P B1 = C 2 .


The theorem is proved.

7. If µ ∈ C is a multiplier for the Hill’s equation (5.1), then 1/µ is also a multiplier. Proof. Due to Liouville’s formula for the determinant W (t) of the matrizant X(t, t0 ) we have W (t) ≡ 1. Hence, det X(1, 0) = 1. From characteristic equation (5.3) of the definition of the multipliers and from Vieta’s Theorem for the products
of the multipliers µ1 and µ2 , we get µ1 µ2 = 1, which proves the statement. If y1 (t) and y2 (t) are two solutions to (5.1), which take initial data y1 (0) = 0 ,

y1 (0) = 1 ,

y2 (0) = 1 ,

y2 (0) = 0 ,

then we can give a “more explicit” formula for the monodromy matrix and for the multipliers. Indeed,       y1 (t) y2 (t) y1 (1) y2 (1) Xλ (t, 0) = = =⇒ Xλ (1, 0) = , y1 (t) y2 (t) y1 (1) y2 (1) and the characteristic equation (5.3) becomes µ2 − [y1 (1) + y2 (1)]µ + 1 = 0 . The function

∆(λ) = y1 (λ, 1) + y2 (λ, 1) = trace Xλ (1, 0) is called the discriminant of Hill’s equation. 8. Consider now the family of equations wtt + λ2 α(t)w = 0, where λ is a complex parameter and α is smooth, π-periodic such that α(t) ≥ 0

for all t ∈ R .

(5.6)

332

Karen Yagdjian

In the following lemma the periodicity assumption can be omitted. Lemma 5.2. The discriminant ∆(λ), ∆(λ) := y1 (λ, π) + y2 (λ, π) = trace Xλ (π, 0), is an entire function such that ∆(0) = 2 and
(∂λ2 ∆)(0) = −2π

π

α(s)ds .

0

Proof. Let y1 (λ; t) and y2 (λ; t) be two solutions to equation (5.6), which take initial data y2 (λ; 0) = 1 , y2 (λ; 0) = 0 . y1 (λ; 0) = 0 , y1 (λ; 0) = 1 , First we note that these functions are holomorphic in λ ∈ C. Then y1 (0; t) = t and y2 (0; t) = 1, so that ∆(0) = 2. By definition we have ∂λ ∆(λ) = ∂λ y1 (λ, π) + ∂λ y2 (λ, π) ,

∂λ2 ∆(0) = (∂λ2 y1 )(0, π) + (∂λ2 y2 )(0, π) .

Further, from equation (5.6) we get ∂λ y1tt + λ2 α(t)∂λ y1 = −2λα(t)y1 ,
t
t ∂λ y1t (λ, t) − ∂λ y1t (λ, 0) + λ2 α(s)∂λ y1 (λ, s)ds = −2λ α(s)y1 (λ, s)ds , 0 0
t
t α(s)∂λ y1 (λ, s)ds = −2λ α(s)y1 (λ, s)ds ∂λ y1t (λ, t) + λ2 0

0

since the initial data for y1 are independent of λ. Hence,
π
2 α(s)y1 (0, s)ds = −2 (∂λ y1t )(0, π) = −2 0

π

α(s)s ds .

0

Similarly ∂λ y2tt + λ2 α(t)∂λ y2 = −2λα(t)y2
t
t 2 ∂λ y2t (λ, t) − ∂λ y2t (λ, 0) + λ α(s)∂λ y2 (λ, s)ds = −2λ α(s)y2 (λ, s)ds 0 0
t
t α(s)∂λ y2 (λ, s)ds = −2λ α(s)y2 (λ, s)ds ∂λ y2t (λ, t) + λ2 0 0
t
τ
t
τ 2 dτ α(s)∂λ y2 (λ, s)ds = −2λ dτ α(s)y2 (λ, s)ds ∂λ y2 (λ, t) + λ 0

implies (∂λ2 y2 )(0, π)

0




0




0


π
τ α(s)y2 (0, s)ds = −2 dτ α(s)ds 0 0 0 0
π
π
π = −2 α(s)ds dτ = −2 α(s)(π − s)ds 0 s 0
π
π α(s)ds + 2 α(s)sds = −2π π

= −2

τ

dτ

0

0

Global Existence for Nonlinear Wave Equations

333


which completes the proof. 9.

Theorem 5.3 (Floquet). (a) Assume that ∆2 > 4 for some λ. Then there exist a real number γ and two continuous functions p1 (t) and p2 (t), which are π-periodic or π-semi-periodic functions as ∆ > 2 or ∆ < −2, in such a way that any solution to equation (5.6) can be written as follows: w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt , with some C1 , C2 ∈ C. (b) Assume that ∆2 = 4. Then equation (5.6) admits at least one 2π-periodic solution. Namely, it has a π-periodic solution when ∆ = 2 and π-semi-periodic solution when ∆ = −2. Proof. (a) Assume that ∆ > 2. Then the eigenvalues µ1 and µ2 of the monodromy matrix Xλ (π, 0), are positive, and there are nontrivial solutions y1 (t) and y2 (t) such that y1 (t + π) = µ1 y1 (t) = eln µ1 y1 (t) ,

y2 (t + π) = µ2 y2 (t) = eln µ2 y2 (t)

for all t ∈ R. Then the functions p1 (t) := e−t(ln µ1 )/π y1 (t) and p2 (t) := e−t(ln µ2 )/π y2 (t) are π -periodic. Indeed, pk (t + π) = =

e−(t+π)(ln µk )/π yk (t + π) = e−(t+π)(ln µk )/π eln µk yk (t) e−t(ln µk )/π yk (t) = pk (t) ,

k = 1, 2 .

t(ln µ1 )/π

and y2 (t) = p2 (t)et(ln µ2 )/π form a fundamenThe functions y1 (t) = p1 (t)e tal system of solutions. Assume now that ∆ < −2, then µ1 and µ2 are negative, so that ln µk = ln |µk | + iπ, k = 1, 2, and defined by p1 (t) := e−t(ln µ1 )/π y1 (t) and p2 (t) := e−t(ln µ2 )/π y2 (t) functions are not π-periodic. Nevertheless one can write yk (t) = pk (t)et(iπ+ln |µk |)/π = pk (t)eit et(ln |µk |)/π , pk (t) = yk (t)e−it e−t(ln |µk |)/π , pk (t)eit = yk (t)e−t(ln |µk |)/π , where the functions pk (t)eit , k = 1, 2, are π-semi-periodic. Indeed, pk (t + π)ei(t+π)

=

yk (t + π)e−(t+π)(ln |µk |)/π

=

µk yk (t)e−(t+π)(ln |µk |)/π

=

eln µk yk (t)e−(t+π)(ln |µk |)/π

=

eln |µk |+iπ yk (t)e−(t+π)(ln |µk |)/π

=

eiπ yk (t)e−t(ln |µk |)/π

=

−pk (t)eit ,

k = 1, 2 .

It remains to denote γ := ln |µ1 | = − ln |µ2 |, when ln |µ1 | > 0.

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Karen Yagdjian

(b) If ∆ = 2, then the characteristic equation has a double zero at µ = 1. The function X(t, 0)c, where c is an eigenvector of C, is a π-periodic solution. If ∆ = −2, then the characteristic equation has a double zero at µ = −1. The function X(t, 0)c, with c chosen as an eigenvector of C, is a π-semi-periodic solution.
The case ∆2 < 4 can be considered similarly, and we leave it to the reader. 5.2. Borg’s Theorem 10. The previous theorem describes a structure of the solutions by means of the discriminant ∆ of Hill’s equation, that is in an implicit way, assuming a knowledge of the two independent solutions. On the other hand one prefers to obtain the same conclusion starting from the coefficients of the equation. This is a more difficult problem and only a partial answer will be given in the next theorem. The following theorem proves the existence of at least one instability zone Λ, that is, the existence of parametric resonance. The main feature of the parametric resonance is the exponentially increasing amplitude of oscillatory systems, whereas in the case of an ordinary resonance they increase with a power law. Theorem 5.4 (G. Borg’46 [6]). Assume that α is non-constant, positive, and πperiodic function. Then there exists an open interval Λ ⊂ (0, ∞) such that for every given λ ∈ Λ any solution of (5.6) can be written in the form w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt ,

C1 , C2 ∈ C ,

where γ is a positive number and both functions p1 (t) and p2 (t) are π-periodic or π-semi-periodic functions. Proof. The proof consists of the following four steps: Step 1. The function ∆2 (λ) − 4 has only real zeros. Indeed, assume that ∆2 (λ0 ) = 4. Then by Floquet’s theorem, there exists a nontrivial solution w0 (t) to the equation wtt + λ20 α(t)w = 0, which is π-periodic or π-semi-periodic. Multiplying by w0 (t) and integrating we derive
π
π w0tt (s)w0 (s)ds + λ20 α(s)w0 (s)w0 (s)ds = 0 . 0

0

It follows




w0t (π)w0 (π) − w0t (0)w0 (0) −




π

w0t (s)w0t (s)ds + 0

Hence,


0

Thus λ0 is real.




π

π

α(s)|w0 (s)|2 ds = 0 . 0

|w0t (s)|2 ds + λ20

−

λ20

π

α(s)|w0 (s)|2 ds = 0 . 0

Global Existence for Nonlinear Wave Equations

335

Step 0 π 2. The entire function ∆(λ) has exponential growth with type not greater than α(s)ds, that is, for every positive δ there is a constant C(δ) such that 0 &  

π α(s)ds + δ |λ| for all λ ∈ C . |∆(λ)| ≤ C(δ) exp 0

For a given λ ∈ C we define the energy E(t; λ) of the solution w of Hill’s equation by E(t; λ) = |λ|2 α(t)|w(t)|2 + |w (t)|2 . Then we get d E(t; λ) = |λ|2 α (t)|w(t)|2 + |λ|2 α(t)2$(w(t)w (t)) + 2$(w (t)w (t)) dt = |λ|2 α (t)|w(t)|2 + |λ|2 α(t)2$(w(t)w (t)) − 2$(w (t)λ2 α(t)w(t))   |α (t)| E(t; λ) + 2|λ| a(t)(|λ| α(t)|w(t)|)|w (t)| ≤ α(t)   +2|λ| α(t)(|λ| α(t)|w(t)|)|w (t)|     |α (t)| + 2|λ| α(t) E(t; λ) . ≤ a(t) It follows



t

E(t; λ) ≤ E(0; λ) exp 0

&   |α (s)| + 2|λ| α(s) ds . α(s)

This inequality holds for any solution w(t). If we set w = y1 and w = y2 , then we get for the energies E1 (π; λ) and E2 (π; λ) of that solutions 
& 
π   |α (s)| + 2|λ| α(s) ds E1 (π; λ) ≤ E1 (0; λ) exp α(s) 0 
& 
π   |α (s)| = exp + 2|λ| α(s) ds , α(s) 0 
& 
π   |α (s)| + 2|λ| α(s) ds E2 (π; λ) ≤ E2 (0; λ) exp α(s) 0 
& 
π   |α (s)| = |λ|2 α(0) exp + 2|λ| α(s) ds , α(s) 0 respectively, since E1 (0; λ) = 1 and E2 (0; λ) = |λ|2 α(0). Hence for the discriminant we have |∆(λ)|

= ≤ ≤

|y1 (λ, π) + y2 (λ, π)| ≤ |y1 (λ, π)| + |y2 (λ, π)| 3  E2 (π; λ) E1 (π; λ) + |λ|2 α(π) 
& 
π

    1 |α (s)| + |λ| α(s) ds . 1 + α(0)/α(π) exp 2 α(s) 0

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Karen Yagdjian

Then for δ = 0 (we reserve δ = 0 in the next estimate for Remark 5.5) we obtain &  

π α(s)ds + δ |λ| , |∆(λ)| ≤ C(δ) exp 0

which proves the claimed statement. Step 3. If ∆2 (λ) ≤ 4 for any real λ, then there exists a complex number B such that ∆(λ) = 2 cos(Bλ) . Indeed, the function 4−∆2 (λ) does not vanish (Step 1) for non-real λ. Assume 2 that it is nonnegative on R. Then all zeros of 4 − ∆ (λ) are of even multiplicity. 2 Hence, 4 − ∆ (λ) is an entire function, too. According to Step 2 the function  4 − ∆2 (λ) has exponential growth. Remind ∆(0) = 2. Denote  f (λ) := 1 − ∆2 (λ)/4 and g(λ) := ∆(λ)/2 , g(0) = 1 . The proof of this step will be completed if we apply the following theorem from complex analysis: Let f and g be entire functions with exponential growth. If f (z)2 + g(z)2 = 1 and g(0) = 1, then there is a complex number B ∈ C such that f (z) = sin(Bz) and g(z) = cos(Bz). To prove the last theorem we consider g + if and g − if , then (g + if )(g − if ) ≡ 1, so that they do not vanish anywhere on C. Therefore g + if = exp(A + Bz) and g − if = exp(−A − Bz) while g(0) = 1 leads to A = 0. Step 4. Completion of the proof. If the statement of the theorem is not true, then by Floquet’s theorem ∆2 (λ) ≤ 4 on [0, ∞). But ∆(λ) is an even function, so that ∆2 (λ) ≤ 4 on R. By Step 3 this implies ∆(λ) = 2 cos(Bλ) for some B ∈ C. It follows 1 B 2 = − (∂λ2 ∆)(0) . 2 Moreover, by Lemma 5.2 we have
π 1 2 2 B = − (∂λ ∆)(0) = π α(s)ds . 2 0 By Step 2 the entire function ∆(λ) has exponential growth with type not greater 0π than 0 α(s)ds. Hence, 
2
π
π
π |B| ≤ α(s)ds =⇒ = π α(s)ds ≤ α(s)ds . 0

0

0

At the same time, by the Cauchy-Schwarz inequality 
2 
 

π
π
π
α(s)ds ≤ α(s)ds 1ds = π 0


π

0





π

π

α(s)ds = 0

0

 α(s)ds

0

2 ,

0

π

α(s)ds

= =⇒

Global Existence for Nonlinear Wave Equations which can hold only for constant function α(t). The theorem is proved.

337


Remark 5.5. Borg’s theorem can be proved for non-negative functions α as well. In this case δ = 0 helps. (See, also, [8].) According to Borg’s theorem for every given λ ∈ Λ ⊂ (0, ∞) any solution of (5.6) can be written in the form w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt ,

C1 , C2 ∈ C ,

where γ is a positive number and p1 (t) and p2 (t) are two π-periodic or π-semiperiodic functions. Then according to the following corollary, ∆(λ) > 2 or ∆(λ) < −2, respectively. Corollary 5.6. If for a given λ ∈ C any solution of (5.6) can be written in the form w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt ,

C1 , C2 ∈ C ,

where γ is a positive number and p1 (t) and p2 (t) are two π-periodic or π-semiperiodic functions, then ∆(λ) > 2 or ∆(λ) < −2, respectively. Proof. Indeed we have w(0) = w (0) =

C1 p1 (0) + C2 p2 (0) , C1 p1 (0) + C1 p1 (0)γ + C2 p2 (0) − C2 p2 (0)γ .

In particular for the solution y1 this implies y1 (0) = y1 (0) =

a1 p1 (0) + a2 p2 (0) = 0 , a1 p1 (0) + a1 p1 (0)γ + a2 p2 (0) − a2 p2 (0)γ = 1 .

One can easily obtain from the last system of equations p2 (0) , p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) p1 (0) , p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0)

a1

= −

a2

=

since p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) = 0. Similarly from y2 (0) = b1 p1 (0) + b2 p2 (0) = 1 , y2 (0) = b1 p1 (0) + b1 p1 (0)γ + b2 p2 (0) − b2 p2 (0)γ = 0 , it follows b1 b2

p2 (0) − γp2 (0) , p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) p1 (0) + γp1 (0) . = − p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) =

338

Karen Yagdjian

We denote D = 1/(p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0)), then ∆(λ)

= y1 (π) + y2 (π) = a1 p1 (π)eγπ + a2 p2 (π)e−γπ + a1 p1 (π)γeγπ − a2 p2 (π)γe−γπ + b1 p1 (π)eγπ + b2 p2 (π)e−γπ = a1 p1 (0)eγπ + a2 p2 (0)e−γπ + a1 p1 (0)γeγπ − a2 p2 (0)γe−γπ + b1 p1 (0)eγπ + b2 p2 (0)e−γπ  −p2 (0)p1 (0)eγπ +p1 (0)p2 (0)e−γπ −p2 (0)p1 (0)γeγπ −p1 (0)p2 (0)γe−γπ =  + (p2 (0) − γp2 (0))p1 (0)eγπ − (p1 (0) + γp1 (0))p2 (0)e−γπ D    = eγπ p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp2 (0)p1 (0)   + e−γπ p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp2 (0)p1 (0) D = eγπ + e−γπ .

Hence we have proved that for the periodic functions p1 (t) and p2 (t) one has ∆(λ) > 2. The case of the anti-periodic functions can be considered in a similar way. The corollary is proved.
5.3. Construction of an exponentially increasing solution to Hill’s equation 11. About the structure of the monodromy matrix. According to the next key lemma one can find in the instability zone Λ a number λ such that a non-diagonal element of the monodromy matrix does not vanish. Moreover, this property is stable under small perturbations of λ. Lemma 5.7 (K. Yagdjian [41], [42]). Let b(t) defined on R be a non-constant, positive, smooth function which is 1-periodic. Then there exists an open subset Λ0 ⊂ Λ such that b21 = 0 for all λ ∈ Λ0 . Proof. If we assume that b21 = 0, then the function w(t) = x21 (t, 0), where x21 (t, 0) is the (2, 1)-element of the matrix   x11 (t, 0) x12 (t, 0) Xλ (t, 0) = , x21 (t, 0) x22 (t, 0) solves equation (5.1) and takes boundary values w(0) = w(1) = 0 . Hence, if λ does not belong to the spectrum of the Sturm-Liouville problem (5.1), w(0) = w(1) = 0, then w(t) vanishes identically. It follows dx21 /dt(0, 0) = x11 (0, 0) = 0 and we get a contradiction to the initial condition x11 (0, 0) = 1 of the definition of the fundamental solution. It remains to take into consideration that the above mentioned spectrum is discrete. The lemma is proved.
Next we use the periodicity of b = b(t) and the eigenvalues µ0 , µ−1 of 0 the matrix Xλ (1, 0) to construct solutions of (5.1) with prescribed values on a discrete set of time. The eigenvalues of the matrix Xλ (1, 0) are µ0 and µ−1 0 with

Global Existence for Nonlinear Wave Equations

339

−1 ∆(λ) = b11 + b22 = µ0 + µ−1 0 . Hence (b11 − µ0 ) + (b22 − µ0 ) = −µ0 + µ0 implies −1 |b11 − µ0 | + |b22 − µ0 | ≥ |(b11 − µ0 ) + (b22 − µ0 )| = |µ0 − µ0 | > 0. This leads to 1 max{|b11 − µ0 |, |b22 − µ0 |} ≥ |µ0 − µ−1 0 | > 0. 2 Without loss of generality we can suppose 1 1 −1 |b22 − µ−1 |b11 − µ0 | ≥ |µ0 − µ−1 0 | > 0, 0 | ≥ |µ0 − µ0 | > 0 , 2 2 because of b11 − µ0 = −(b22 − µ−1 0 ). Further,

1−

b21 b12 − b22 µ0 − b11

µ−1 0

=

1 − b22 µ0 − b11 µ−1 0 + b11 b22 − b21 b12 (µ−1 − b 22 )(µ0 − b11 ) 0

=

1 − b22 µ0 − b11 µ−1 0 +1 −1 (µ0 − b22 )(µ0 − b11 )

=

−1 −1 µ0 µ−1 0 − b22 µ0 − b11 µ0 + µ0 µ0 −1 (µ0 − b22 )(µ0 − b11 )

=

−1 µ0 (µ−1 0 − b22 ) − (b11 − µ0 )µ0 (µ−1 0 − b22 )(µ0 − b11 )

=

µ0 (b11 − µ0 ) − (b11 − µ0 )µ−1 0 (µ−1 − b )(µ − b ) 22 0 11 0

= =

(µ0 − µ−1 0 )(b11 − µ0 ) −1 (µ0 − b22 )(µ0 − b11 ) 1 (µ0 − µ−1 = 0 . 0 ) b22 − µ−1 0

Thus, 1−

b12 b21 1 = (µ0 − µ−1 = 0 . 0 ) µ − b µ−1 − b b − µ−1 11 22 0 22 0 0

Lemma 5.8. [41] Let W = W (t) be a solution to (5.1) with parameter λ such that b21 = 0. Suppose that W = W (t) takes the initial data W (0) = 0 ,

Wt (0) = 1.

Then for every positive integer number M ∈ N one has b21 −M W (M ) = (µM ). 0 − µ0 µ0 − µ−1 0

(5.7)

Proof. Let w = w(t) and z = z(t) be the solutions of (5.1) with initial data w(0) = 1 ,

wt (0) = b12 /(µ0 − b11 ) ,

z(0) = b21 (µ−1 0 − b22 ) ,

Then for every positive integer number M ∈ N they satisfy

wt (M ) = µM w(M ) = µM 0 , 0 b12 /(µ0 − b11 ) , −M z(M ) = µ0 b21 /(µ−1 − b zt (M ) = µ−M . 22 ) , 0 0

zt (0) = 1.

(5.8)

340

Karen Yagdjian

Indeed, for the function w(t) we have  d   d  w(0) dt w(M ) dt = X(M, 0) w(M ) w(0)



X(M, M − 1)X(M − 1, M − 2) · · · X(1, 0) 566 7 4

=

M−multipliers



X(1, 0)X(1, 0) · · · X(1, 0) 4 566 7

=

d dt w(0)

d dt w(0)



w(0)

 .

w(0)

M−multipliers

The matrix

 B=

b12 /(µ0 − b11 ) 1 1 b21 /(µ−1 0 − b22 )

 (5.9)

with det B

=

B −1

=

b21 b12 1 − 1 = (µ−1 = 0 , 0 − µ0 ) µ − b µ−1 − b b − µ−1 0 11 22 22 0 0   1 −1 b21 /(µ−1 0 − b22 ) , −1 b12 /(µ0 − b11 ) det B

is a diagonalizer for X(1, 0), that is,   µ0 0 B −1 X(1, 0)B = =: M , 0 µ−1 0 It follows 

d dt w(M )

 X(1, 0)B = B

µ0 0



= X(1, 0)X(1, 0) · · · X(1, 0) 4 566 7

d dt w(0)

.



w(0)

M−multipliers

= B (B 4





w(M )

−1

0 µ−1 0

X(1, 0)B)(B

−1

X(1, 0)B) · · · (B 566

−1

X(1, 0)B) B 7

−1



d dt w(0)



w(0)

M−multipliers

= B MM ·56 ·6· MM7 B 4

−1



d dt w(0)



w(0)

M−multipliers M

= BM B  M µ0 =B 0

 b12 /(µ0 − b11 ) 1    0 b12 /(µ0 − b11 ) −1 . B 1 µ−M 0

−1



On the other hand   −1  b12 b21 − b22 ) −1 b21 /(µ−1 −1 0 −1 B = , −1 b12 /(µ0 − b11 ) µ−1 0 − b22 µ0 − b11

Global Existence for Nonlinear Wave Equations so that B

−1



b12 /(µ0 − b11 ) 1



−1 b12 b21 −1 = −1 µ0 − b22 µ0 − b11   1 = . 0 

Hence, 

d dt w(M )





341

µM 0 0



b21 (µ−1 0 −b22 )

−1

0 µ−M 0



1 0

−1



b12 (µ0 −b11 )

b12 (µ0 −b11 )



1



= B   M  b12 /(µ0 − b11 ) 1 µ0 = 0 1 b21 /(µ−1 0 − b22 )   b12 /(µ0 − b11 ) = µM , 0 1

w(M )

and the first line of (5.8) is proved. The solution z(t) can be considered in a similar way which leads to  d  d   dt z(M ) dt z(0) = BMM B −1 z(M ) z(0) ⎛ ⎞   M 1 µ0 0 ⎠. b21 B −1 ⎝ = B 0 µ−M −1 0 (µ0 − b22 ) On the other hand   1 −1 B b21 (µ−1 0 −b22 )



−1 b12 b21 = − 1 µ−1 0 − b22 µ0 − b11   0 = 1 implies 

d dt z(M )

z(M )





b21 (µ−1 0 −b22 )

−1

−1 b12 (µ0 −b11 )



1 b21 (µ−1 0 −b22 )

  0 µM 0 0 1 0 µ−M 0    b12 /(µ0 − b11 ) 1 0 µ−M . 0 1 1 b21 /(µ−1 − b ) 22 0 

= =

B

It follows the second line of (5.8). Further, we have W (t) = cw w(t) + cz z(t), and W (0) = cw w(0) + cz z(0) = 0 ,

Wt (0) = cw wt (0) + cz zt (0) = 1 ,



342

Karen Yagdjian

where the numbers cw , cz are easily calculated: z(0) w(0) , cz = − , wt (0)z(0) − w(0)zt (0) wt (0)z(0) − w(0)zt (0) b21 b21 −1 b21 (µ−1 − b ) (µ 22 0 0 − b22 ) cw = , = =− 1 b12 b21 (µ0 − µ−1 −1 0 ) (µ − µ ) − 1 0 0 (µ0 − b11 ) (µ−1 b22 − µ−1 0 0 − b22 ) cw =

cz = −

1 b12 b21 −1 (µ0 − b11 ) (µ−1 0 − b22 )

=

(b22 − µ−1 0 ) −1 . (µ0 − µ0 )

Hence (5.8) and the representation W (t) = w(t)

b21 (b22 − µ−1 0 ) + z(t) −1 −1 (µ0 − µ0 ) (µ0 − µ0 )

imply W (M ) = = =

w(M )

b21 (b22 − µ−1 0 ) −1 + z(M ) −1 (µ0 − µ0 ) (µ0 − µ0 )

b21 b21 (b22 − µ−1 −M 0 ) + µ 0 −1 −1 −1 (µ0 − µ0 ) (µ0 − b22 ) (µ0 − µ0 ) b21 −M (µM ). 0 − µ0 µ0 − µ−1 0

µM 0




The lemma is proved. 5.4. Construction of blow-up solutions In fact equation (4.4) is transformed into the linear wave equation by 

u
s v = G(u) := exp f (r)dr ds . 0

(5.10)

0

Since G ∈ C 2 (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C 2 (a, b)), where we denote a := lim G(u) , u→−∞

b := lim G(u) . u→∞

Since condition (4.2) does not hold, one has a > −∞ or b < ∞. If u(t, x) is a solution of (4.4) and takes initial values (4.6), then the function (5.10) solves the linear equation vtt − b2 (t)∆v = 0 ,

(5.11)

Global Existence for Nonlinear Wave Equations and takes initial values 


u0 (x) exp v(0, x) = 0






s

f (r)dr ds ,

vt (0, x) = u1 (x) exp

0

343 

u0 (x)

f (r)dr

.

0

(5.12) Now let us choose initial data (the positive number S > 2n is fixed) 1  x  u0 (x) = ∈ C0∞ , χ MS M2 
 u0 (x) A  x  exp − u1 (x) = χ f (r)dr cos(x · y) ∈ C0∞ , MS M2 0 where y ∈ Rn , |y|2 = λ while χ ∈ C0∞ (Rn ) is a non-negative cut-off function, χ(x) = 1 when |x| ≤ 1. The number A ∈ R, |A| = 1, independent of the large parameter M ∈ N, will be chosen later. Let u = u(t, x) be the  classical  solution which takes these initial data. Then the function v(t, x) = G u(t, x) solves the equation (5.11) and takes values 

1S χ( x2 )
s M M v(0, x) = exp f (r)dr ds ∈ C0∞ , 0

0

vt (0, x)

=

Consider the function 

1S
M V (t, x) = exp 0

0

A  x  cos (x · y) ∈ C0∞ . χ MS M2 s

 A f (r)dr ds + W (t) S cos(x · y) ∈ C ∞ ([0, ∞] × Rn ) . M

The function V (t, x) solves equation (5.11) while 

s
1S M A exp f (r)dr ds, Vt (0, x) = S cos(x · y) for all x ∈ Rn . V (0, x) = M 0 0 On the other hand for v(t, x) we have 

s
1S M A exp f (r)dr ds, vt (0, x) = S cos(x · y) when |x| ≤ M 2 . v(0, x) = M 0 0 The finite propagation speed in the Cauchy problem (5.11), (5.12) implies V (t, x) = v(t, x)

ΠM := [0, M ] × {x ∈ Rn ; |x| ≤ M 3/2 }

in

for large M . Hence 


1S M exp v(t, x) =



s

f (r)dr ds + W (t)

0

0

In particular,


1 MS

v(M, 0) = 0



exp

s

 f (r)dr ds +

0

A cos(x · y) in ΠM . MS

b21 A −M (µM ). 0 − µ0 M S µ0 − µ−1 0

344

Karen Yagdjian

Assume now that b < ∞. Then global existence of u means 

u(t,x)
s v(t, x) = exp f (r)dr ds < b for all t ≥ 0, x ∈ Rn . 0

0

We choose A ∈ R, |A| = 1, and S such that for M large enough one has v(0, x) < b for all x ∈ Rn and all M ∈ N, while v(M, 0) > b as well as (4.5) for u0 , u1 . The theorem is proved. Open problems: (i) Find all u0 , u1 (possible with small Sobolev norms) such that with these data a classical solution exists globally. (ii) Prove that if u0 ≡ 0, then for every u1 ∈ C0∞ (Rn ) there is no global solution.

6. Coefficient stabilizing to a periodic one. Parametric resonance dominates After studying the periodic case the next question is that for corresponding results for equations with a coefficient stabilizing to a periodic one. Equations with coefficient being a product of a periodic function and a function stabilizing to a constant, are considered in many papers and books (see, e.g., Ch.4 [12]). Following this let us restrict to the model equation   (6.1) utt − exp(2tα )b2 (t)∆u + f (u) (ut )2 − exp(2tα )b2 (t)|∇ ∇x u|2 = 0 , t ∈ [1, ∞), x ∈ Rn , for α ∈ R, α < 0. Here f ∈ C ∞ (R) is a real-valued function while b = b(t) is defined on R, a periodic, non-constant, smooth, and positive function. We study the global solvability of the Cauchy problem with data prescribed on t = 1: u(1, x) = u0 (x), ut (1, x) = u1 (x) . (6.2) First we note that condition (4.2), 

∞
s exp f (r)dr ds = ∞ 0

0








0

and

s

exp −∞

 f (r)dr ds = ∞,

0

is equivalent to the following: for every given numbers n1 , n2 , n3 , n4 ∈ R, 

 
s
n3
s
∞ exp f (r)dr ds = ∞ and exp f (r)dr ds = ∞. n1

n2

−∞

n4

(6.3) In Section 11 we prove the global existence for small data for the equation (6.1) with at most fast oscillating coefficients, that is for the case α ∈ [1/2, ∞). The next theorem shows that oscillations, which approach for large time the pure periodic behavior, in general break the global existence. For the equation (6.1) with f (u) ≡ 1 see [41], [43]. Theorem 6.1. Let α ∈ (−∞, −1) while b = b(t) defined on R, be a periodic, non-constant, smooth, and positive function. Suppose that (6.3) does not hold.

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345

Then for every n, s, and for every positive δ there are data u0 ∈ C0∞ (Rn ) and u1 ∈ C0∞ (Rn ) such that the inequality u0 (s+1) + u0 (s) ≤ δ

(6.4)

is fulfilled, but a solution u ∈ C 2 ([1, ∞) × Rn ) to the Cauchy problem (6.1), (6.2) does not exist. In [30] this statement is proved for equation (6.1) with α ∈ [−1, 0). Open problem: Consider the case α ∈ (0, 1/2). The parametric resonance can be a mechanism of the blow-up phenomenon in hyperbolic systems, too. To illustrate this consider the system

∇x u|2 + cu (ut )2 + du u = 0 , utt − (1 + v 2 )∆u − (1 + v 2 )|∇ 2 −1 vtt − (1 + (1 + u ) )∆v − (1 + (1 + u2 )−1 )|∇ ∇x v|2 + cv (vt )2 + dv v = 0, (6.5) where cu = dv = 1, du = cv = 0. Then for every n, s, p, and for every positive δ there are data u0 , u1 , v0 , v1 ∈ C0∞ (Rn ) such that u1 s,2 + ∇u0 s,2 + u1 s,p + ∇u0 s,p + v1 s,2 + ∇v0 s,2 + v1 s,p + ∇v0 s,p < δ, but a solution (u, v), u, v ∈ C 2 (R+ × Rn ), to the Cauchy problem u(0, x) = u0 (x), ut (0, x) = u1 (x), v(0, x) = v0 (x), vt (0, x) = v1 (x), does not exist. (Here ϕs,p denotes the Sobolev norm of ϕ, the element of the Sobolev space W s,p (Rn ), 1 ≤ p ≤ ∞.) We will prove this in a forthcoming paper.

7. Proof of Theorem 6.1: Perturbation theory 1. Perturbation theory for the fundamental solutions and conclusions Consider again the auxiliary ordinary differential equation (5.1) with a periodic coefficient b = b(t). As in the previous section, X(t, t0 ) denotes the fundamental solution to the equation (5.1). Then X(t + 1, t) is independent of t ∈ N. Set for t0 ∈ N   b11 b12 X(t0 + 1, t0 ) = X(1, 0) = . (7.1) b21 b22 The continuous dependence of the fundamental solution X(t, t0 ) on the coefficient b2 (t) on the interval [t0 , t0 + 1] leads together with Lemma 5.7 to the following conclusions. Lemma 7.1. Let us consider instead of (5.1) the ordinary differential equation wtt + λ0 c2 (t)w = 0, where c = c(t) is a real-valued smooth function defined on R. Then to each ε > 0 there exists a δ = δ(ε) such that if the continuous function c(t) is a perturbation of b(t) on the interval [T, ∞) in the sense that supt∈[T,∞) |b(t) − c(t)| < δ , then the fundamental solution X(t, t0 ) has at the point t = t0 + 1 a real-valued eigenvalue µ = µ(t0 ) which satisfies |µ − µ0 | < ε uniformly for all t0 ∈ [T, ∞). Moreover, one has |b21 | ≥ const > 0 uniformly for all t0 ∈ [T, ∞).

346

Karen Yagdjian

2. Representation of solutions of special ordinary differential equations We use Lemma 7.1 to estimate the solutions for a class of Cauchy problems for special ordinary differential equations. We prescribe Cauchy data on t = t0 , where for M ∈ N the number t0 = t0 (M ) will be denoted by tM , while the integer number tM is defined as follows: !

1 "  &− |α| 1 1 tM := ln 1 + o(1) + 1 , o(1) → 0 as M → ∞ . (7.2) 2 M The sequence o(1) will be specified more exactly later in Lemma 7.4 by the condition o(1)M |α|−1 → ∞ as M → ∞. The constant M , M > 1, will be chosen later such that tM ∈ [T, ∞), T = T (α, δ), T is large and max t∈[T (α,δ),∞)

| exp(2tα ) − 1| = | exp(2T (α, δ)α ) − 1| < δ(ε),

where |µ0 | − ε > 1, ε > 0. It is obvious that for α ∈ (−∞, −1) 1 max | exp(2tα ) − 1| ≤ o(1) , o(1) → 0 as M → ∞ . t∈[tM ,∞) M Next we construct solutions with prescribed asymptotics on a discrete set of time. Lemma 7.2. Consider the ordinary differential equation wtt + λ exp(2tα )b2 (t)w = 0 ,

(7.3)

where b(t) defined on R, is a 1-periodic, non-constant, smooth, and positive , (t) be a solution to equation (7.3) with initial data function. Let W , (tM ) = 0 , W

,t (tM ) = 1 . W

Then there are positive numbers C0 , µ ,0 , 1 < µ ,0 < µ0 , and a real-valued function C(M ) such that one has M , (tM + M ) = C(M ), W µ0 ,

|C(M )| ≥ C0 > 0 ,

for every sufficiently large positive integer number M ∈ N. Proof. Let the matrix-valued function Xk (t, 0), k = 1, . . . , M , be the solution of the Cauchy problem   d 0 −λ0 exp(2(tM + k − 1 + t)α )b2 (tM + t) X= X, X(0, 0) = I, 1 0 dt where I is identity matrix. We set Xk (1, 0) =



b11 (k) b12 (k) b21 (k) b22 (k)

 .

It is easily seen for the differences ∆k (t) := Xk (t, 0) − X(t, 0) that with X(t, 0) from (5.2) the following estimate 1 for all t ∈ [0, 1] , ∆k (t) = Xk (t, 0) − X(t, 0) ≤ o(1) M

Global Existence for Nonlinear Wave Equations

347

with o(1) → 0 as M → ∞, is fulfilled. Using Corollary 7.1 with a given arbitrary small positive ε and T = T (α, δ(ε)) leads to the following two properties for the matrices Xk (1, 0), k = 1, . . . , M : • Each matrix Xk (1, 0) has a real-valued eigenvalue νk = νk (M ) which satisfies |νk (M ) − µ0 | ≤ γ0 , where γ0 = (|µ0 | − 1)/2 > 0 uniformly for all M ∈ N sufficiently large. • For all k = 1, 2, . . . , M , M ∈ N sufficiently large, the estimate |bij (k) − bij | < ε, i, j = 1, 2, is fulfilled. Further, the matrix (5.9) is a diagonalizer for X(1, 0) of (7.1). On the other hand each matrix Xk (1, 0) also has a diagonalizer   b12 (k)/(νk − b11 (k)) 1 Bk = . 1 b21 (k)/(νk−1 − b22 (k)) The properties of Xk (1, 0), k = 1, . . . , M , lead to the following estimate for the diagonalizers Bk , k = 1, . . . , M with some constant C2 : Bk − B ≤ o(1/M ) and Bk  + Bk−1  ≤ C2 for k = 1, . . . , M.

(7.4)

For any solution w

= w(t)

of the equation (7.3) one can write    d  d

M + M) w(t

M) dt w(t dt = XM (1, 0)XM−1 (1, 0) · · · X1 (1, 0) . w(t

M + M) w(t

M) If we denote Gk := Bk−1 Bk−1 − I , k ≥ 2 , then  d  dt w(tM + M ) w(tM + M )     νM−1 0 νM 0 ) = BM (I + G (I + GM−1 ) · · · M −1 −1 0 νM−1 0 νM     d   ν2 0 ν1 0 w(tM ) −1 dt . (I + G2 ) B1 ×(I + G3 ) 0 ν2−1 0 ν1−1 w(tM ) Taking into consideration the above properties for the diagonalizers Bk , then sufficiently small ε , sufficiently large T = T (α, δ(ε)), respectively, lead to Gk  = Bk−1 Bk−1 − I ≤ o(1)

1 M

for all k = 2, . . . , M ,

with o(1) → 0 as M → ∞. From (7.4) and from the special choice of the initial data, we see that in order to prove the statement of this lemma we have to consider the (1, 1)-element x11 and the (2, 1)-element x21 of the matrix       νM 0 νM−1 0 ν1 0 (I + GM ) · · · (I + G2 ) . −1 −1 0 νM 0 νM−1 0 ν1−1

348

Karen Yagdjian

To find out the properties of the matrix    νM−1 νM 0 ) (I + G M −1 0 νM 0 = νM νM−1 . . . ν2    1 1 0 × ) (I + G M −2 0 0 νM

0 −1 νM−1



0 −2 νM−1

 ···



ν2 0

 ···

0 ν2−1

1 0 0 ν2−2

 (I + G2 )  (I + G2 )

it is sufficient to consider the matrix       1 0 1 0 1 0 (M) a := (I + GM ) · · · (I + G2 ) (I + G1 ). −2 −2 0 νM−1 0 νM 0 ν1−2 Lemma 7.3. Assume that the matrices Gk , k = 1, . . . , M , satisfy Gk  ≤ M −1 o(1)

where

o(1) → 0

as

M → ∞,

(7.5)

while |νk | ≥ const > 1, k = 1, . . . , M . Then there is a positive constant such that for sufficiently large M one has (M)

(M)

(M)

(M)

a11 = 1 + o(1) , a21 = o(1) , a12 = o(1) , |a22 | ≤ q ,

q 0 , |ν −1 − b22 (1)| ≥ C −1 > 0 , 0 1 0 ν1 − b11 (1) ν1−1 − b22 (1) b21 (M ) ν1−1 − b22 (1) z |z22 | ≤ q < 1 b21 (1) ν −1 − b (M ) 22 ≤ q1 < 1 , 22 M


for M sufficiently large. Lemma 7.2 is proved. 3. Estimate in [1, tM ] of the energy of the solutions

Lemma 7.4. Consider for a positive number λ the ordinary differential equation wtt + λ exp(2tα )b2 (t)w = 0 ,

(7.7)

where the parameter α ∈ (−∞, −1) while the function b = b(t) defined on R, is 1-periodic, smooth, and positive. Let w(t) be a solution to equation (7.7). Then with o(1) of (7.2) appropriately chosen, for the energy E(w; t) := |wt (t)|2 + λ exp(2tα )b2 (t)|w(t)|2 of the solution w the following estimate holds: E(w; 1) ≤ E(w; tM ) exp (o1 (1)M ) 1 − |α|

with o1 (1) = o(1)

M

1 −1 |α|

(7.8)

→ 0 as M → ∞.

Proof. If we denote a2 (t) := λ exp(2tα )b2 (t), then for the energy E(w; t) := (|wt (t)|2 + a2 (t)|w(t)|2 )/2 we have d E(w; t) = a (t)a(t)w(t)w(t) ¯ . dt Hence,

 d E(w; t) ≤ |a (t)| E(w; t) dt a(t)

leads to



E(w; 1) ≤ E(w; tM ) exp

tM

1

|a (t)| dt a(t)

 .

Further,
1

tM

|a (t)| dt a(t)


≤ 2|α|




tM

t

α−1

tM

dt + 2

1

1

|b (t)| dt b(t)

 ≤ 2tα M + 2tM max |b (t)|/b(t) [0,1]

1 − |α|

≤ o(1) implies (7.8). The lemma is proved.

M

1 |α|




352

Karen Yagdjian

Lemma 7.5. For a given positive number ε the function , (t)/(|W , (1)| + |W ,t (1)|) W (ε) (t) := εW solves the equation (7.3) and there exist positive constants q, q1 such that |W (ε) (tM + M )| ≥ εq1 exp(qM ) for all sufficiently large M . Proof. We use Lemma 7.2 to write with C0 > 0 , (1)| + |W ,t (1)|) |C(M )|ε, µ0 M /(|W # , ; 1) . C0 ε, µ0 M / E(W

|W (ε) (tM + M )| ≥ ≥

, ; 1) we use Lemma 7.4 applied to W ,: Then to estimate E(W #   M , ; tM ) exp o1 (1)M |W (ε) (tM + M )| ≥ C0 ε, µ0 / E(W   ,0 − o1 (1)/2) . ≥ C0 ε exp M (ln µ It remains to note that µ ,0 − ε1 > 1 with ε1 > 0. The lemma is proved.




4. Construction of blow-up solutions If u(t, x) is a solution of (6.1) and takes initial values u(1, x) = u0 (x) , then the function

ut (1, x) = u1 (x) ,







u(t,x)

s

exp

v(t, x) = G(u(t, x)) := 0

 f (r)dr ds

0

solves the linear equation vtt − exp(2tα )b2 (t)∆v = 0 , and takes initial values 

u0 (x)
v(1, x) = exp 0

s




 f (r)dr ds ,

(7.9)

vt (1, x) = u1 (x) exp

0



u0 (x)

f (r)dr

.

0

Since G (u) > 0, the inverse transformation,   u(t, x) = H v(t, x) , exists, is smooth, and has the property H(0) = 0. Now let us choose the initial data (the positive number S will be chosen later)   1  x  A  x  (ε) u0 (x) = H + W (1) S χ cos (x · y) ∈ C0∞ (Rn ), χ MS M2 M M2 
u0 (x)  A  x  (ε) W χ (1) cos (x · y) exp f (r)dr ∈ C0∞ (Rn ), − u1 (x) = t MS M2 0

Global Existence for Nonlinear Wave Equations

353

where y ∈ Rn , |y|2 = λ while χ ∈ C0∞ (Rn ) is a non-negative cut-off function, χ(x) = 1 when |x| ≤ 1. The parameter A ∈ R, |A| = 1, independent of M , will be chosen later. Due to Lemma 7.5 we have u0 , u1 ∈ C0∞ (Rn ). Moreover, for every given s, p, and δ the inequality (6.4) is fulfilled for all sufficiently large M and S. Assume that the solution u = u(t, x) to the Cauchy problem for equation (6.1) with above chosen initial data u0 (x) and  u1 (x) exists for all t ≥ 1 and all x ∈ Rn . For the function v(t, x) = G u(t, x) this leads to the global existence in the Cauchy problem for (7.9) with the initial data 1  x  A  x  + W (ε) (1) S χ cos (x · y) ∈ C0∞ , χ v0 (x) = S 2 M M M M2 A  x  (ε) cos (x · y) ∈ C0∞ v1 (x) = Wt (1) S χ M M2 prescribed at t = 1. Consider now the function 1 A + W (ε) (t) S cos (x · y) . S M M The function V (t, x) solves the equation V (t, x) =

Vtt − exp(2tα )b2 (t)∆V = 0

in

[1, ∞) × Rn ,

(7.10)

while A 1 + W (ε) (1) S cos (x · y) , S M M A (ε) Vt (1, x) = Wt (1) S cos (x · y) M for all x ∈ Rn . Compare the last relations with v0 and v1 we conclude V (1, x)

=

V (1, x) = v(1, x) ,

Vt (1, x) = vt (1, x) when

|x| ≤ M 2 .

Hence, due to the finite propagation speed in the Cauchy problem for (7.10), V (t, x) = v(t, x)

in ΠM := [1, tM + M ] × {x ∈ Rn ; |x| ≤ M 3/2 }

for large M . Thus v(t, x) =

1 A + W (ε) (t) S cos(x · y) in MS M

ΠM .

In particular, 1 A + W (ε) (tM + M ) S . MS M
∞ 
s  b := exp f (r)dr ds < ∞ .

v(tM + M, 0) = Now assume that

0

0

Due to Lemma 7.5 we can choose A ∈ R, |A| = 1, such that for M large enough one has v(tM + M, 0) > b as well as (6.4) for u0 , u1 . The theorem is proved.


354

Karen Yagdjian

8. Nonexistence for equations with permanently restricted domain of influence In this section we give an example of the influence of the decay of a time-dependent coefficient on the global existence of the solution to nonlinear hyperbolic equations. Namely, for arbitrary small initial data we will construct blowing up solutions. To this end we consider in R+ × Rn the equation utt − t−4 exp(−2t−1 )∆u + f (u)(u2t − t−4 exp(−2t−1 )|∇ ∇x u|2 ) = 0 ,

(8.1)

where u = u(t, x) is a real-valued unknown function, f is a real-valued smooth function. Thus the equation (8.1) has multiple characteristics at infinity, or, in other words, is weakly hyperbolic at infinity, or degenerates at infinity. The conditions on f for which the Cauchy problem for equation (8.1) has a global smooth solution for any smooth initial data are 

 
s
0
s
∞ exp f (r)dr ds = ∞ and exp f (r)dr ds = ∞. (8.2) 0

−∞

0

0

The Cauchy problem for (8.1) does not have a global solution for small initial data when f ≡ 1 [36]. The theorem below implies blow-up of the solution of equations with non-constant functions f . To formulate that theorem we need some auxiliary functions v0 and v1 defined by initial data u0 and u1 . We transform the equation (8.1) into the linear equation by change 

u(t,x)
s v(t, x) = G(u(t, x)) := exp f (r)dr ds . 0 ∞

0



Since G ∈ C (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C ∞ (a, b) , where we denote a := lim G(u) ≥ −∞ , u→−∞

b := lim G(u) ≤ ∞ . u→∞

(8.3)

The above transformation applied to equation (8.1) leads to the linear equation vtt − t−4 exp(−2t−1 )∆v = 0

(8.4)

for the function v = v(t, x). Moreover, v takes the initial values v(0, x) = v0 (x) and vt (0, x) = v1 (x), where 
 

u0 (x)
s u0 (x) v0 (x) = exp f (r)dr ds, v1 (x) = u1 (x) exp f (r)dr . 0

0

0

(8.5) It is easily seen that the supports of v0 and v1 are compact sets, and that v0 (s+1) + v1 (s) is small if and only if u0 (s+1) + u1 (s) is small, provided that s is large enough.

Global Existence for Nonlinear Wave Equations

355

Further, in the next theorem we use notations of Section 6.11.2 from [4] for the confluent hypergeometric function,
(0+) 1 −iaπ e Ψ(a, c; z) = Γ(1 − a) e−zt ta−1 (1 + t)c−a−1 dt (8.6) 2πi ∞eiφ −π/2 < φ + arg z < π/2, arg t = φ at the starting point, and Γ(α) is Euler’s function. In what follows the constant γ is Euler’s constant (a positive number belonging to the interval (0,1)) and ψ is the digamma function (psi function of Gauss): ψ(z) := Γ (z)/Γ(z). Theorem 8.1. Suppose that (8.2) does not hold. Assume also that the real-valued functions u0 , u1 ∈ C0∞ (Rn ) fulfill the condition

 . / π ei(x−y)·ξ Im ln |2ξ| + i + ψ(1/2) − 2γ ei|ξ| Ψ(1/2, 1; −2i|ξ|) v0 (y) dy dξ 2 Rn Rn

. / + ei(x−y)·ξ εIm ei|ξ| Ψ(1/2, 1; −2i|ξ|) v1 (y) dy dξ =: vst (x) ≡ 0 . Rn Rn

In both cases, if −∞ < a and there is a x0 ∈ Rn such that vst (x0 ) < 0, or if b < ∞ and there is a x0 ∈ Rn such that vst (x0 ) > 0, a solution u ∈ C 2 (R+ × Rn ) to the Cauchy problem with data u(0, x) = u0 (x),

ut (0, x) = u1 (x) ,

x ∈ Rn ,

(8.7)

does not exist. In particular, if we set u0 ≡ 0, then the second assumption of the theorem reads as

. / ei(x−y)·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u1 (y) dy dξ ≡ 0 , Rn Rn

or


Rn

. / eix·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u 1 (ξ) dξ ≡ 0 .

The last condition is due to relation (15) of Section 6.9.1 [4] between the confluent hypergeometric function Ψ(1/2, 1; −2i|ξ|) and the Bessel function J0 equivalent to
eix·ξ J0 (|ξ|) u1 (ξ) dξ ≡ 0 . Rn

Hence we get an evident consequence of the theorem. Corollary 8.2. Suppose that −∞ < a and b < ∞. For any real-valued function u1 ∈ C0∞ (Rn ), u1 ≡ 0, a solution u ∈ C 2 (R+ × Rn ) to the Cauchy problem with data x ∈ Rn , u(0, x) = 0, ut (0, x) = u1 (x) , does not exist.

356

Karen Yagdjian

Proof. If for a given function u1 ∈ C0∞ (Rn ), u1 ≡ 0, we assume that
. / eix·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u 1 (ξ) dξ ≡ 0 , .

Rn

/ then.Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u 1 (ξ) ≡ 0 implies u1 (x) ≡ 0, since the functions / iz
Im e Ψ(1/2, 1; −2iz) and J0 (z) have isolated zeros in z ∈ R+ . According to the next corollary the choice of a function u0 ≡ 0 does not improve essentially the situation even for small data. Corollary 8.3. Suppose that (8.2) does not hold. Then for every n, s, and for every positive δ there are data u0 ∈ C0∞ (Rn ) and u1 ∈ C0∞ (Rn ) such that u0 (s+1) + u1 (s) ≤ δ

(8.8)

but a solution u ∈ C 2 (R+ × Rn ) to the Cauchy problem for (8.1) with data (8.7) does not exist. The next statement shows that for the Cauchy problem for (8.1) with “large datum” u1 one can manage a blow-up at a prescribed time t = t0 > 0. Corollary 8.4. Suppose that −∞ < a and b < ∞. For every given positive t0 one can find a real-valued function u1 ∈ C0∞ (Rn ), u1 ≡ 0, such that the solution u(t, x) to the Cauchy problem for (8.1) with data u(0, x) = 0,

ut (0, x) = u1 (x) ,

x ∈ Rn ,

develops a singularity not later than at time t0 . Proof of Theorem 8.1. Consider the ordinary differential equation y  + λ2 (t)|ξ|2 y = 0 ,

(8.9)

where t ∈ [0, ∞), ξ is a parameter ranging over R, while Λ(t) = exp(−t−1 ), λ(t) = t−2 exp(−t−1 ). In accordance with the discussion in Subsection 2.1.2 [40], we apply Liouville transformation y(t, |ξ|) = tf (t, |ξ|) and τ = |ξ|Λ(t) to the unknown function and to the independent variables, respectively. Then equation (8.9) becomes 1 (8.10) fττ + fτ + f = 0 . τ One can use either the properties of the Bessel equation (8.10) and the Bessel function J0 (z), or reduce it to the Kummer equation and make use of hypergeometric functions. We follow the last approach since it gives also a parametrix as well as the propagation of singularities [2], [40]. Let us put z z exp , where z = 2iτ , w(z) = f 2i 2 then (8.10) becomes zw (z) + (1 − z)w (z) − αw(z) = 0,

(8.11)

Global Existence for Nonlinear Wave Equations

357

that is the Kummer equation with γ = 1,

α=

1 . 2

This is the so-called logarithmic case (see, Section 6.7.1 from [4]). In the logarithmic case linear independent solutions can be chosen as follows: w1 = Ψ(α, 1, z) ,

w2 = ez Ψ(1 − α, 1, −z) ,

(8.12)

that is y5 , y7 , respectively, in notations of Section 6.7 [4]. Lemma 8.5 (G. Alexandrian [2]). The functions y1 (t, |ξ|)

= te−iΛ(t)|ξ| Ψ(1/2, 1; 2iΛ(t)|ξ|) ,

y2 (t, |ξ|)

= teiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|)

form for t ≥ 0 a fundamental system of solutions of (8.9). Hence, any solution y(t, ξ) to equation (8.9) can be written as follows: y(t, |ξ|) = p0 (t, |ξ|)y(0, |ξ|) + p1 (t, |ξ|)y  (0, |ξ|) . Taking into account of the consideration of Subsection 2.1.2 from [40] we easily obtain for p0 (t, |ξ|) and p1 (t, |ξ|) :  π t '  − ln |2ξ| − i + ψ(1/2) − 2γ e−iΛ(t)|ξ| Ψ(1/2, 1; 2iΛ(t)|ξ|) p0 (t, |ξ|) = √ i π 2   ( π + ln |2ξ| + i + ψ(1/2) − 2γ eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) , 2 ( t ' iΛ(t)|ξ| e Ψ(1/2, 1; −2iΛ(t)|ξ|) − e−iΛ(t)|ξ| Ψ(1/2, 1; 2iΛ(t)|ξ|) . p1 (t, |ξ|) = √ i π Thus according to page 108 of [40] we conclude  ( ' 2t π p0 (t, |ξ|) = √ Im ln |2ξ| + i + ψ(1/2) − 2γ eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) , 2 π ( ' 2t p1 (t, |ξ|) = √ Im eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) , π and we arrive at v(t, x)

=

2t √ (2π)−n/2 π


Rn

2t + √ (2π)−n/2 π

eix·ξ Im


Rn

'

ln |2ξ| + i

 π + ψ(1/2) − 2γ 2

( ×eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|)  v0 (ξ) dξ ' ( eix·ξ Im eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) v1 (ξ) dξ .

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Since Λ(t) → 1 as t → ∞, for every v0 , v1 ∈ C0∞ (Rn ) we conclude
 ' π −n/2 (2π) eix·ξ Im ln |2ξ| + i + ψ(1/2) − 2γ 2 Rn

−n/2




+(2π)

Rn

( ×eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) v0 (ξ) dξ ' ( eix·ξ Im eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) v1 (ξ) dξ

→ vst (x) ∈ C0∞ (Rn ) , where vst (x)

:=

(2π)−n/2


Rn

+(2π)−n/2

 π + ψ(1/2) − 2γ 2 ( i|ξ| ×e Ψ(1/2, 1; −2i|ξ|) v0 (ξ) dξ ' ( eix·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) v1 (ξ) dξ .

eix·ξ Im


Rn

'

ln |2ξ| + i

Further, since the condition (8.2) is not fulfilled, for the numbers a and b of (8.3) we have min{|a|, b} < ∞. Hence, for large t due to the condition vst (x) ≡ 0, we / (a, b) for some x0 ∈ Rn . Then a global solution u ∈ C 2 (R+ × Rn ) to get v(t, x0 ) ∈ the Cauchy problem for (8.1) does not exist. The theorem is proved.
Concluding remark. We make a conjecture that the results of this section are valid also for the equation (with a(t) ≥ 0)   ∇x u|2 = 0 utt − a2 (t)∆u + f (u) u2t − a2 (t)|∇ 0∞ provided that R := 0 a(t)dt < ∞, that is, if the domain of influence for any point x0 ∈ Rn is permanently restricted. In particular that implies that possible singularities of the initial data at x = x0 are trapped in the ball of radius R centered at x0 ∈ Rn . In that sense the hyperbolic equation behaves like an ordinary differential equation utt + f (u)u2t = 0.

9. Global existence for a model equation with a polynomially growing coefficient We consider the equation

  ∇x u|2 = 0 utt − a2 (t)∆u + f (u) u2t − a2 (t)|∇

(9.1)

with a(t) = tl , where l > 0. Here f ∈ C ∞ (R) is a real-valued function. Theorem 9.1. For the Cauchy problem for the equation (9.1) with initial conditions u(0, x) = u0 (x) ,

ut (0, x) = u1 (x) ,

(9.2)

Global Existence for Nonlinear Wave Equations

359

small data solutions exist globally. That is, there are s ∈ N and δ > 0 such that for every given u0 , u1 ∈ C0∞ (Rn ) satisfying the inequality u0 (s+1) + u1 (s) ≤ δ

(9.3)

the solution u ∈ C ([0, ∞) × R ) to the Cauchy problem (9.1), (9.2) exists. 2

n

Proof. The proof of this theorem consists of four steps. Step 1: Reduction to a linear equation We transform the equation (9.1) into the linear equation by change 

s
u exp f (r)dr ds . v = G(u) := 0

0

Since G ∈ C ∞ (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C ∞ (a, b) , where we denote a := lim G(u) ,

b := lim G(u) .

u→−∞

u→∞

Then the equation (9.1) leads to the linear equation vtt − a2 (t)∆v = 0

(9.4)

for the function v = v(t, x). Moreover, v takes initial values 


s
u0 (x) exp f (r)dr ds , vt (0, x) = u1 (x) exp v(0, x) = 0

0



u0 (x)

f (r)dr

.

0

(9.5) It is easily seen that v0 (s+1) + v1 (s) is small if and only if u0 (s+1) + u1 (s) is small, provided that s is large enough, s > n/2. For the Cauchy problem for the hyperbolic equation (9.4) with the data prescribed at t = 0, v(0, x) = v0 (x) ,

vt (0, x) = v1 (x) ,

(9.6)

there is a representation of its solution (see [40]) v(t, x) = V1 (t, (−∆)1/2 ; l)v0 (x) + V2 (t, (−∆)1/2 ; l)v1 (x) ,

(9.7)

where ∆ is the Laplace operator in Rn .We derive (9.7) in the next step. Step 2: Ordinary differential equations with a parameter Consider the ordinary differential equation y  + t2l |ξ|2 y = 0

(9.8)

with a positive integer l, while ξ is a parameter ranging over R . Set τ = ωtl+1 |ξ| or τ = Λ(t)|ξ| with ω = 1/(l + 1), Λ(t) := ωtl+1 . Then the function v(τ ; l) = y(t, ξ; l) is a solution of the equation n

d2 v dv +v = 0, + lωτ −1 dτ 2 dτ

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provided that y solves (9.8) (t = 0, ξ = 0). If we introduce the new unknown function z  z exp (z = 2iτ ) , w(z) = v 2i 2 then for w(z) equation (9.8) becomes 1 zw (z) + (lω − z)w (z) − ωlw(z) = 0 . 2 If we use the notations γ = lω = l/(l + 1), α = coincides with the following one

1 2 ωl,

(9.9)

then the last equation

zw (z) + (γ − z)w (z) − αw(z) = 0 ,

(9.10)

which is called confluent hypergeometric equation (see, for example, Section 6.3 of [4]). Any solution of this equation is called confluent hypergeometric function. There is a solution F (α; γ; z) to equation (9.10) which can be represented in the form F (α; γ; z) =

1 Γ(γ) Γ(α)Γ(γ − α) (1 − e2πi(γ−α) )(1 − e2πiα )
(1+,0+,1−,0−) × ezζ ζ α−1 (1 − ζ)γ−α−1 dζ

(9.11)

C

with F (α; γ; 0) = 1. The function F (α; γ; z) is an entire function of z. In (9.9), (9.10) the parameter γ is not an integer, so (see Section 6.3 of [4]) another linear independent solution is z 1−γ F (α − γ + 1; 2 − γ; z). Thus for the symbols V1 (t, |ξ|; l) and V2 (t, |ξ|; l), which are introduced in (9.7), we obtain  l+1 V1 (t, |ξ|; l) = e−iωt |ξ| F (α; γ; 2iωtl+1 |ξ|) , l+1 V2 (t, |ξ|; l) = te−iωt |ξ| F (1 + α − γ; 2 − γ; 2iωtl+1 |ξ|), or



V1 (t, |ξ|; l) = e−iΛ(t)|ξ| F (α; γ; 2iΛ(t)|ξ|) , V2 (t, |ξ|; l) = ω −ω Λ(t)ω e−iΛ(t)|ξ| F (1 + α − γ; 2 − γ; 2iΛ(t)|ξ|),

where γ = l/(l + 1), α = l/(2(l + 1)), 1 + α − γ = (l + 2)/(2(l + 1)), while F (a; c; z) =

N  (a)n (a − c + 1)n −a−n z + O(|z|−a−N −1 ) n! n=0

3 3 − π < arg z < π , 2 2

as |z| → ∞ ,

(a)n := Γ(a + n)/Γ(a) .

Hence the estimates |F (α; γ; 2iΛ(t)|ξ|)| ≤ C|Λ(t)ξ|−l/(2(l+1))

as

−(l+2)/(2(l+1))

|F (1 + α − γ; 2 − γ; 2iΛ(t)|ξ|)| ≤ C|Λ(t)ξ|

Λ(t)|ξ| → ∞ , as

Λ(t)|ξ| → ∞ ,

Global Existence for Nonlinear Wave Equations

361

imply for the multipliers V1 and V2 |V V1 (t, |ξ|; l)| ≤

C|Λ(t)ξ|−l/(2(l+1))

|V V2 (t, |ξ|; l)| ≤

CΛ(t) |Λ(t)ξ| ω

as Λ(t)|ξ| → ∞ ,

−(l+2)/(2(l+1))

as Λ(t)|ξ| → ∞ .

Step 3: L1 − L∞ estimate for the linear equation Theorem 9.2. Assume v0 , v1 ∈ S. Then the solution v = v(t, x) to (9.4) belongs to C ∞ and satisfies the a priori estimate   v(t, ·)∞ ≤ Cε Dn+ε v0 1 + Dn−ω+ε v1 1 for all ε > 0. Proof. To obtain the required inequality one has to consider two cases (zones) : small Λ(t)|ξ| and large Λ(t)|ξ|. Thus let t ≤ 1 and Λ(t)|ξ| ≤ 1. Then by Plancherel’s formula v(t, ·)2

= V V1 (t, (−∆)1/2 ; l)v0 (x) + V2 (t, (−∆)1/2 ; l)v1 (x)2 = V V1 (t, |ξ|; l)ˆ v0 (ξ) + V2 (t, |ξ|; l)ˆ v1 (ξ)2 ≤ V V1 (t, |ξ|; l)∞ ˆ v0 2 + V V2 (t, |ξ|; l)∞ ˆ v1 2 ≤ C(v0 2 + v1 2 ) .

If t ≤ 1 and Λ(t)|ξ| ≥ 1, then |ξ| ≥ l + 1, and we derive v(t, ·)2

≤

V V1 (t, |ξ|; l)∞ v0 2 + |ξ|ω V2 (t, |ξ|; l)∞ |ξ|−ω vˆ1 2

≤

C(v0 2 + v1 (−ω) ) .

Now the Sobolev inequalities w2 ≤ Cε (Dε+n/2 w1

and w∞ ≤ Cε (Dε+n/2 w2 ,

for w ∈ S lead to the desired estimate. If t > 1 we estimate directly between L1 and L∞ and obtain v(t, ·)∞

≤

V V1 (t, |ξ|; l)ˆ v0 (ξ) + V2 (t, |ξ|; l)ˆ v1 (ξ)1

≤

V V1 (t, |ξ|; l)ξ−n 1 ξn vˆ0 (ξ)∞ +ξ−n+ω V2 (t, |ξ|; l)1 ξn−ω vˆ1 ∞   C Dn v0 1 + Dn−ω v1 1 .

≤ The theorem is proved.




Step 4: Completion of the proof of Theorem 9.1 If we choose u0 and u1 small enough in the corresponding Sobolev norms, then the norms of v0 and v1 are small. Hence, according to Theorem 9.2 the solution v = v(t, x) to the linear Cauchy problem (9.4), (9.5) is small and takes values in the domain of the operator H, v(t, x) ∈ (a, b) for all t ∈ [0, ∞), x ∈ Rn . Due

362

Karen Yagdjian

to (4.8) the smooth function u = H(v) is a global solution to the Cauchy problem. The theorem is proved.


10. An example with an exponentially growing coefficient In this section we show that the approach of the previous section works out also for a model equation with an exponentially growing coefficient. Namely, we consider the equation   (10.1) ∇x u|2 = 0 utt − a2 (t)∆u + f (u) u2t − a2 (t)|∇ with a(t) = exp(t). Here f ∈ C ∞ (R) is a real-valued function. Thus we have the following global existence theorem. Theorem 10.1. For the Cauchy problem for the equation (10.1) with initial conditions (10.2) u(0, x) = u0 (x) , ut (0, x) = u1 (x) , small data solution exists globally. That is, there are s ∈ N and δ > 0 such that for every given u0 , u1 ∈ C0∞ (Rn ) satisfying the inequality u0 (s+1) + u1 (s) ≤ δ

(10.3)

the solution u ∈ C ([0, ∞) × R ) to the Cauchy problem (10.1), (10.2) exists. 2

n

Proof. The proof of this theorem consists of six steps. Step 1: Reduction to a linear equation We transform the equation (10.1) into the linear equation by change  

s
u exp f (r)dr ds . v = G(u) := 0

(10.4)

0

Since G ∈ C ∞ (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C ∞ (a, b), where we denote a := lim G(u) , u→−∞

b := lim G(u) . u→∞

Then the equation (10.1) leads to the linear equation vtt − exp(2t)∆v = 0

(10.5)

for the function v = v(t, x). Moreover, v takes initial values 


u0 (x)
s v(0, x) = exp f (r)dr ds , vt (0, x) = u1 (x) exp 0

0

u0 (x)

 f (r)dr .

0

(10.6) It is easily seen that v0 (s+1) + v1 (s) is small if and only if u0 (s+1) + u0 (s) is small, provided that s is large enough.

Global Existence for Nonlinear Wave Equations

363

Step 2: Ordinary differential equations with a parameter Galstian in [13], [14] suggested the model equation (10.5) to get precise Lp − Lq decay estimates. She gave a representation of the solution to the Cauchy problem by means of Bessel functions. Wirth [35] extended this approach to a model equation depending on a parameter µ vt = 0,
v + µ > 0, 1+t that is, a wave equation with weak dissipation, which is obtained from (10.5) after transformation of the time variable if µ = 1. We also follow the papers [13], [14] and do appeal to 0the Bessel equation. If v solves (10.5), then the partial Fourier transform Fp (v) = Rn eixξ v(t, x)dx solves Fp (v)tt + e2t |ξ|2 Fp (v) = 0.

(10.7)

Fp (v)(ξ, 0) = F (v0 )(ξ), Fp (v)t (ξ, 0) = F (v1 )(ξ).

(10.8)

One has also If V = V (t, |ξ|) is a solution of (10.7), then the function W(τ, |ξ|)) = |ξ|V (t, |ξ|) with τ = et |ξ| satisfies the Bessel equation 1 Wτ τ + Wτ + W = 0 . τ If J0 (τ ) and Y0 (τ ) are the first and second kind Bessel functions, then for the Wronskian W (J J0 (τ ), Y0 (τ )) according to formula (28) of [4], vol. 2, page 91, one has 2 . W (J J0 (τ ), Y0 (τ )) = πτ Hence, the fundamental system for the equation (10.7) can be written in the form  π  V1 (t, |ξ|) = − |ξ| J0 (et |ξ|)Y Y1 (|ξ|) − Y0 (et |ξ|)J J1 (|ξ|) , (10.9) 2  π V2 (t, |ξ|) = − J0 (et |ξ|)Y Y0 (|ξ|) − Y0 (et |ξ|)J J0 (|ξ|) , (10.10) 2 where V1 (0, |ξ|) = 1, V1,t (0, |ξ|) = 0,

V2 (0, |ξ|) = 0, V2,t (0, |ξ|) = 1 .

Finally, for the solution to the Cauchy problem for the equation (10.5) with the initial conditions v(0, x) = v0 (x) ∈ C0∞ (Rn ) ,

vt (0, x) = v1 (x) ∈ C0∞ (Rn )

(10.11)

we obtain the representation: ' π   − |ξ| J0 (et |ξ|)Y Y1 (|ξ|) − Y0 (et |ξ|)J J1 (|ξ|) F (v0 )(ξ) v(t, x) = Fp−1  (  π2 J0 (et |ξ|)Y Y0 (|ξ|) − Y0 (et |ξ|)J J0 (|ξ|) F (v1 )(ξ) . (10.12) + − 2 The last representation allows to get an L1 − L∞ estimate for the solution v = v(t, x) if we make use of the following proposition.

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Karen Yagdjian

Proposition 10.2. The symbol V1 (t, |ξ|) satisfies the estimates ⎧ for et |ξ| ≤ 1 , ⎪ C ⎪ ⎪ ⎨ t |V V1 (t, |ξ|)| ≤ for et |ξ| ≥ 1 and |ξ| ≥ 1 , Ce− 2 ⎪   ⎪ 1/2 ⎪ ⎩ ≤ C for et |ξ| ≥ 1 and |ξ| ≤ 1 , C πe2t |ξ|

(10.13)

while for V2 (t, |ξ|) one has ⎧ C(1 + t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ t t C|ξ|−1 e− 2 ≤ Ce− 2 |V V2 (t, |ξ|)| ≤ ⎪ ⎪ ⎪  ⎪  ⎪ ⎩ C 1 − ln |ξ|

ffor

et |ξ| ≤ 1 ,

for

et |ξ| ≥ 1

and

|ξ| ≥ 1 ,

for

et |ξ| ≥ 1

and

|ξ| ≤ 1 .

(10.14)

The proof of this proposition is given in the next two steps. Step 3: Consideration in the pseudo-differential zone: et |ξ| ≤ 1 To find out the asymptotic behavior at zero of the symbol V1 (t, |ξ|) of (10.9) we use the following standard formulas in a neighborhood of the origin: ∞ 

(−1)m = m!Γ(m + 2) m=0

J1 (|ξ|)

∞ 

(−1)m = m!Γ(m + 1) m=0

J0 (|ξ|)

 

|ξ| 2 |ξ| 2

2m+1 =

|ξ| + O(|ξ|3 ) , 2

|ξ| ∈ [0, 1] ,

2m = 1 + O(|ξ|2 ) ,

|ξ| ∈ [0, 1] ,

and Y0 (|ξ|)

=

Y1 (|ξ|)

=

=

      |ξ| |ξ| 2 2 γ + ln J0 (|ξ|) + O(|ξ|2 ) = γ + ln + O(|ξ|2 ) , π 2 π 2 2 |ξ| J1 (|ξ|) ln π 2  1+2l  −1 ∞ |ξ| 1 |ξ| ψ(l + 2) + ψ(l + 1) 1 − − (−1)l π 2 π 2 l!(l + 1)! l=0  −1 2 |ξ| |ξ| 1 |ξ| |ξ| ln + O(|ξ|3 ) ln − π 2 2 2 π 2 1 |ξ| (ψ(2) + ψ(1)) + O(|ξ|3 ) , − |ξ| ∈ [0, 1] . π 2

Global Existence for Nonlinear Wave Equations

365

Here the function ψ is the digamma function : ψ(z) := Γ (z)/Γ(z). Thus, for the first solution V1 (t, |ξ|) we obtain in the zone et |ξ| ≤ 1  −1 1 |ξ| π 2 |ξ| |ξ| − (γ + t) − |V V1 (t, |ξ|)| = 2 π 2 π 2  1+2l ∞ |ξ| ψ(l + 2) + ψ(l + 1) |ξ| 1 l t 2 + O((e |ξ|) ) , − (−1) π 2 l!(l + 1)! 2 l=0

which leads with |ξ| ≤ e−t to the final estimate   −1  1 |ξ| 1 |ξ| π (C + t) + ≤C |V V1 (t, |ξ|)| ≤ |ξ| 2 π 2 π 2

for et |ξ| ≤ 1 ,

(10.15)

in the pseudo-differential zone. Consider now the second solution V2 (t, |ξ|):      |ξ| π 2 t 2 |V V2 (t, |ξ|)| = γ + ln J0 (|ξ|) + O(|ξ|) ) J0 (e |ξ|) 2 π 2     t  e |ξ| 2 t t 2 − γ + ln J0 (e |ξ|) + O((e |ξ|) ) J0 (|ξ|) π 2 π π J0 (et |ξ|)J = − tJ J0 (|ξ|) + J0 (et |ξ|)O(|ξ|)2 ) − J0 (|ξ|)O((et |ξ|)2 ) . 2 2 Hence, |V V2 (t, |ξ|)| ≤ C(1 + t)

for

et |ξ| ≤ 1 .

(10.16)

The last estimate completes the consideration in the pseudo-differential zone. Step 4: Consideration in the hyperbolic zone: et |ξ| ≥ 1 We have to split that consideration into two cases. Case et |ξ| ≥ 1, |ξ| ≥ 1. For ν = 0, 1 and large z we have  1/2  ∞  2 1 1 (−1)m (ν, 2m) cos(z − νπ − π) · Jν (z) ∼ πz 2 4 (2z)2m m=0

Yν (z) ∼

∞  1 (−1)m (ν, 2m + 1)  1 , − sin(z − νπ − π) · 2 4 (2z)2m+1 m=0  1/2  ∞  2 1 1 (−1)m (ν, 2m) sin(z − νπ − π) · πz 2 4 (2z)2m m=0 ∞  1 1 (−1)m (ν, 2m + 1)  . + cos(z − νπ − π) · 2 4 (2z)2m+1 m=0

(10.17)

(10.18)

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Karen Yagdjian

Here, following Hankel, we write

(ν, m) :=

Γ(ν + m + 12 ) . m!Γ(ν − m + 12 )

Using (10.17), (10.18) we obtain for the first solution after substitution

V1 (t, |ξ|)

 1/2  ∞ 2 π π  (−1)m t cos(e ) · ∼ − |ξ| |ξ| − 2 πet |ξ| 4 m=0 (2et |ξ|)2m − sin(et |ξ| −  ×

2 π|ξ|

1/2 

sin(|ξ| −

∞ 3π  (−1)m (1, 2m) )· 4 (2|ξ|)2m m=0

+ cos(|ξ| −  −

2 πet |ξ|

1/2 

sin(et |ξ| −

∞ 3π  (−1)m (1, 2m + 1)  )· 4 (2|ξ|)2m+1 m=0

∞ π  (−1)m )· 4 m=0 (2et |ξ|)2m

+ cos(et |ξ| −  ×

2 π|ξ|

1/2 

∞ (−1)m  π  )· 4 m=0 (2et |ξ|)2m+1

∞ (−1)m  π  )· 4 m=0 (2et |ξ|)2m+1

∞ 3π  (−1)m (1, 2m) )· cos(|ξ| − 4 (2|ξ|)2m m=0

 ∞ 3π  (−1)m (1, 2m + 1)  )· − sin(|ξ| − . 4 (2|ξ|)2m+1 m=0

Therefore, |V V1 (t, ξ)| ∞  ≤2e−t/2

∞ ∞ ∞     1 1 2m 2m + 1  . + + (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0 (2|ξ|)2m m=0 (2|ξ|)2m+1 m=0

It follows the second estimate for the function V1 (t, ξ) stated in Proposition 10.2.

Global Existence for Nonlinear Wave Equations

367

Consider now the second solution V2 (t, |ξ|). Using (10.17), (10.18) we obtain V2 (t, |ξ|)  1/2  1/2 2 2 π ∼− 2 πet |ξ| π|ξ|  ∞ ∞  (−1)m  π  (−1)m π  t ) · × cos(et |ξ| − ) · − sin(e |ξ| − t 2m 4 m=0 (2e |ξ|) 4 m=0 (2et |ξ|)2m+1 ∞ ∞ π  (−1)m π  (−1)m  × sin(|ξ| − ) · + cos(|ξ| − ) · 4 m=0 (2|ξ|)2m 4 m=0 (2|ξ|)2m+1



∞ ∞  (−1)m  π  (−1)m π  t ) · + cos(e |ξ| − − sin(et |ξ| − ) · 4 m=0 (2|et ξ|)2m 4 m=0 (2et |ξ|)2m+1  ∞ ∞  π  (−1)m π  (−1)m  × cos(|ξ| − ) · − sin(|ξ| − ) · . 4 m=0 (2et |ξ|)2m 4 m=0 (2|ξ|)2m+1

Therefore, |V V2 (t, |ξ|)|  1/2  1/2 2 2 ≤π πet |ξ| π|ξ|   ∞ ∞ ∞ ∞       1 1 1 1 . × + + (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0 (2|ξ|)2m m=0 (2|ξ|)2m+1 m=0 It follows the second estimate for the function V2 (t, ξ) stated in Proposition 10.2.

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Karen Yagdjian

Case et |ξ| ≥ 1, |ξ| ≤ 1. Using (10.17) , (10.18) and the formulas from the beginning of Step 3 we obtain V1 (t, |ξ|) π ∼ − |ξ| 2



2 πet |ξ|

1/2 

∞ π  (−1)m cos(e |ξ| − ) · 4 m=0 (2et |ξ|)2m t

− sin(et |ξ| −  2 |ξ|

∞ (−1)m  π  )· 4 m=0 (2et |ξ|)2m+1

|ξ| |ξ| + O(|ξ|3 ) ln 2 2  −1  1 |ξ| 1 |ξ| − (ψ(2) + ψ(1)) + O(|ξ|3 ) − π 2 π 2 1/2   ∞ π  (−1)m 2 t )· sin(e |ξ| − − πet |ξ| 4 m=0 (2et |ξ|)2m ×

π 2

ln

+ cos(et |ξ| − Therefore, |V V1 (t, ξ)| ≤

≤



∞  (−1)m   |ξ| π  × )· + O(|ξ|3 ) . t 2m+1 4 m=0 (2e |ξ|) 2

1/2   ∞

∞   1 1 + (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0  −1  2 |ξ| |ξ| 1 |ξ| |ξ| 3 ln + O(|ξ| ) ln + × π 2 2 2 π 2  1 |ξ| (ψ(2) + ψ(1) + π) + O(|ξ|3 ) + π 2  1/2 2 . C πet |ξ|

π |ξ| 2

2 πet |ξ|

It follows the last estimate for the function V1 (t, ξ) stated in Proposition 10.2.

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Consider now the second solution V2 (t, |ξ|). Using (10.17), (10.18) and the formulas from the beginning of Step 3 we obtain V2 (t, |ξ|)   1/2      2 2 |ξ| π 2 2 ∼− γ + ln (1 + O(|ξ| )) + O(|ξ| ) 2 πet |ξ| π 2 ∞ ∞  (−1)m  π  (−1)m π  t × cos(et |ξ| − ) · ) · − sin(e |ξ| − 4 m=0 (2et |ξ|)2m 4 m=0 (2et |ξ|)2m+1 ∞  π  (−1)m − sin(et |ξ| − ) · 4 m=0 (2|et ξ|)2m

 ∞  m  (−1) π (1 + O(|ξ|2 )) . + cos(et |ξ| − ) · 4 m=0 (2et |ξ|)2m+1

Therefore, |V V2 (t, |ξ|)|  1/2   ∞ ∞   π 1 1 2 ≤ + 2 πet |ξ| (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0    |ξ| 2 2 2 2 × γ + ln (1 + O(|ξ| )) + O(|ξ| ) + (1 + O(|ξ| )) π 2   1/2   |ξ| 2 2 2 2 2 ≤C γ + ln (1 + O(|ξ| )) + O(|ξ| ) + (1 + O(|ξ| )) . πet |ξ| π 2 It follows the last estimate for the function V2 (t, ξ) stated in the proposition. This completes the proof of Proposition 10.2.

Step 5: L1 − L∞ estimate for the linear equation Theorem 10.3. Assume v0 , v1 ∈ S. Then the solution v = v(t, x) to the Cauchy problem for the equation (10.5) with initial conditions (10.11) belongs to C ∞ and satisfies the a priori estimate   v(t, ·)∞ ≤ Cε Dn+ε v0 1 + Dn+ε v1 1 , for all ε > 0.

370

Karen Yagdjian

Proof. According to Proposition 10.2 one can write for every given positive ε v(t, ·)∞

≤ ≤

V V1 (t, |ξ|)ˆ v0 (ξ) + V2 (t, |ξ|)ˆ v1 (ξ)1 V V1 (t, |ξ|)ˆ v0 (ξ)1 + V V2 (t, |ξ|)ˆ v1 (ξ)1

≤

V V1 (t, |ξ|)ξ−n−ε 1 ξn+ε vˆ0 (ξ)∞ +V V2 (t, |ξ|)ξ−n−ε 1 ξn+ε vˆ1 (ξ)∞   Cε Dn+ε v0 1 + Dn+ε v1 1 ,

≤

that completes the proof of the theorem.




Step 6: Completion of the proof of Theorem 10.1 If we choose u0 and u1 small enough in the corresponding Sobolev norms, then the norms of v0 and v1 are small. Hence, according to Theorem 10.3 the solution v = v(t, x) to the linear Cauchy problem (10.5), (10.6) is small and takes values in the domain of the operator H, v(t, x) ∈ (a, b) for all t ∈ [0, ∞), x ∈ Rn . Due to (4.8) the smooth function u = H(v) is a global solution to the Cauchy problem (10.1), (10.2). The theorem is proved.


11. Fast oscillating coefficients: no resonance ?! The counterexample from [7] shows that if we want the Cauchy problem for linear weakly hyperbolic equations to be C ∞ well-posed, then we have to restrict the oscillations. On the other hand in the theory of linear hyperbolic equations with multiple characteristics one can allow some oscillations in the coefficients. These oscillations are described by means of the monotonic part of the coefficient (see, for instance, [40]). In this section we show how the approach of the previous sections and of [38], [40], works out also for model quasi-linear equations with coefficients simultaneously containing monotonically increasing and oscillating factors. To track down the condition on the uniformly positive and oscillating function b(t) in the equation   (11.1) ∇x u|2 = 0 , utt − λ2 (t)b2 (t)∆u + f (u) u2t − λ2 (t)b2 (t)|∇ where f = f (u) is a smooth and real-valued function, λ = λ(t) is a smooth, positive, monotone function, λ (t) ≥ 0, we appeal to the linear weakly hyperbolic equation vtt − λ2 (t)b2 (t)∆v = 0 . In the case λ(0) = 0 a condition on the oscillating term is according to Section 3.11.1 [40] k  λ(t) (k) | ln λ(t)| for all t ∈ (0, T ] , k = 0, 1, 2 . (11.2) |b (t)| ≤ ck Λ(t)

Global Existence for Nonlinear Wave Equations

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0t Here Λ(t) := 0 λ(τ )dτ . This estimate is a key to the next conditions which guarantee a global existence result for the Cauchy problem for the quasi-linear equation (11.1). The logarithmic term in the condition (11.2) points also on a critical case in the theory of operators with multiple characteristics (Section 3.11.1 [40]). Analogously there is a corresponding critical case in the global solvability problem for the quasi-linear equation (11.1), which reveals itself in the dependence of the global solvability on the dimension of the x variables. That is an interesting point about that critical case that only a micro-local consideration allows to prove the wellposedness for the linear weakly hyperbolic equation with fast oscillations. That micro-local approach developed for linear weakly hyperbolic equations hints at the tools which will be used in this section for nonlinear wave equations. Since some steps of this section are similar to those which are already introduced in the previous section, we will skip the details of many calculations trying to minimize the unnecessary reiterations. To formulate the result we need some assumptions on the functions λ = λ(t) and b = b(t). Following Section 3.11.1 [40] we assume concerning the monotone factor that the derivatives of order k = 0, 1, 2 , satisfy  k λ(t) λ (t) λ(t) λ(t) (k) ≤ ≤ c0 for all t ∈ (t0 , ∞), (11.3) , c |λ (t)| ≤ ck λ(t) Λ(t) Λ(t) λ(t) Λ(t) 0t where Λ(t) := 0 λ(s)ds, t0 > 0. The oscillations are narrowed by the condition k  λ(t) | ln λ(t)| for all t ∈ (t0 , ∞) , k = 0, 1, 2 . (11.4) |b(k) (t)| ≤ ck Λ(t) Because of the last condition the oscillations are called fast. If λ(0) = 0 (that is the equation (11.1) is weakly hyperbolic), then we assume additionally that (11.3) and (11.4) are fulfilled in some neighborhood of t0 = 0, with possibly other constants. Theorem 11.1. Assume that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3) and (11.4), respectively. Moreover, suppose that the dimension n is large enough, that is    1 16  3 + N C12 + n> inf Cb2 Cb,sec + 2C , (11.5) {N | N >2Cb , N >2} 2 3 N where 0 < C0 := min b(t), and the constants C1 , Cb , and Cb,sec are determined by  1 Dt b(t) Λ(t) C1 := max b(t), Cb := inf sup (11.6) , {T | 1≤T 1. Theorem 11.2. Assume that n > 1 and that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3), (11.9), and (11.10). Then for the Cauchy problem for the equation (11.1) with initial conditions u(0, x) = u0 (x) ,

ut (0, x) = u1 (x) ,

(11.11)

small data solutions exist globally. That is, there are s ∈ N and δ > 0 such that for every given u0 , u1 ∈ C0∞ (Rn ) satisfying the inequality u0 (s+1) + u1 (s) ≤ δ a unique solution u ∈ C 2 ([0, ∞)×Rn ) to the Cauchy problem (11.1), (11.11) exists. The case without oscillations, that is b = const, follows immediately from Theorem 11.2. The proof of Theorem 11.2 can be derived from Theorem 11.1 if we use the “λ − b scaling”. That means the following. For the equation (11.1) with the property (11.10) the numbers Cb and Cb,sec are vanishing. Then, the equation is invariant with respect to the transformation λ(t) → ε−1 λ(t), b(t) → εb(t), where ε is a positive number. That allows to make C1 arbitrary small and to reach condition (11.5) for every given n > 1. After that scaling one can apply Theorem 11.1.

Global Existence for Nonlinear Wave Equations

373

Example. Consider the equation

  utt − exp(2tα )b2 (t)∆u + f (u) (ut )2 − exp(2tα )b2 (t)|∇ ∇x u|2 = 0 ,

(11.12)

t ∈ [0, ∞), x ∈ Rn , with α ∈ R, α > 0. Here f ∈ C ∞ (R) is a real-valued function while b = b(t) is defined on R, a periodic, non-constant, smooth, and positive function. The condition (11.10) is satisfied if and only if α > 1/2. According to Theorem 11.2 for such α the Cauchy problem (11.11) with small data has a global solution. The case α = 1/2 is a critical case. According to Theorem 11.1 for that value of α global existence can be guaranteed only for large dimension n of the spatial variable x ∈ Rn . Open problems: It was mentioned in Section 6 that the Cauchy problem for equation (6.1) with α ≤ 0 has no global solution. There is no any Lp − Lq decay estimate (see [29]) for the corresponding linear equation if α ∈ (−∞, 1/2). According to Theorem 11.2 the problem for equation (6.1) has a global solution for small data if α ∈ (1/2, ∞), n > 1. For α = 1/2 it is known for large n only, Theorem 11.1. Thus for the equation (11.12) the case α ∈ (0, 1/2] is still open. The one-dimensional case deserves special attention. The proof of Theorem 11.1 is given in the next section.

12. Linear wave equations with oscillating coefficients The equation (11.1) leads to the linear equation vtt − λ2 (t)b2 (t)∆v = 0

(12.1)

for the function v(t, x) = G(u(t, x)) defined by means of the operator G of (10.4). For the Cauchy problem with the data prescribed at t = t0 v(t0 , x) = v0 (x) ,

vt (t0 , x) = v1 (x) ,

(12.2)

we apply Fourier transform and obtain a representation vˆ(t, ξ) = V1 (t, ξ)ˆ v0 (ξ) + V2 (t, ξ)ˆ v1 (ξ) . The next proposition proved in Section 3.11.1 [40] allows to overcome the difficulties generated by the multiplicity of the characteristics at the point t = 0. Proposition 12.1. Suppose that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3) and (11.4), respectively, with t0 = 0. Then for every given T > 0 there exist constants k and CT such that the multipliers V1 (t, ξ) and V2 (t, ξ) of the Cauchy problem with t0 = 0 satisfy the estimate |V V1 (t, ξ)| + |V V2 (t, ξ)| ≤ CT (1 + |ξ|)k

for all t ∈ [0, T ] , ξ ∈ Rn .

374

Karen Yagdjian

Keeping in mind the last proposition we focus on the case with data given at large t0 . Step 1: Zones. Next we split the dual space into the pseudo-differential zone Zpd and the hyperbolic zone Zhyp , defined for a given positive number N ≥ 2 by (compare with (2.1.71)[40]) Zpd (N )

= {(t, ξ) ∈ (t0 , ∞) × Rn | Λ(t)|ξ| ≤ N ln λ(t)} ,

(12.3)

Zhyp (N )

= {(t, ξ) ∈ (t0 , ∞) × Rn | Λ(t)|ξ| ≥ N ln λ(t)} .

(12.4)

Then for every given point ξ ∈ Rn we define the function t = tξ as a solution to the equation Λ(tξ )|ξ| = N ln λ(tξ ) .

(12.5)

Because of the monotonicity of the function Λ(t)/ ln λ(t) the point t = tξ is well defined for all ξ ∈ Rn and depends on N . Indeed, for t0 large enough one has λ2 (t) ln λ(t) > Λ(t)λ (t)

for all t ≥ t0 /2 .

Evidently one has ln Λ(tξ ) ≤ 2 ln N − ln |ξ| for all tξ ≥ t0 /2 , for all tξ ≥ t0 /2 . ln |ξ| ≤ ln N , − ln |ξ| ≤ ln Λ(tξ ) In what follows we also need the ball BR(t0 ) (0) with the radius R(t0 ) < 1 determined by the equation Λ(t0 )R(t0 ) = N ln λ(t0 ) .

(12.6)

Proposition 12.2. Suppose that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3) and (11.4), respectively. Then the multipliers V1 (t, ξ) and V2 (t, ξ) satisfy for every given small ε > 0 with sufficiently large positive t0 and N , N > 2C Cb , the following estimates: in the pseudo-differential zone, 2 |V V1 (t, ξ)| + |ξ||V V2 (t, ξ)| ≤ C(N, t0 , ε)|ξ|−ε − N C1

for all (t, ξ) ∈ Zpd (N ),

while in the hyperbolic zone, for all (t, ξ) ∈ Zhyp (N ), |V V1 (t, ξ)| + |ξ||V V2 (t, ξ)| ⎧  1 1 16  ⎪ ⎪ − − ε − N C12 − Cb,sec + 2C Cb2 + ε ⎪ ⎪ 3 N ⎨ C(N, t0 , ε)|ξ| 2 for all |ξ| ≤ R(t0 ) , ≤ ⎪ ⎪ 1 ⎪ ⎪ − ⎩ C(N, t0 )λ(t) 2 for all |ξ| ≥ R(t0 ) . The last proposition provides us with the key tool to prove the next theorem that brings an L1 − L∞ estimate for solutions to the linear equation (12.1).

Global Existence for Nonlinear Wave Equations

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Theorem 12.3. Assume that the dimension n is sufficiently large, that is the condition (11.5) is satisfied. Then for sufficiently large t0 the solution v = v(t, x) to the Cauchy problem for equation (12.1) with the conditions v(t0 , x) = v0 (x) ∈ C0∞ (Rn ) ,

vt (t0 , x) = v1 (x) ∈ C0∞ (Rn ) ,

satisfies the following a priori estimate:   v(t, ·)∞ ≤ C(t0 , ε) Dn+ε v0 1 + Dn−1+ε v1 1 for every ε > 0 with a suitable constant C(t0 , ε). Proof of Theorem 12.3. The Fourier image vˆ = vˆ(t, ξ) of the function v = v(t, x) satisfies the inequality ˆ v (t, ·)∞ ≤ v(t, ·)1 . Further, one has V1 (t, ξ)ˆ v0 1 + V V2 (t, ξ)ˆ v1 1 . v(t, ·)∞ ≤ v(t, ·)1 ≤ V On the other hand, v0 1 V V1 (t, ξ)ˆ
=


|V V1 (t, ξ)ˆ v0 | dξ +

(t,ξ )∈Zpd (N )

|V V1 (t, ξ)ˆ v0 | dξ (t,ξ )∈Zhyp (N )∩{|ξ| 0 imply the existence of a solution to the linear equation (12.1) and an estimate in the Sobolev norms like   v(t, ·)2(s+1) + Dt v(t, ·)2(s) ≤ C v(0, ·)2(s+1+k) + Dt v(0, ·)2(s+k) for all t ∈ [0, t0 ]. Then we apply Theorem 12.3 to choose v0 = v(0, x) and v1 = vt (0, x) such that v(t, x) belongs to the domain of the operator H for all t and x. Then u(t, x) = H(v(t, x)) is a global solution to the Cauchy problem (11.1), (11.8). The theorem is proved.


References [1] R. Agemi, Y. Kurokawa, and H. Takamura, Critical curve for p-q systems of nonlinear wave equations in three space dimensions. J. Differential Equations 167 (2000), no. 1, 87–133 [2] G. R. Aleksandrian, The parametrix and propagation of the wave front of the solution of the Cauchy problem for a model equation. Izv. Akad. Nauk Armyan. SSR Ser. Mat. 19 (1984), no. 3, 219-232, [3] S. Alinhac, Blow-up for nonlinear hyperbolic equations. Progress in Nonlinear Differential Equations and their Applications, 17. Birkh¨ ¨ auser Boston, Inc., Boston, MA, 1995. [4] H. Bateman and A. Erdelyi, Higher Transcendental Functions. vol. 1, 2, McGrawHill, New York 1953. [5] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hungar. 7 (1956), 81–94. [6] G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78 (1946). 1–96. [7] F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in C ∞ . Acta Math. 148 (1982), 243–253 [8] F. Colombini and S. Spagnolo, Hyperbolic equations with coefficients rapidly oscillating in time: a result on nonstability. J. Differential Equations 52 (1984), 24–38.

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[9] D. Del Santo, Global existence and blow-up for a hyperbolic system in three space dimensions. Rend. Istit. Mat. Univ. Trieste 29 (1997), no. 1-2, 115-140 (1998). [10] D. Del Santo, V. Georgiev, and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems. in “Geometrical optics and related topics (Cortona, 1996)”, 117–140, Progr. Nonlinear Differential Equations Appl., 32, Birkh¨ ¨ auser Boston, Boston, MA, 1997. [11] D. Del Santo and E. Mitidieri, Blow-up of solutions of a hyperbolic system: the critical case. Differential Equations 34 (1998), no. 9, 1157–1163 (1999). [12] M.S.P. Eastham, The asymptotic solution of linear differential systems. Clarendon Press, Oxford, 1989. [13] A. Galstian, Lp -Lq decay estimates for the equation with exponentially growing coefficient. Preprint 2001/24, ISSN 1437-739X, Arbeitsgruppe “Partielle Differentialgleichungen und Komplex Analysis”, Institut f¨ fur Mathematik, Univ. Potsdam, 2001. [14] A. Galstian, Lp -Lq decay estimates for the wave equations with exponentially growing speed of propagation. Appl. Anal. 82 (2003), no. 3, 197-214. [15] H. Hochstadt, Instability intervals of Hill’s equation. Comm. Pure Appl. Math., vol. 17 (1964), 251–255. [16] H. Hochstadt, On the determination of a Hill’s equation from its spectrum. Arch. Ration. Mech. Anal. 19 (1965), 353–362. [17] L. H¨ ¨ ormander, Lectures on nonlinear hyperbolic equations. Springer-Verlag, Berlin et al. 1997. [18] F. John, Nonlinear wave equations, formation of singularities. Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. University Lecture Series, 2. American Mathematical Society, Providence, RI, 1990. [19] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28 (1979), no. 1-3, 235–268. [20] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations. Comm. Pure Appl. Math. 33 (1980), no. 4, 501–505. [21] S. Klainerman, Global existence for nonlinear wave equations. Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101. [22] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = −Au + F (u). Trans. Amer. Math. Soc. 192 (1974), 1–21. [23] W. Magnus and S. Winkler, Hill’s equation. Interscience Publishers, New York/London/Sydney, 1966. [24] E. Mitidieri and S. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234 (2001), 1–384. [25] S. Mizohata, Theory of Partial Differential Equations. Cambridge University Press, Cambridge 1973. [26] K. Nakanishi and M. Ohta, On global existence of solutions to nonlinear wave equations of wave map type. Nonlinear Anal. 42 (2000), no. 7, Ser. A: Theory Methods, 1231–1252.

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[43] K. Yagdjian, Parametric resonance and nonexistence of global solution to nonlinear wave equations. Journal of Mathematical Analysis and Applications 260 (2001), no.1, 251–268. [44] V.A. Yakubovich and V.M. Starzhinskii, Linear Differential Equations with Periodic Coefficients, v.I, II. J.Wiley&Sons, New York, 1975. Karen Yagdjian Department of Mathematics University of Texas – Pan American 1201 W. University Drive Edinburg, TX 78541-2999 USA and Institute of Mathematics National Academy of Sciences of Armenia Marshal Baghramian Avenue 24B Yerevan, 375019 Armenia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 159, 387–448 c 2005 Birkhauser
¨ Verlag Basel/Switzerland

On the Nonlinear Cauchy Problem Massimo Cicognani and Luisa Zanghirati Abstract. Our aim is to describe how to obtain, with the same procedure, several results of local existence, uniqueness and propagation of regularity for the solution of a quasilinear hyperbolic Cauchy Problem. Mathematics Subject Classification (2000). 35L75; 35B65; 35A20. Keywords. Nonlinear hyperbolic Cauchy problem, regularity of solutions.

1. Introduction Let us consider the quasilinear Cauchy problem ⎧

⎨ P (t, x, Dm u, Dt , Dx )u = f (t, x, Dm u), (QCP ) ⎩ j Dt u|t=0 = gj , 0 ≤ j < m, (t, x) ∈ [0, T ] × Rn , for the operator


P (t, x, Dm u, Dt , Dx ) =

(P )






α aα (t, x, Dm u)Dt,x ,

|α|≤m


α α α u; |α| ≤ m ), m < m, Dt,x = −i∂ ∂t,x . Dm u := (Dt,x We are concerned with the well-posedness of Problem (QCP) and with the propagation of regularity of the solution. For a space X of functions v(x) in Rn , we say that Problem (QCP) is well posed in X if for every gj ∈ X there is T ∗ > 0 such that (QCP) has a unique solution u ∈ C m−1 ([0, T ∗ ]; X). The coefficients aα (t, x, w) and the function f (t, x, w) are defined for t ∈ [0, T ], x ∈ Rn , w ∈ W , W a neighborhood in RN of Cauchy data. Their reg
ularity has to be such that for every u ∈ C m ([0, T ]; X) the composed func
tion f (t, x, Dm u) belongs to C([0, T ]; X) and such that the linear operator P =
P (t, x, Dm u, Dt , Dx ) maps C m ([0, T ]; X) into C([0, T ]; X) continuously.

388

Massimo Cicognani and Luisa Zanghirati

Here X will be C ∞ (Rn ), the Sobolev space H ∞ (Rn ), a Gevrey class γ s , s > 1, ) s s γs = γA , γA = {v(x); vγAs := sup |Dα v(x)|A−|α| |α!|−s < ∞}, x∈Rn , α∈Zn +

A>0

or a Sobolev-Gevrey space (the definition of such spaces is given in Section 3). In view of the Lax-Mizohata Theorem, recently proved also for nonlinear
equations in [51] and [50], we assume that P = P (t, x, Dm u, Dt , Dx ) is hyperbolic, that is the principal symbol  Pm (t, x, w, τ, ξ) = aα (t, x, w)(τ, ξ)α |α|=m

does not vanish for (τ, ξ) = (1, 0), so we may assume aα = 1 for α = (m, 0), and has only real roots in the variable τ . For any parametrization of the roots, we have m 1 (τ − λk (t, x, w, ξ)), λk ∈ R (H) Pm (t, x, w, τ, ξ) = k=1

for every t ∈ [0, T ], x ∈ Rn , w ∈ W , ξ ∈ Rn . When all the roots are simple for every (t, x, w, ξ), ξ = 0, we have a strictly hyperbolic operator and the Cauchy problem (QCP) is well posed in all the spaces X that we consider under suitable regularity assumptions with respect to t. In proving this well-known result, e.g., [48], by pseudodifferential operators calculus, one uses the fact that in the parametrization λ1 (t, x, w, ξ) < λ2 (t, x, w, ξ) < . . . < λm (t, x, w, ξ),

ξ = 0

the roots inherit the same regularity of the coefficients aα , in particular they are symbols after a modification in a neighborhood of ξ = 0. This fact holds true if Pm (t, x, w, τ, ξ) has d < m distinct roots λ1 (t, x, w, ξ) < λ2 (t, x, w, ξ) < · · · < λd (t, x, w, ξ),

ξ = 0

each one of constant multiplicity mk , k = 1, . . . , d, so that (H) becomes ⎧ d 1 ⎪ ⎪ ⎪ (τ − λk (t, x, w, ξ))mk , λk ∈ R, ⎨ Pm (t, x, w, τ, ξ) = (CM ) k=1 ⎪ ⎪ ⎪ ⎩ λk (t, x, w, ξ) = λk
(t, x, w, ξ), k = k  , ξ = 0. Linear equations, that is with aα and f independent of w in (QCP) and (P), have been widely studied. The Cauchy problem for a non-strictly hyperbolic (weakly hyperbolic) linear operator P may not be well posed and the lower order terms play an essential role, remaining arbitrary only when dealing with Gevrey classes γ s of index s ≤ r/(r − 1), r being the largest multiplicity, [7], [31]. Every condition for well-posedness is called a Levi condition after the pioneering paper [39]. General Levi conditions for any weakly hyperbolic linear operator have not yet been obtained, but the case of constant multiplicity is quite well understood

On the Nonlinear Cauchy Problem

389

both in C ∞ and in Gevrey classes. Among many others, we quote [42], [27], [49], [8], [9], [34], [30]. The necessary and sufficient Levi condition for well-posedness in C ∞ stated in [27] says that for every real solution ϕ(t, x) of ∂t ϕ = λj (t, x, ∇x ϕ)

(E) we have (L)

e−iϕ P (eiϕ ) = O(m−mj ),  → +∞, j = 1, . . . , d.

The sufficient Gevrey-Levi condition for well-posedness in γ s stated in [30] says that for every real solution ϕ(t, x) of (E), for every real C ∞ function ψ(t, x) and every C ∞ function h(t, x) with compact support we have (Ls)

P (heiϕ+i

1/s

ψ

) = O(m−mj (1−1/s) ),  → +∞, j = 1, . . . , d.

We also use the equivalent forms of these conditions given in [41], [34], [30]. A proof of the equivalence is given in [43]. Still dealing with linear equations, another kind of degenerating problem is the case of less than Lipschitz continuous coefficients with respect to the time variable t whose study begins with the paper [20]. Even for strictly hyperbolic operators the Cauchy problem may be not well posed in C ∞ . For instance, taking Holder ¨ continuous coefficients of exponent χ ∈]0, 1[ one has well-posedness only in Gevrey classes γ s with s < 1/(1 − χ) and the threshold for the modulus of continuity in order to have a C ∞ well-posed problem is the Log-Lipschitz regularity |a(t + τ ) − a(t)| ≤ C|τ || log |τ ||, t, t + τ ∈ [0, T ], τ = 0, [20], [24], [11], [1]. Counterexamples show that these results cannot be improved, [20], [24]. As it concerns linear weakly hyperbolic equations with coefficients in C 0,χ , χ ∈]0, 1], the Cauchy problem is well posed in Gevrey classes γ s for s ≤ 1 + (χ/r), see [23], also for counterexamples, and [44],[45]. When the characteristic roots have the same Holder ¨ continuity of coefficients, in particular when they are of constant multiplicity, the bound for s can be enlarged to s < r/(r − χ), [45], [10], [13]. Recently, after the paper [21], another way to weaken the Lipschitz regularity has been considered. Namely, to compare the growth of the ∂t derivative with q powers 1/ |t − t¯| , q ≥ 1, as t tends to a point t¯ ∈ [0, T ], say t¯ = T . For strictly hyperbolic linear problems, we have C ∞ well-posedness assuming that each coefficient in the principal part satisfies a condition of the type a ∈ C 1 (0, T [), |a (t)| ≤ C/(T − t), t ∈ [0, T [ whereas we have Gevrey well-posedness in classes γ s , s ≤ q/(q − 1), assuming a ∈ C 1 ([0, T [), |a (t)| ≤ C/(T − t)q , q > 1, t ∈ [0, T [, [21],[36], [22], [29], [12], [13].

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Now, if on the one hand we find a wide range of literature on the degenerating hyperbolic linear Cauchy problem, weakly hyperbolic or with non-Lipschitz coefficients in time, on the other hand this is not true for nonlinear equations. Problem (QCP) was considered in [38] assuming that the full operator P is a product of strictly hyperbolic factors. For such an operator the roots are smooth even if they are not necessarily of constant multiplicity and, more importantly, the linear Cauchy problem in the variable v ⎧
⎨ P (t, x, Dm u, Dt , Dx )v = f (t, x), (LCP ) ⎩ j Dt v|t=0 = gj , 0 ≤ j < m, is well posed in X = C ∞ and in every Gevrey class X = γ s , s > 1, given any
function u such that Dm u ∈ C k ([0, T ]; X), k sufficiently large to allow all the compositions in the factorization of P . As far as we know, the only paper, before [14], dealing with a class of quasilinear operators whose linearized problem satisfies the Levi condition (L) is [28]. As far as Gevrey classes are concerned, we quote [32] where γ s well-posedness was proved for s ≤ r/(r − 1) without any condition on lower order terms, which is the same result as for linear equations. We considered Gevrey-Levi conditions for quasilinear problems in [15]. Nonlinear problems with less than Lipschitz regularity in the time variable have been considered in [33], [35] and, with the method we are going to describe, in [16], [13], [3]. Here we show how to obtain with the same procedure several results of local existence, uniqueness and propagation of regularity of the solution of the quasilinear problem (QCP) under different assumptions on the lower order terms of
P (t, x, Dm u, Dt , Dx ) and on the regularity of the coefficients aα in (P) with respect to the time variable t. We have a unified approach to linear equations which allows us to solve the quasilinear problems with a standard argument of fixed point based on energy estimates of strictly hyperbolic type, [46], after having controlled the degeneracy of the problem by means of a reduction to a first-order system. Given any sufficiently smooth u(t, x), following [26], [41] and [37], from (CM) we get a factorization (F )

P = P1 ◦ P2 ◦ . . . ◦ Pd + R,

where each Pk has principal symbol (τ −λk )mk and the remainder R can be taken of any fixed negative order −M performing a finite number of operations depending on M , m, r and the dimension n. From (F), the scalar problem (QCP) is equivalent to a first order system ⎧ ⎨ Dt U − ∆(t, x, U, Dx )U + R(t, x, U, Dx )U = F (t, x, QU ), (S) ⎩ U|t=0 = G,

On the Nonlinear Cauchy Problem

391

with ∆(t, x, w, ξ) a real diagonal matrix of symbols of order 1, R(t, x, w, ξ) of order  ∈ [0, 1[, Q = Q(t, x, U, Dx ) of order m − m + r(1 − ) + . The index  is determined in the factorization procedure (F) by the Levi conditions on the lower-order terms in P or by the regularity of the coefficients in the principal part and it identifies the largest space of well-posedness for the linear Cauchy problem (LCP) in the scale A ⊂ γ s ⊂ C ∞ , A being the space of all analytic functions. Precisely, we have that (LCP) is well posed in C ∞ for  = 0, in γ 1/ for  > 0. Then we find a unique local solution of (S) in • • C ∞ if the orders of R and Q are not positive that is if  = 0 and m ≤ m−r; • • γ s if the orders of R and Q are not larger than 1/s. We obtain the largest s = 1/ assuming m ≤ m − r(1 − 1/s). So, the nonlinear Levi conditions (N LL)

m ≤ m − r

and (N LLs)

m ≤ m − r(1 − 1/s)

appear in a natural way. We describe here five realizations of our general scheme: A) The operator P satisfies conditions (CM), (NLL) and, in addition, (L) for any
u ∈ C ∞ ([0, T ] × Rn ), Dm u ∈ W . In particular the linear problem (LCP) is well posed in C ∞ . Then the nonlinear Problem (QCP) is also well posed in H ∞ and C ∞ , see Theorem 2.5 and Theorem 2.19. B) The operator P satisfies conditions (CM), (NLLs) and (Ls), this latest for
any u ∈ C ∞ ([0, T ]; γ s ), Dm u ∈ W . In particular the linear problem (LCP) is well posed in γ s . Then the nonlinear Problem (QCP) is also well posed in Sobolev-Gevrey spaces of index s and in γ s , see Theorem 3.5 and Theorem 3.15. C) The operator P is strictly hyperbolic and the coefficients aα , |α| = m, of the principal part are smooth in the variables x, w but only Log-Lipschitz ∂t aα | ≤ C/t. In this case the in the variable t or only C 1 in ]0, T ] with |∂
T matrix R in K = ∆ + R in (S) is such that |R(t, x, w, ξ)|dt ≤ δ logξ, 0

ξ = (1 + |ξ|2 )1/2 , and we get the well-posedness of (QCP) in H ∞ and C ∞ (with a δ-loss of derivatives), see Theorem 4.1. D) The operator P satisfies condition (CM) and the coefficients aα , |α| = m, of the principal part are regular in the variables x, w, ξ but only H¨ o¨lder continuous of exponent χ < 1 in the variable t. In this case we find wellposedness of (2.3) in Sobolev-Gevrey spaces of index s and in γ s , provided that s ≤ r/(r − χ), see Theorem 5.1.

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E) In all these situations we can prove the propagation of analytic regularity of the solution u in domains of influence whenever the Cauchy data and the equation are locally analytic, Theorems 2.20, 2.22, 3.16. In A), one needs only a finite differentiability of the coefficients to solve (QCP) in a fixed Sobolev space H µ . In both cases A) and B), the conditions (L) or (Ls) are satisfied for any given function u if and only if the coefficients aα (t, x, w) fulfill some universal relations, so that it is possible to check them looking only at the symbol P (t, x, w, τ, ξ) without
taking care of the values after the substitution w = Dm u(t, x), see Examples 2.9, 2.10, 3.9 and Remarks 2.12, 3.10. We : can apply our method to a product of operators of constant multiplicity P = j Aj simply performing (F) for each Aj , so we recover the operators of variable multiplicity of [38] as a particular case, taking also into account that the conditions of nonlinear type in [38] are exactly (NLL) and (NLLs). In both cases C) and D) we have no Levi conditions either of linear or of nonlinear type, in particular we can take m = m − 1. In D) the hypothesis (CM) is not essential. What we really need is to be able to choose a regular parametrization of the roots in (H), see [10] and [13] for the linear problem under this more general assumption. As far as the propagation of analytic regularity is concerned, the reduction to the system (S) allows us to apply here the technique used in [2] for strictly hyperbolic equations. We refer to [17] for the case of Gevrey solutions in γ s , s ≤ r/(r − 1), where Levi conditions are not needed, and to [18], [19] for solutions in γ s , s > r/(r − 1), of some semilinear equations. Classes of semilinear weakly hyperbolic equations with non-regular roots are considered in [47], [18], [25].

2. Well-posedness in C ∞ 2.1. Function and symbol spaces

; We use the standard notation H µ (Rn ), µ ∈ R, H ∞ (Rn ) = µ H µ (Rn ), for the usual Sobolev spaces in Rn , often in the shortened form H µ and H ∞ . For k ∈ Z+ , we denote by B k (Rn ) the space of multipliers for H k (Rn ) consisting of all C k functions f in Rn which are bounded together with all their derivatives ∂ α f , |α| ≤ k. We use bounded pseudodifferential operators p(x, D), D = (1/i)∂, in Sobolev spaces with symbol p(x, ξ) ∈ Sm , m ∈ R,  ∈ Z+ , defined as the class of all functions in Rn × Rn such that pSm :=

sup

sup |∂ ∂xβ ∂ξα a(x, ξ)|ξ|α|−m < ∞, ξ = (1 + |ξ|2 )1/2

|α|+|β|≤ R2n

On the Nonlinear Cauchy Problem

393

and, as usual, we write Sm . S m = lim ← →+∞

We need to consider the composition





p(x, Dm u(x), ξ), Dm u = (∂ α u; |α| ≤ m ) for a family of symbols p(x, w, ξ) ∈ B ∞ (W ; S m ) depending on a parameter w ∈ W ,
W is a neighborhood of the origin in RN , and the derivatives Dm u of a function
u ∈ H µ+m . Provided µ > n/2, we can write p(x, u(x), ξ) = p0 (x, ξ) + p1 (x, ξ), p0 (x, ξ) = p(x, 0, ξ) ∈ S m , where p1 has a Sobolev regularity H µ with respect to the variable x. This leads us to consider the following spaces as in [48], Chapter IV: Definition 2.1. We say p(x, ξ) ∈ H M S m provided pH M S m := sup sup Dξα p(·, ξ)H M ξ|α|−m < ∞ |α|≤M

ξ

One needs only finitely differentiability of a symbol for continuity in Sobolev spaces H s , when s is bounded, and to perform any finite number of operations like compositions, adjoint operators, commutators. Besides the well-known results for p(x, ξ) ∈ Sm , in a similar way one can prove: Proposition 2.2. If p(x, ξ) ∈ H M S 0 and M > (n/2) + µ, then p(·, D)uH s ≤ Cµ pH M S m uH s , u ∈ H s , |s| ≤ µ. Proposition 2.3. Given any M , m1 , m2 there is a µ such that, if pj (x, D) ∈ OP H µ S mj , j = 1, 2, we have p1 (x, D)p2 (x, D) ∈ OP H M S m1 +m2 , pj (x, D)∗ ∈ [ 1 (x, D), p2 (x, D)] ∈ OP H M S m1 +m2 −1 . OP H M S mj , and [p So, for a symmetric hyperbolic system ([48]), ⎧ ⎨ ∂t U (t, x) = K(t, x, Dx )U (t, x) + F (t, x), ⎩

(2.1) U (0, x) = G(x),

by the usual energy method one gets: Proposition 2.4. Given any µ there is an M such that, if   K(t, x, ξ) ∈ B 0 [0, T ]; H M S 1 and K(t) + K ∗ (t) has a symbol

  a(t, x, ξ) ∈ B 0 [0, T ]; H M S 0 ,

then the system (2.1) has a unique solution U ∈ C ([0, T ]; H s) to given G ∈ H s , F ∈ C ([0, T ]; H s ), and supposing |s| ≤ µ. Such a solution satisfies for |s| ≤ µ the estimate 
t 2 2 F (τ )2H s dτ , t ∈ [0, T ], (2.2) U (t)H s ≤ (1 + Ct) GH s + 0

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Massimo Cicognani and Luisa Zanghirati

where C depends on finitely many seminorms of K(t) in H M S 1 and of K(t)+K ∗(t) in H M S 0 , on µ and on n. M depends on µ and n. 2.2. Levi conditions Let us consider the quasilinear Cauchy problem ⎧ 

α ⎪ aα (t, x, Dm u)Dt,x u = f (t, x, Dm u), ⎪ ⎨ ⎪ ⎪ ⎩

|α|≤m

(2.3)

Dtj u(0, x) = gj (x), 0 ≤ j < m,

where t ∈ [0, T ], x ∈ Rn , m < m, gj ∈ H ∞ (Rn ), j = 1, . . . , m − 1, f, aα ∈ B ∞ ([0, T ] × Rn × W ), |α| ≤ m, W a neighborhood in RN of the set {∂ ∂xβ gj (x); x ∈ n  R , j + |β| ≤ m }. With a change of the variable u, one can reduce to null Cauchy data, so we may assume that the origin of RN belongs to W without loss of generality. Moreover, since we look for local solutions, we may assume that f (t, x, 0) has compact support. For the principal symbol Pm of the linear operator P (t, x, w, Dt , Dx ) =

m 

Pj (t, x, w, Dt , Dx ),

j=0

Pj (t, x, w, Dt , Dx ) =



α aα (t, x, w)Dt,x

|α|=j

we assume

⎧ d 1 ⎪ ⎪ ⎪ P (t, x, w, τ, ξ) = (τ − λj (t, x, w, ξ))mj , ⎪ m ⎪ ⎨ j=1

λ ∈ R, j = 1, . . . , d, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎩ λh = λk for every (t, x, w, ξ), ξ = 0, h = k,

(2.4)

so, in particular, P is hyperbolic. The largest multiplicity r = max{mj ; j = 1, . . . , d} plays an important role. Notice that r = 1 means that P is strictly hyperbolic and the well-posedness in H ∞ of the Cauchy problem (2.3) is well known in this case. When r ≥ 2, assumption (2.4) is not sufficient in general, even for linear problems that is with aα and f independent of w. So we impose further conditions on the lower-order terms, starting from the nonlinear one m ≤ m − r.

(2.5)

On the Nonlinear Cauchy Problem

395




Then, for any smooth u such that Dm u takes values in W , we require that the linear problem in the unknown v ⎧
⎨ P (t, x, Dm u, Dt , Dx )v(t, x) = f0 (t, x), ⎩

Dtj v(0, x) = gj (x), 0 ≤ j < m,

satisfies the necessary and sufficient Levi condition for well-posedness in C ∞ stated in [27], that is for every real solution ϕ(t, x) of


∂t ϕ = λj (t, x, Dm u, ∇x ϕ) we have e−iϕ P (eiϕ ) = O(m−mj ),  → +∞, j = 1, . . . , d. (2.6) There are several equivalent forms of the linear Levi condition, e.g., [42], [41], [34]. We use some of them in (2.11) and in the Examples 2.9, 2.10 after Theorem 2.5. There we show that to require (2.6) for any given u leads to analytic relations for the coefficients aα (t, x, w), |α| > m − r. It is clear that the terms Pj of P with j ≤ m − r are not involved in (2.6). Now we can state the main result of this section: Theorem 2.5. Assume that conditions (2.4), (2.5) and (2.6) are fulfilled. Then, for every gj ∈ H ∞ , j = 1, . . . , m − 1, there is T ∗ ≤ T such that the Cauchy problem (2.3) has a unique solution u ∈ C ∞ ([0, T ∗ ]; H ∞ ) . Remark 2.6. In fact we prove that there is s0 such that, for any gj ∈ H s+m−1−j , j = 1, . . . , m − 1, with s > s0 , there is a unique solution u∈

m−1 2

  C j [0, T ∗ ]; H s+m−r−j .

j=0

The solution satisfies the estimate ⎧ m−1 ⎪ ⎪  ∂ ⎪ ∂tj u(t, ·)2H s+m−r−j ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎡ ⎤
t m−1  ⎪ ≤C⎣ ⎪ gj 2H s+m−1−j + f (τ, ·, Dm−r u(τ, ·))2H s dτ ⎦ , ⎪ ⎪ ⎪ 0 ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t ∈ [0, T ∗].

(2.7)

Remark 2.7. Using the condition (2.4) and the estimate (2.7), one can prove that the equation has cones of dependence, e.g., [32]. In fact, there is a positive constant c such that for every (t0 , x0 ), 0 < t0 < T , if u1 and u2 are solutions in the backward cone Γ = {(t, x); 0 ≤ t ≤ t0 , |x − x0 | ≤ c(t0 − t)} (2.8)

396

Massimo Cicognani and Luisa Zanghirati

with base ω0 = {x; |x − x0 | ≤ ct0 }, then Dtj u1 = Dtj u2 in ω0 , j = 0, . . . , m − 1 =⇒ = u1 = u2 in Γ.

(2.9)

Remark 2.8. Theorem 2.5 holds with finite differentiability in the time variable of the coefficients, f, aα ∈ B k ([0, T ]; B ∞(Rn × W )), |α| ≤ m, with k sufficiently large.


Example 2.9. Let Qm−1 = Qm−1 (t, x, Dm u, τ, ξ) denote the subprincipal symbol of P : n  Qm−1 = Pm−1 + (i/2)∂ ∂τ ∂t Pm + (i/2) ∂xj ∂ξj Pm . j=1

For r = 2, the linear Levi condition (2.6) is equivalent to Qm−1 ≡ 0 for τ = λj , mj = 2. From this analytic relation for the coefficients, one can construct examples. For instance, P = (τ 2 − a2 (u)ξ 2 )2 + b(u, Du)τ 3 + c(u, Du)ξ 3 (n = 1, a(u) = 0) is the symbol of an operator which satisfies all the assumptions of Theorem 2.5 if and only if b(u, Du) = −(4/a(u))a (u)Dt u,

c(u, Du) = 4a3 (u)a (u)Dx u.

Example 2.10. From [40], the operator P satisfies condition (2.4) if and only if it can be factorized as follows: r

P = Ar11 · · · Add

+ Rm−1 , where the Aj ’s are strictly hyperbolic differential operators and the remainder Rm−1 is of order m−1. We have {m1 , . . . , md } = {r1 , . . . , rd
}, so r = max1≤j≤d
rj . For 1 ≤ k ≤ r, let us define (r −k)+

Qk = A1 1

(r −k)+

· · · Ad
d


, (z)+ = max{z, 0}, A0 = I.

P satisfies also the Levi condition (2.6) if (and only if in case of analytic coefficients) P = Q0 +

m 

Bj Qj ,

j=1

where Bj denotes an arbitrary differential operator of order m − j − ord Qj , [26], [34], [30], [43]. For instance, an operator with triple and double roots which satisfies all the assumptions of Theorem 2.5 is given by P = A31 A22 + B1 A21 A2 + B2 A1 + B3 , where A1 = A1 (t, x, Dµ1 −1 u, Dt , Dx ) and A2 = A2 (t, x, Dµ2 −3 u, Dt , Dx ) are strictly hyperbolic operators of respective orders µ1 and µ2 , m = 3µ1 + 2µ2 , and

On the Nonlinear Cauchy Problem

397

Bj = Bj (t, x, Dm−3 u, Dt , Dx ), j = 1, 2, 3, are arbitrary operators of respective orders µ1 + µ2 − 1, 2µ1 + 2µ2 − 2, m − 3. The orders of the nonlinear terms µ1 in A1 , µ2 − 3 in A2 , m − 3 in Bj , j = 1, 2, 3, are determined by the nonlinear Levi condition (2.5). There is a small intersection between Theorem 2.5 and the results of [28]. The proof that we are going to give through Subsections 2.3, 2.4, 2.5 and 2.6, is a slight modification of that one given in [14]. 2.3. Factorization Let u∈

k0 +m−r 2

  C j H µ0 +m−r−j , µ0 > n/2,

j=0

be a given function with Dm−r u(t, x) ∈ W , W is a neighborhood of the origin in RN . For a family of symbols p(t, x, w, ξ) ∈ B ∞ (W ; S m ) depending on parameters w ∈ W , t ∈ [0, T ], we have p(t, x, Dm−r u(x), ξ) = p0 (t, x, ξ) + p1 (t, x, Dm−r u, ξ), p0 (t, x, ξ) = p(t, x, 0, ξ) with p0 (t) ∈ S m , p1 (t) ∈ H µ0 S m so p(t) ∈ S m + H µ0 S m . Let the linear operator P , P (t, x, Dm−r u, Dt , Dx ) =



α aα (t, x, Dm−r u)Dt,x ,

|α|≤m

aα ∈ B k0 ([0, T ]; B ∞(Rn × W )) , satisfy the hyperbolicity condition (2.4) and the linear Levi condition (2.6). After a modification in a neighborhood of ξ = 0, the d distinct characteristic roots are symbols λj (t) ∈ S 1 + H µ0 S 1 , so one can carry out the factorization (2.4) to the operators level. From Proposition 2.3, for any fixed µ > n/2 one can take k0 and µ0 = µ + M0 + m − r large enough in order to have P = (Dt − λd )md ◦ · · · ◦ (Dt − λ2 )m2 ◦ (Dt − λ1 )m1 + R(0) , R(0) =

m 

(0)

(0)

rj Dtm−j , rj

(0)

= rj (t, x, DM0 +m−r u, Dx ),

j=1 (0)

rj

    ∈ C [0, T ]; S j−1 + C [0, T ]; H µS j−1 ,

398

Massimo Cicognani and Luisa Zanghirati

where k0 depends only on m, r and n, M0 depends only on µ, m, r and n. This is the first step of a well-known factorization performed in [41] and [37] for P with smooth coefficients till to obtain a regularizing remainder of order −∞. In order to allow finitely many operations, we are going to assume each time k0 ≥ kj ≥ kj−1 , µ0 ≥ µj ≥ µj−1 , µj = µ + Mj , j = 1, 2, . . ., where kj depends only on m, r and n, Mj depends only on µ, m, r and n. In particular, performing finitely many steps of the factorization of [41] and [37], we obtain: Proposition 2.11. Let P satisfy conditions (2.4) and (2.6). For every µ there are k1 and µ1 = µ + M1 such that, if k0 ≥ k1 and µ0 ≥ µ1 , then ⎧ m  ⎪ ⎪ P = P ◦ · · · ◦ P ◦ P + R, R = r Dtm− , ⎪ d 2 1 ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (j) (j) Pj = (Dt − λj )mj + a1 (Dt − λj )mj −1 + · · · + amj , ⎪ ⎪ ⎪     ⎪ (j) (j) ⎪ M +m−r ⎪ u, Dx ) ∈ C [0, T ]; S 0 + C [0, T ]; H µ S 0 , ⎪ ak = ak (t, x, D 1 ⎪ ⎪ ⎪ ⎪     ⎩ r = r (t, x, DM1 +m−r u, Dx ) ∈ C [0, T ]; S −m + C [0, T ]; H µ S −m . (2.10) Remark 2.12. One needs only condition (2.4) to get the factorization of P with (j)

 = max{(ord ak )/(mj − k); 1 ≤ k ≤ mj , 1 ≤ j ≤ d} strictly smaller than 1. The index  is invariant under permutations of the indices 1, 2, . . . , d and the Levi condition (2.6) is equivalent to  ≤ 0.

(2.11)

2.4. The linear problem From the factorization (2.10), we can reduce the linear scalar equation in the unknown v P (t, x, Dm−r u, Dt , Dx )v = f (2.12) to an equivalent system. Without loss of generality, but only to have a simpler notation, let us consider the case d = 2 of an operator with two multiple characteristic roots. Applying Proposition 2.11, we have both P = P2 ◦ P1 + R and, by permutation, ˜ P = P˜1 ◦ P˜2 + R. (j) (j) The factors Pj and P˜j have respective coefficients ak and a ˜k of order 0 whereas ˜ have respective coefficients r and ˜ of order  − m. Then, the remainders R and R

On the Nonlinear Cauchy Problem

399

given the scalar function v, let us define the vector V = (v0 , . . . , vm−1 , vm , . . . , v2m−1 ) by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

v0 = v v1 = (Dt − λ1 )v .. .

vm1 −1 = (Dt − λ1 )m1 −1 v vm1 = P1 v P1 v vm1 +1 = (Dt − λ2 )P .. . vm−1 = (Dt − λ2 )m2 −1 P1 v

vm = v vm+1 = (Dt − λ2 )v .. . vm+m2 −1 = (Dt − λ2 )m2 −1 v vm+m2 = P˜2 v vm+m2 +1 = (Dt − λ1 )P˜2 v .. . v2m−1 = (Dt − λ1 )m1 −1 P˜2 v.

(2.13)

The scalar equation (2.12) is equivalent to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(Dt − λ1 )vj = vj+1 (0 ≤ j ≤ m1 − 2) (Dt − λ1 )vm1 −1 = vm1 −

*m1

(1)

k=1

ak vm1 −k

(Dt − λ2 )vj = vj+1 (m1 ≤ j ≤ m − 2) (Dt − λ2 )vm−1 = f − Rv −

*m2 k=1

(2)

ak vm−k

⎪ ⎪ (Dt − λ2 )vm+j = vm+j+1 (0 ≤ j ≤ m2 − 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ *m2 (2) ⎪ ⎪ (Dt − λ2 )vm+m2 −1 = vm+m2 − k=1 a ˜k vm+m2 −k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (D − λ )v ⎪ t 1 m+j = vm+j+1 (m2 ≤ j ≤ m − 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ * ⎩ (1) ˜ − m1 a (Dt − λ1 )v2m−1 = f − Rv k=1 ˜k v2m−k ,

(2.14)

˜ We have v = v0 and the remainders R, R ˜ so we need to consider the terms Rv, Rv. are of order 0 in Dt,x but they contain derivatives Dtj v0 up to j = m − 1. Anyway, from (2.13), by induction on j = 0, . . . , m − 1, it is easy to obtain: Lemma 2.13. Let u and P (t, x, Dm−r u, Dt , Dx ) be as in Proposition 2.11 and, given the scalar function v, let the vector V = (v0 , . . . , vm−1 , vm , . . . , v2m−1 )

400

Massimo Cicognani and Luisa Zanghirati

M2 such that, if k0 ≥ k2 be defined by (2.13). For every µ there are k2 and µ2 = µ+M and µ0 ≥ µ2 , then ⎧ j  ⎪ (j) ⎪ j ⎪ D v = c (t, x, DM2 +m−r u, Dx )vj− + vj , ⎪ t ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎨ j (2.15)  (j) j ⎪ ⎪ Dt v = c˜ (t, x, DM2 +m−r u, Dx )vm+j− + vm+j , ⎪ ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎩ j = 0, . . . , m − 1, with

    (j) (j) c , c˜ ∈ C [0, T ]; S  + C [0, T ]; H µS  . This gives immediately ⎧ m−1  ⎪ ⎪ Rv = ⎪ bj (t, x, DM3 +m−r u, Dx )vj , ⎪ ⎪ ⎪ ⎪ j=0 ⎨ ⎪ Rv ˜ = *m−1 ˜bj (t, x, DM3 +m−r u, Dx )vm+j , ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪     ⎩ bj , ˜bj ∈ C [0, T ]; S 0 + C [0, T ]; H µ S 0

(2.16)

for any given µ, provided that k0 ≥ k3 , and µ0 ≥ µ3 = µ + M3 . So, from (2.16) and (2.14), we have that the scalar problem ⎧ ⎨ P (t, x, Dm−r u, Dt , Dx )v(t, x) = f0 (t, x), (2.17) ⎩ j Dt v(0, x) = gj (x), 0 ≤ j < m, is equivalent to

⎧ ⎨ ∂t V − K(t, x, DM3 +m−r u, Dx )V = F0 , ⎩

(2.18) V (0) = G,

*m−1 F0 = (0, . . . , 0, f0 , 0, . . . , 0, f0 ), GH µ ≤ C j=0 gj H µ+m−1−j , for a symmetric hyperbolic 2m × 2m system ∂t − K to which we can apply Proposition 2.4 since K = i∆ + R0 with ∆ is a real diagonal matrix pseudodifferential operator of order 1, R0 is a matrix pseudodifferential operator of order 0. In order to get an energy estimate for the scalar function v from (2.2), we use the following proposition. During its proof, it becomes clear why we need to consider ˜ so obtaining a both the factorizations P = P2 ◦ P1 + R and P = P˜1 ◦ P˜2 + R 2m × 2m system (in general a dm × dm system, d the number of the characteristic roots, using the cyclic permutations of (1, 2, . . . , d)).

On the Nonlinear Cauchy Problem

401

Proposition 2.14. Let the vector V = (v0 , . . . , vm−1 , vm , . . . , v2m−1 ) be defined by (2.13). For every µ there are k4 and µ4 = µ + M4 such that, if k0 ≥ k4 and µ0 ≥ µ4 , then there is a matrix Q = Q(t, x, DM4 +m−r u, Dx ) such that     Dm−r v = QV, Q ∈ C [0, T ]; S 0 + C [0, T ]; H µ S 0 . (2.19) Proof. During this proof, all the symbols denoted by letters a, b, c, q depend  on  (t, x, DM4 +m−r u, ξ) and belong to spaces C [0, T ]; S δ + C [0, T ]; H µS δ . The order δ will be specified for each of them. The finitely many operations we need to perform, will be allowed taking k0 ≥ k4 and µ0 ≥ µ4 = µ + M4 with sufficiently large k4 and M4 . From condition (2.4) we may assume |λ2 − λ1 | > cξ, c > 0 after a modification in a neighborhood of ξ = 0. So, for 0 ≤ k ≤ m − 1, we can write  τ −λ τ − λ2 m−1−k  τ − λ1 τ − λ2 k 1 τk = − λ2 − λ1 , λ2 − λ1 λ2 − λ1 λ2 − λ1 λ2 − λ1 and, by Newton’s formula, we have τk =

m−1−k 

k 

ak1 ,k2 (τ − λ1 )k1 +k2 (τ − λ2 )m−1−k1 −k2 , (k)

k1 =0 k2 =0 (k)

where ord ak1 ,k2 ≤ k + 1 − m. In this sum, for k1 + k2 < m1 , let us write (τ − λ2 )m−1−k1 −k2 = (τ − λ2 )m2 [(τ − λ1 ) + (λ1 − λ2 )]m1 −1−k1 −k2 and, for k1 + k2 ≥ m1 , k1 +k2 −m1 (τ − λ1 )k1 +k2 = (τ − λ1 )m . 1 (τ − λ2 ) + (λ2 − λ1 )]

Applying again Newton’s formula, we obtain τk =

m 1 −1 

(k)

b1,j (τ − λ1 )j (τ − λ2 )m2 +

j=0

m 2 −1 

(k)

b2,j (τ − λ2 )j (τ − λ1 )m1 ,

j=0

(k)

ord bi,j ≤ k + mi − m − j ≤ k − m + r. From (2.13), this gives ⎧ m m m−1 1 −1 2 −1    (k) ⎪ (k) (k) k ⎪ ⎪ D v = b v + b v + c Dt v, ⎨ t 1,j m+m2 +j 2,j m−1+j ⎪ ⎪ ⎪ ⎩

j=0 (k)

ord c

≤ k −  − 1.

j=0

=0

(2.20)

402

Massimo Cicognani and Luisa Zanghirati

In the third sum, we can substitute Dt v with the expression given by (2.20) itself. Repeating this process k + 2m − r − 1 times, we obtain ⎧ m m m−1 1 −1 2 −1    (k) ⎪ (k) (k) k ⎪ ˜ ˜ ⎪ v = v + v + c˜ Dt v, b b D m+m +j m−1+j ⎨ t 2 1,j 2,j j=0 j=0 =0 (2.21) ⎪ ⎪ ⎪ ⎩ (k) (k) ord ˜bi,j ≤ k − m + r, ord ˜ ≤ −2m + r + 1. Now, we use (2.15) for Dt v in the third sum of (2.21) in order to get for all k = 0, . . . , m − 1 Dtk v =

2m−1 

(k)

(k)

qj vj , ord qj

≤ k − m + r.

(2.22)

j=0

We have (2.19) applying Dxα , |α| ≤ m − r − k, to both sides of the above equality (2.22).
Up to now, from Proposition 2.4 applied to the system (2.18), and taking Proposition 2.14 into account, we have proved the following result of well-posedness in Sobolev spaces for the linear equivalent problems (2.17) and (2.18): Proposition 2.15. For every µ there are k5 and µ5 = µ + M5 , such that, if k0 ≥ k5 and µ0 ≥ µ5 , then the Cauchy problem (2.18) has a unique solution V ∈ C ([0, T ]; H µ ) to any given G ∈ H µ and F0 ∈ C ([0, T ]; H µ ). The solution V satisfies the estimate 
t Cu ) G2H µ + F F0 (τ ))2H µ dτ , t ∈ [0, T ]. V (t)2H µ ≤ (1 + tC 0

From this, the scalar Cauchy problem (2.17) has a unique solution v∈

m−1 2

  C j [0, T ]; H µ+m−r−j

j=0

to any given gj ∈ H µ+m−1−j , j = 0, . . . , m − 1, and f0 ∈ C ([0, T ]; H µ ). The solution v satisfies the estimate ⎡ ⎤
t m−1 m−1  j  ∂ ∂t v(t)2H µ+m−r−j ≤ Cu ⎣ gj 2H µ+m−1−j + ff0 (τ ))2H µ dτ ⎦ , t ∈ [0, T ]. j=0

j=0

0

In the above both estimates, Cu denotes a locally bounded function of the norms ∂ ∂tj u(t, ·)H µ0 +m−r−j , 0 ≤ j ≤ k0 .

On the Nonlinear Cauchy Problem

403

2.5. Commutators Proposition 2.15 shows that the reduction of the scalar problem (2.17) to the system (2.18) is not yet sufficient to look for a fixed point. In order to solve (2.18) in H µ , we need u(t) ∈ H µ+M5 +m−r even if the symbol of K depends only on Dµ+M3 +m−r with M3 < M5 . As it concerns the equivalent scalar problem, we loose M5 derivatives in the map u
→ v. So we introduce a further reduction.
For µ0 = µ + M + m − r, β = (β0 , β  ), we apply derivatives ∂ β = ∂tβ0 ∂xβ , β0 ≤ k0 , |β| ≤ M , in (2.17) so obtaining problems ⎧ ⎪ P v (β) + [∂ β , P ]v = f (β) , ⎨ (2.23) ⎪ j (β) (β) ⎩ Dt v|t=0 = gj , 0 ≤ j < m, |β| ≤ M, for v (β) = ∂ β v, [∂ β , P ] denotes the commutator ∂ β P − P ∂ β . Then, we introduce (β) V˜ = (vj ; 0 ≤ j ≤ 2m − 1, β0 ≤ k0 , |β| ≤ M )

the vector of components have the following:

(β) vj

(2.24)

defined by (2.13) taking there v (β) in place of v. We

Proposition 2.16. For every µ there are k6 and M6 , such that for all M ≥ M6 , if k0 ≥ k6 and µ0 = µ+M , then there is a matrix of functions F = F (t, x, DM+m−r u) ˜ = Q(t, ˜ x, DM6 +m−r u, Dx ) of operators such that and a matrix Q ⎧  β  ˜ V˜ , ⎪ [∂ , P ]v; β0 ≤ k0 , |β| ≤ M = F Q ⎪ ⎪ ⎪ ⎨ F ∈ C ([0, T ]; B ∞) + C ([0, T ]; H µ) , (2.25) ⎪ ⎪ ⎪ ⎪     ⎩ ˜ Q ∈ C [0, T ]; S 0 + C [0, T ]; H µ+M−M6 S 0 . Proof. During this proof, all the symbols denoted by letters  a, b, q depend  on  (t, x, DM6 +m−r u, ξ) and belong to spaces C [0, T ]; S h + C [0, T ]; H µ S h . The order h will be specified for each of them. The finitely many operations we need to perform, will be allowed taking k0 ≥ k6 and µ0 = µ + M + m − r with M ≥ M6 . For |β| ≤ M , β0 ≤ k0 , let us define (β)

V (β) = (vj ; 0 ≤ j ≤ 2m − 1), so that (2.24) becomes V˜ = (V (β) ; β0 ≤ k0 , |β| ≤ M ). Applying (2.19) to v (β) we have Dm−r ∂ β v = QV (β) , so in [∂ β , P ]v =

 |α|≤m 0 M − r + 1 neglecting all the terms with |γ| ≥ r in equality (2.27). In doing so, we need the following lemma. Lemma 2.17. We have  % $ akj (Dt − λj )−k , 1 ≤  ≤ mj , ord akj = h (i) a, (Dt − λj ) = 1≤k≤

for every operator a of order h; $ % (ii) (Dt − λ1 ) , (Dt − λ2 )d =



aij (Dt − λ1 )−i (Dt − λ2 )d−j ,

1≤i≤,1≤j≤d

1 ≤  ≤ m1 , 1 ≤ d ≤ m2 , ord aij = i + j − 1; (iii)

(Dt − λ1 )m1 − (Dt − λ2 )m2 −d =



ai (Dt − λ1 )m1 −i (Dt − λ2 )m2

1≤i≤

+



bi (Dt − λ2 )m2 −i (Dt − λ1 )m1

1≤i≤d

+



cij (Dt − λ1 )i (Dt − λ2 )j ,

0≤i≤m1 −1 0≤j≤m2 −1

0 ≤  < m1 , 0 ≤ d < m2 ,  + d > 1, ord (ai , bi ) = − − d + i, ord cij = m −  − d − i − j − 1. Proof of the lemma. Let P, Q, R be three operators. We have the identity [P, QR] = [P, Q]R + Q[P, R]

(2.28)

provided that all the compositions are well defined. Using this, one can prove (i) by induction on . As it concerns the equality (ii), at first one can prove it with  = 1 by induction on d and then complete the induction on  with a fixed d. In both steps one uses (i) and (2.28). Finally, from the representation of the identity operator 1 = q(Dt − λ1 ) − q(Dt − λ2 ) + a, where q has the symbol (λ2 − λ1 )−1 and a is another suitable pseudodifferential operator of order −1, one can prove (iii) by induction on  + d using (i), (ii) and

On the Nonlinear Cauchy Problem

405

(2.28) in (Dt − λ1 )m1 − (Dt − λ2 )m2 −d = (Dt − λ1 )m1 − (q(Dt − λ1 ) − q(Dt − λ2 ) + a)(Dt − λ2 )m2 −d .




Continuation of the proof of Proposition 2.16. Let β be such that M − r + 1 < |β| ≤ M (r > 1) and let us consider the factorization P = P2 P1 + R given by (2.10). Terms of order larger than M + m − r in [∂ β , P ] can only appear from [∂ β , P2 P1 ] = [∂ β , P2 ]P P1 + P2 [∂ β , P1 ]. Using again (2.28) and (2.10), we have ⎧ β [∂ , P2 ]P P1 = [∂ β , (Dt − λ2 )m2 ](Dt − λ1 )m1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ (2) (1) ⎪ ⎪ + [∂ β , aj ](Dt − λ2 )m2 −j ai (Dt − λ1 )m1 −i ⎪ ⎪ ⎪ ⎨ i,j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+



(2.29) aj [∂ β , (Dt − λ2 )m2 −j ]ai (Dt − λ1 )m1 −i , (2)

(1)

i,j

0 ≤ i ≤ m1 , 0 ≤ j ≤ m2 , i + j > 0.

A repeated use of (2.28) gives ⎧ β ⎪ [∂ , (Dt − λj )mj −d ] = ⎪ ⎨  ⎪ (Dt − λj )mj −d−h [∂ β , (Dt − λj )](Dt − λj )h−1 , ⎪ ⎩

(2.30)

1≤h≤mj −d

so all the terms in the right-hand side of equality (2.29) are compositions of operators a, [∂ β , q], (Dt − λ1 )m1 −i , (Dt − λ2 )m2 −j with a of order 0, q of order less or equal to 1. We have to consider only the truncated expansion  aγ ∂ β−γ , ord aγ ≤ 0, 0≤γ n/2 there are k7 and M7 , such that for all M ≥ M7 , if k0 ≥ k7 and µ0 = µ + M , then the Cauchy problem (2.33) has a unique solution V˜ ∈ C ([0, T ]; H µ ) ˜ ∈ H µ and F˜0 ∈ C ([0, T ]; H µ ). The solution V˜ satisfies the estimate to any given G 
t 2 ˜ 2H µ + ˜ Cu ) G F˜0 (τ ))2H µ dτ , t ∈ [0, T ], V (t)H µ ≤ (1 + tC 0

where Cu is a positive locally bounded function of the norms ∂ ∂tj u(t, ·)H µ0 +m−r−j , 0 ≤ j ≤ k0 . Now, the derivatives of u that we need to assume in C ([0, T ]; H µ) in order to solve (2.33) in H µ , are exactly the same DM+m−r u that appears in the symbol ˜ + F˜ Q. ˜ of ∂t − K 2.6. The equivalent quasilinear system For |β| ≤ M , β0 ≤ k0 , M > M7 , k0 > k7 , let us denote (β)

U (β) = (uj ; 0 ≤ j ≤ 2m − 1)

(2.34)

(β)

the vector of components uj defined by (2.13) taking there ∂ β u in place of v, and, for k ≤ M , let us denote ˆk = (U (β) ; |β| ≤ k), U ˆ =U ˆM . U

(2.35)

ˆk ⊂ U ˆ , |γ| ≤ M − k. ∂xγ U

(2.36)

By definition, we have

ˆ = Q(t, ˆ x, U ˆM4 +m−r , Dx ) such that From (2.26), there is Q ⎧ M+m−r ˆU ˆ, u=Q ⎨ D     ⎩ ˆ M4 0 Q ∈ C [0, T ]; S 0 + C [0, T ]; H µ+M−M S ,

(2.37)

408

Massimo Cicognani and Luisa Zanghirati

so, by (2.33) and (2.37), we have that the quasilinear scalar problem (2.3) can be reduced to a quasilinear symmetric hyperbolic system ⎧ ˆ − K(t, ˆ x, U ˆ , Dx )U ˆ = Fˆ (t, x, Q ˆU ˆ ), ⎨ ∂t U (2.38) ⎩ ˆ ˆ U(0) = G, where ˆ = K(t, ˜ x, U ˆM6 +m−r , Dx ) + F˜ (t, x, Q ˆU ˆ )Q(t, ˜ x, U ˆM6 +m−r , Dx ), K

(2.39)

˜ F˜ , Q ˜ as in (2.33), and Fˆ is a smooth function of its arguments such that K, Fˆ (t, x, 0) has compact support. We prove Theorem 2.5 after fixing µ > n/2 + 1 and showing that for every M > M7 there is a unique solution ˆ ∈ C ([0, T ∗]; H µ ) U ˆ H µ . From Proposition 2.18, for every ˆ ∈ H µ . Let us take R > G for any given G µ ˆ ˆ U ∈ C ([0, T ]; H ) with U(t) ≤ R, 0 ≤ t ≤ T , there is a unique solution Vˆ ∈ C ([0, T ]; H µ ) of the linear problem ⎧ ˆ x, U ˆ , Dx )Vˆ = Fˆ (t, x, Q ˆ ), ˆU ⎨ ∂t Vˆ − K(t, ⎩ ˆ ˆ V (0) = G. This solution satisfies


t 2 ˆ ˆ 2H µ + V (t)H µ ≤ (1 + tCR ) G Fˆ (τ ))2H µ dτ , t ∈ [0, T ],

(2.40)

0

so there is a T ∗ < T such that Vˆ (t) ≤ R, 0 ≤ t ≤ T ∗ . ˆ , Uˆ ∈ C ([0, T ∗ ]; H µ ) with U ˆ (t), Inequality (2.40) implies also that for every U ∗   ˆ ˆ ˆ U (t) ≤ R, 0 ≤ t ≤ T , the corresponding solutions V , V satisfy
t 2  ˆ ˆ V (t) − V (t)H µ−1 ≤ LR Vˆ (t) − Vˆ (τ )2H µ−1 dτ, t ∈ [0, T ∗]. 0

ˆ ). So, the usual approximation sequence defined by Let us denote Vˆ = S(U ˆ0 = G, ˆ U ˆk+1 = S(U ˆk ), k = 0, 1, . . . U

(2.41)

ˆk ∈ C ([0, T ∗ ]; H µ ), U ˆk (t) ≤ R, 0 ≤ t ≤ T ∗ , for every k, and, is such that U   ∗ ˆ of (2.38). taking a smaller T , it converges in C [0, T ∗ ]; H µ−1 to a solution U This solution satisfies 
t ˆ 2 µ−1 + ˆ (t)2 µ−1 ≤ (1 + tCR ) G Fˆ (τ ))2 µ−1 dτ , (2.42) U H

H

H

0

t ∈ [0, T ∗ ], hence it is unique. The proof of Theorem 2.5 is complete. The estimate (2.7), with s = µ + M , s0 = µ + M7 , follows from (2.37) and (2.42).

On the Nonlinear Cauchy Problem

409

2.7. Local C ∞ solutions Let Ω ⊂ Rn+1 be an open set where the variable is denoted by y and let S = {ϕ(y) = 0} be an hypersurface with a real-valued C ∞ function ϕ, ∇ϕ = 0 in S, ν = ∇ϕ/∇ϕ is the normal vector. Let us consider the Cauchy problem ⎧ m
m
⎪ ⎪ ⎨ P (y, D u, Dy )u = f (y, D u), (2.43) ⎪ ∂j u ⎪ ⎩ = g in S, 0 ≤ j < m, j ∂ν j gj ∈ C ∞ (S), j = 1, . . . , m − 1, f ∈ C ∞ (Ω × W ), W is a neighborhood of the Cauchy data, for an operator 

P (y, Dm u, Dy ) = aα (y, Dm u)Dyα (2.44) |α|≤m 

∞

with m < m, aα ∈ C (Ω × W ), |α| ≤ m. Given y0 , ν0 with y0 ∈ S and ν0 the normal vector to S at y0 , we say that the
operator P (y, Dm u, Dy ) satisfies the condition of hyperbolicity (2.4) and the Levi conditions (2.5), (2.6) at (y0 , ν0 ) if there is a neighborhood of y0 in Rn+1 where these conditions are fulfilled taking local coordinates y = (t, x) with dt = ν0 . It is well known that, in this case, the same holds true for (y0 , −ν0 ). From Theorem 2.5 and Remark 2.7 we have the following result of local existence and uniqueness of C ∞ solutions. Theorem 2.19. Let the condition of hyperbolicity (2.4) and the Levi conditions (2.5), (2.6) be satisfied at (y, ν) for every y ∈ S. Then, for every gj ∈ C ∞ (S), j = 1, . . . , m − 1, there is a neighborhood A of S in Ω such that problem (2.43) has a unique solution u ∈ C ∞ (A). In particular, uniqueness can be proved in domains of influence. Let D ⊂ Ω be a compact domain ) D= S¯µ , 0≤µ≤1

where {Sµ ; 0 ≤ µ ≤ 1} is a continuous family of surfaces satisfying for local coordinates z = (z0 , z1 , . . . , zn ) = (z0 , z  ): (i) S¯0 is the closure of a bounded open set S0 ⊂ {z0 = 0} with C ∞ boundary; (ii) S¯µ = {(z0 , z  ); z0 = ϕµ (z  ), z  ∈ S¯0 }, where ϕ0 = 0 and {ϕµ } is a family of C ∞ functions defined on S¯0 such that ϕµ = 0, ϕµ (z  ) < ϕµ
(z  ), z  ∈ S0 , 0 ≤ µ < µ ≤ 1. ∂S0

We say that D is a domain of influence based on S¯0 if for all µ ∈ [0, 1] the condition of hyperbolicity (2.4) and the Levi conditions (2.5), (2.6) are satisfied at every point (y, ν), y ∈ S¯µ , ν is a normal vector to S¯µ at y.

410

Massimo Cicognani and Luisa Zanghirati

For such a domain D and solutions u1 , u2 of





P (y, Dm u, Dy )u = f (y, Dm u)

(2.45)

in a neighborhood of D, we have ∂ j u1 ∂ j u2 = in S0 , j = 0, . . . , m − 1 = =⇒ u1 = u2 in D. j ∂ν ∂ν j 2.8. Analytic regularity Now, let us consider a fixed C ∞ real solution u(t, x) of ⎧

P (t, x, Dm u, Dt,x )u = f (t, x, Dm u), ⎪ ⎪ ⎨ 
m
α ⎪ P (t, x, D u, D ) = aα (t, x, Dm u)Dt,x , t,x ⎪ ⎩

(2.46)

(2.47)

|α|≤m

for (t, x) ∈ [0, T ] × M, M is a real analytic compact manifold of dimension n. Let us assume that aα (t, x, w), |α| ≤ m, and f (t, x, w) are real analytic functions of their arguments in a neighborhood of (2.48)
the set {(t, x, Dm u(t, x)); t ∈ [0, T ], x ∈ M}. We prove the following result of propagation of analytic regularity of the solution u. Theorem 2.20. Let the condition (2.48) be fulfilled and let the operator P (t, x, Dm−r u, Dt,x ) satisfy the conditions (2.4), (2.5) and (2.6) for the fixed function u. If the traces ∂tj u(0, x), j = 0, . . . , m−1 are analytic, then u is analytic in [0, T ]×M. Remark 2.21. If the functions f and aα , |α| ≤ m, are respectively continuous and C k with respect to t with k large enough and if we assume for them analytic regularity only with respect to (x, w), then we can prove that the solution u(t, x) is analytic with respect to x for all t ∈ [0, T ]. From the Cauchy-Kovalevsky Theorem, there is an analytic solution v in a right-sided neighborhood of {t = 0}, say for 0 < t < T0 , with the same traces of u and there we have u = v. The above theorem means that any C ∞ solution with T0 , T ]. lifespan T > T0 cannot develop analytic singularities at t ∈ [T We use the reduction of the scalar equation (2.47) to the equivalent system ˆ hence its first (2.38) in the cylinder [0, T ] × M and prove that the vector U, β component u, is analytic. In doing so, we take derivatives ∂ = ∂xβ and estimate ˆβ = ∂ β U by induction on |β| using the energy inequality (2.42). U As in [2], one can localize this kind of proof using cutoff functions χN (t, x), N = |β| + N0 , such that α |∂ ∂t,x χN (t, x)| ≤ (CN )|α| , |α| ≤ N,

On the Nonlinear Cauchy Problem

411

N0 is a suitable fixed positive integer depending on m, r, n. So, in the same way, in the equation (2.45), one can prove the propagation of analytic regularity in domains of influence for a fixed C ∞ solution u(y) in an open set A ⊂ Ω, provided that the coefficients satisfy aα (y, w), |α| ≤ m, f (y, w) are analytic functions of their
arguments in a neighborhood of {(y, Dm u(y)); y ∈ A}.

(2.49)

Theorem 2.22. Let the condition (2.49) be fulfilled, S be an analytic hypersurface, u ∈ C ∞ (A) be a fixed solution in A of equation (2.45) and let D ⊂ A be a domain of influence determined by u and based on S¯0 , S0 is a bounded open set in S with C ∞ boundary. ∂j u , 0 ≤ j ≤ m − 1, are analytic in S¯0 , this means, analytic in an If the traces ∂ν j S open neighborhood of S¯0 , then u is analytic in D. The remaining part of the section is devoted to the proof of Theorem 2.20. We are going to deal with symbols p(t, x, w, ξ) in S m (M × Rn ) depending on
parameters (t, w) ∈ [0, T ]×W , W is a neighborhood of the set {Dm u(t, x); (t, x) ∈ [0, T ] × M}. From (2.48), these symbols are analytic functions of (t, x, w) ∈ [0, T ] × M × W , so they satisfy ⎧ j γ ⎪ p(t, x, w, ξ) ≤ Cα Aj+|β|+|γ| j!β!γ!ξm−|α| , ⎨ ∂t ∂ξα ∂xβ ∂w (2.50) ⎪ ⎩ (t, x, w) ∈ [0, T ] × M × W. From (2.38), (2.37) and (2.39), the equation


P u = f (t, x, Dm u), (t, x) ∈ [0, T ] × M, is equivalent to the system ⎧ ˆ −K ˜U ˆ = Φ, ⎪ ∂t U ⎪ ⎪ ⎪ ⎨ ˆ )Q ˜U ˆ + Fˆ (t, x, QU ˆ ), Φ = F˜ (t, x, QU ⎪ ⎪ ⎪ ⎪ ⎩ (t, x) ∈ [0, T ] × M,

(2.51)

(2.52)

where now: • the matrix F˜ and the vectors Fˆ are analytic functions of the variable (t, x, w) ∈ C¯ × W ; • the symbols of ˜ = K(t, ˜ x, DM6 +m
u, Dx ) K (of order 1) and of ˆ = Q(t, ˆ x, DM6 +m
u, Dx ), Q ˜ = Q(t, ˜ x, DM6 +m
u, Dx ) Q

412

Massimo Cicognani and Luisa Zanghirati

(of order 0) satisfy (2.50); • we can choose M as larger than M6 as we need, in particular for any fixed M∗ we may assume
ˆ. DM6 +m +M∗ +1 u ⊂ U (2.53) ˆj = Applying derivatives ∂j = ∂xj , j = 1, . . . , n, we obtain equations for U ˆ ∂j U , ⎧ ˜U ˆj + * Aij U ˆj − K ˆ i = Φj , ⎪ ∂t U i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˜ ∂j ] = * Aij ∂i , ⎨ [K, i (2.54) ⎪
⎪ M +m +1 6 ⎪ Ai,j = Ai,j (t, x, D , Dx ), ord Aij = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Φj = ∂j Φ, 1 ≤ i, j ≤ n. Our aim now is to take further derivatives ∂ β = ∂xβ and to prove that the function u is analytic with respect to x ∈ M estimating ˆβ,j = ∂ β Uj , U hence its first component ∂ β ∂j u, by induction on |β|. We obtain equations ⎧ * ˆβ,j − K ˆβ,i = Φβ,j , ˜U ˆβ,j + Aij U ⎨ ∂t U i

⎩

˜ U ˆj + Φβ,j = −[∂ β , K]

(2.55)

*

β β ˆ i [∂ , Aij ]Ui + ∂ ∂j Φ,

for which the energy inequality gives ⎧  ˆβ,j (t)2 µ ⎪ U ⎪ H (M) ≤ ⎪ ⎪ ⎪ j ⎨
t ⎪  ⎪ ⎪ ˆβ,j (0)2 µ ⎪ C {U + Φβ,j (τ )2H µ (M) dτ }. ⎪ H (M) ⎩ 0

(2.56)

j

ˆβ,j (t)H µ (M) by Gronwall’s method. This will allow us to estimate U We need to introduce some notation. Let us consider the sequence mp = p! , where the constant c is chosen in order to satisfy c (p + 1)2  α  α m|β| m|α−β| ≤ m|α| , m|β| m|α−β|+1 ≤ |α|m|α| . β β 0≤β≤α

0 (n/2) be a fixed integer. For ε > 0, p ≥ 1, w ∈ C ∞ (M), let us denote wµ = wH µ (M) , Mp = ε1−p mp and |w|q . 0 0,

j = k.

Then, for each B ∈ MN ×N (C), there is a constant ε0 > 0 with the property that the eigenvalues µjε of A + εB for |ε| ≤ ε0 are distinct, c0 |µjε − µkε | ≥ , j = k, |ε| ≤ ε0 , 2 and depend analytically on ε and the entries of A  and B. The bound ε0 depends analytically on c0 and the norms A, B, and M −1  , where M with M  = 1 is a diagonalizer of A. Here,   denotes the row-sum matrix norm. Proof. By definition of M , we have M AM −1 = diag(µ1 , . . . , µN ), and the eigenvalues µj as well as the entries of M depend analytically on the entries of A. Therefore, we can assume that A = diag(µ1 , . . . , µN ) is diagonal. By Gerschgorin’s theorem, each of the N balls ' (  |Bjk | Ωjε = z ∈ C |z − (µj + εBjj )| ≤ |ε| k=  j

contains exactly one eigenvalue µjε of A + εB provided that these balls do not intersect. This happens when |ε| ≤ ε0 and 2ε0 B ≤ c0 /2. The eigenvalues µjε of A + εB are solutions to the polynomial equation 0 = φε (µ) = det (A + εB − µ1N ) , and are given by the integral µjε =

1 2πi

@ ∂ Ωjε

µφε (µ) dµ, φε (µ)

which completes the proof.




We likewise need the following results: 0 Lemma A.3. For each N × N matrix symbol q ∈ S1,0 (Rn × Rn ) that satisfies 2n | det q(x, ξ)| ≥ c for all (x, ξ) ∈ R , |ξ| ≥ C, and some constants C, c > 0, there 0 (Rn ) such that is an invertible operator Q ∈ Op S1,0 −1 (Rn ). Q − q(x, Dx ) ∈ Op S1,0

502

Michael Dreher and Ingo Witt

Proof. We construct two invertible operators Q1 = q1 (x, Dx ), Q2 = q2 (x, Dx ) ∈ 0 (Rn ) such that Op S1,0 −1 (Rn × Rn ), q1 (x, ξ) ≡ χ(|ξ|/(2C))q(x, ξ)q(x0 , ξ)−1 mod S1,0 −1 mod S1,0 (Rn × Rn ).

q2 (x, ξ) ≡ q(x0 , ξ)

Here, the point x0 ∈ Rn is chosen arbitrarily. Then the composition Q1 Q2 possesses the desired properties. Construction of Q1 . We employ the parameter calculus of Grubb [22]. Rename (1 − χ(|ξ|/(2C)))1N + χ(|ξ|/(2C))q(x, ξ)q(x0 , ξ)−1 to q(x, ξ). Then | det q(x, ξ)| ≥ c for |ξ| ≥ 2C and some c > 0 and q(x0 , ξ) = 1N for all ξ ∈ Rn . By a standard application of the mapping degree and homotopy theory, we obtain 0 (Rn × Rn ) such that h(x, ξ) = q(x, ξ) for an N × N matrix function h ∈ S1,0  |ξ| ≥ 2C and | det h(x, ξ)| ≥ c /2 for all (x, ξ) ∈ R2n , by changing q(x, ξ) for |ξ| ≤ 2C if necessary. We then further get an N × N matrix function p0 (x, ξ, µ) ∈ 0 S1,0 (Rn × Rn × R+ ), where µ ≥ 0 enters as additional covariable, such that | det p0 (x, ξ, µ)| ≥ c /2, p0 (x, ξ, 0) = q(x, ξ),

(x, ξ, µ) ∈ R2n × R+ ,

|ξ, µ| ≥ 2C,

(x, ξ) ∈ R , 2n

For that it suffices to set |ξ, µ| ≥ 2C, µ ≥ |ξ|, √ √ √ and then to connect p0 (x, ξ, 0) = q(x, ξ) to p0 (x, ξ/ √2, |ξ|/ 2) = h(x, ξ/( 2|ξ|)) for |ξ| ≥ 2C along√ the curve [0, 1] & σ
→ (x, (1−κσ)ξ, 2κσ − κ2 σ 2 |ξ|) ∈ R2n ×R+ , where κ = 1 − 1/ 2, appropriately:   ξ ξ +σ p˜0 (x, ξ, µ) = h x, (1 − σ) , |ξ, µ| ≥ 2C, µ < |ξ|, 1 − κσ |ξ, µ| p˜0 (x, ξ, µ) = h(x, ξ/|ξ, µ|),

where σ = κ−1 (1 − |ξ|/|ξ, µ|), i.e., ξ/(1−κσ) = |ξ, µ| ξ/|ξ|. The symbol ˜0 thus obtained (appropriately continued into |ξ, µ| < 2C) is continuous, but only piecewise C ∞ ; smoothing ˜0 along |ξ, µ| ≥ 2C, µ = |ξ|, whilst keeping the symbol estimates and invertibility, yields the symbol p0 . We now set    p(x, ξ, µ) = χ(|ξ, µ|) p1 (x, ξ, µ) + χ(|ξ|) p0 (x, ξ, µ) − p1 (x, ξ, µ) , * where p1 (x, ξ, µ) = |α| 0, see Grubb [22, Remark 2.1.13]. According to Grubb [22, Theorem 3.2.11], there is a µ0 ≥ 0 such that, for all µ ≥ µ0 , the operator p(x, Dx , µ) : L2 (Rn ) → L2 (Rn ) is invertible. Eventually, it suffices to set Q1 = p(x, Dx , µ), where µ ≥ µ0 . Construction of Q2 . Rename q(x0 , ξ) to q(ξ). The task to construct a symbol 0 q2 ∈ S1,0 such that q2 (x, Dx ) is invertible and q(Dx ) − q2 (x, Dx ) ∈ Op S −∞ can be fulfilled within the framework of SG-calculus, where one has symbols which fulfil

Sharp Energy Estimates

503

independently symbol estimates in both the x- and the ξ-variables. Recall that S m;η = S m;η (Rn × Rn ) is the class of all a ∈ C ∞ (R2n ) such that |∂ ∂xα ∂ξβ a(x, ξ)| ≤ Cαβ xη−|α| ξm−|β| ,

(x, ξ) ∈ R2n .

Recall also that a(x, Dx ) : xr H s (Rn ) → xr+η H s−m (Rn ) for a ∈ S m;η is a Fredholm operator if and only if, for some R > 0, det a(x, ξ) = 0 (x, ξ) ∈ R2n ,

|x| + |ξ| ≥ R,

(A.2)

and xm ξη a−1 (x, ξ) ≤ c,

(x, ξ) ∈ R2n ,

|x| + |ξ| ≥ R.

(A.3)

Now we choose a symbol q2 ∈ S 0;0 that is elliptic in the sense of (A.2), (A.3) such that (A.4) q2 (x, ξ) ≡ q(ξ) mod S −1;0 . The choice of q2 (x, ξ) relies on the split exactness of the short sequence (σm ,ση )

0 −−−−→ S m−1;η−1 −−−−→ S m;η −−−−−e→ ΣS m;η −−−−→ 0 where σ m (a) = a + S m−1;η is the principal symbol of a ∈ S m;η , σeη (a) = a + S m;η−1 is its principal exit symbol, and both symbols are. subject to the condition σ m (a)+S m;η−1 = σeη (a)+S m−1;η . Accordingly,/ΣS m;η = (a, ae ) ∈ S m;η /S m−1;η × S m;η /S m;η−1 a ≡ ae mod S m−1;η + S m;η−1 . (A.4) says that σ 0 (q2 ) = σ 0 (q), while the choice of σe0 (q2 ) is restricted by the requirement σe0 (h) ≡ σ 0 (q) mod S −1;0 + S 0;−1 and is free otherwise (except that σe0 (q2 ) needs to be elliptic). Then q2 (x, Dx ) : L2 (Rn ) → L2 (Rn ) is a Fredholm operator. It follows from standard SG–calculus that, upon an appropriate choice of σe0 (q2 ), one can achieve each integer as index of this operator. We choose q2 (x, ξ) in such a way that q2 (x, Dx ) : L2 (Rn ) → L2 (Rn ) has index 0. Then, by adding a contribution from Op S −∞;−∞ (Rn × Rn ) = Op S(R2n ) if necessary, we finally arrive at an operator Q2 = q2 (x, Dx ) that is invertible as operator on L2 (Rn ). For more on SG-calculus we refer to Cordes [15], Parenti [57], Schrohe [62], and Schulze [64].
Proposition A.4. For each N × N matrix symbol q ∈ S 0,0;λ that satisfies | det q(t, x, ξ)| ≥ c,

(t, x, ξ) ∈ [0, T ] × R2n , |ξ| ≥ C,

for some C, c > 0, there exists an invertible operator Q ∈ Op S 0,0;λ such that Q − q(t, x, Dx ) ∈ Op S −1,−(l∗ +1);λ . Proof. A parameter version of Lemma A.3 yields the existence of an invertible 0 operator Q ∈ C ∞ ([0, T ]; Op S1,0 ) ⊂ Op S 0,0;λ such that −1 Q − q(t, x, Dx ) ∈ C ∞ ([0, T ]; Op S1,0 ) ⊂ Op S −1,−(l∗ +1);λ .




504

Michael Dreher and Ingo Witt There is the following improvement of Proposition A.4:

Proposition A.5. Given N ×N matrix symbols q0 ∈ C ∞ ([0, T ]; S (0) ) and q1 ∈ S (−1) such that |det q0 (t, x, ξ)| ≥ c for all (t, x, ξ) ∈ [0, T ] × R2n and some c > 0, there is an invertible operator Q ∈ Op S˜0,0;λ that satisfies σ 0 (Q) = q0 ,

σ ˜ −1,0 (Q) = q1 .


Proof. See Dreher–Witt [18, Proposition 3.6 (b)].

Finally, we show that, for special functions, certain cut-off operators can be estimated from below: Lemma A.6. Define v0,j ∈ Cc∞ (Rn ; R) and ϕj , ϕj for j = 1, 2, . . . as in the proof of Theorem 2.10. Then, for each s ∈ Cb∞ (Rn ; R), there are constants C and j0 such that     (k) C −1 v0,j H s(x) (Rn ) ≤ ϕj (x, Dx )v0,j  s(x) n ≤ C v0,j H s(x) (Rn ) , (1)

(2)

(R )

H

for k = 1, 2 and j ≥ j0 . Proof. The Fourier transform vˆ0,j is given by vˆ0,j (ξ) = vˆ0 (ξ − 2j ξ0 ),

|ξ0 | = 1,

and decays rapidly, |ˆ v0,j (2j ξ0 + η)| ≤ CN η−N ,

N ∈ N.

For s ∈ R, we split the norm v0,j H s (R) : 2

v0,j H s (Rn ) = I1,s + I2,s
=


ξ |ˆ v0,j (ξ)| dξ + 2s

|ξ −2j ξ0 |≤2j−1

Since vˆ0 decays rapidly, we have
2js I1,s ∼ 2

|η | (n + 1)/2, s + β∗ δ(x)l∗ ≥ (n + 1)/2.

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Acknowledgment We would like to thank Prof. Reissig und Prof. Schulze for the invitation of our contribution to this volume. Michael Dreher Fachbereich Mathematik und Statistik Universit¨ ¨ at Konstanz D-78457 Konstanz Germany e-mail: [email protected] Ingo Witt Institut f¨ fur Mathematik Universit¨ ¨ at Potsdam 14415 Potsdam Germany Current address: Department of Mathematics Imperial College 180 Queen’s Gate London SW7 2BZ United Kingdom e-mail: [email protected]