Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity [1st ed.] 9789811585777, 9789811585784

Citation preview

Alexander A. Lokshin

Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity

Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity

Alexander A. Lokshin

Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity

Alexander A. Lokshin Department of Mathematics and Informatics in Primary School Moscow Pedagogical University Moscow, Russia

ISBN 978-981-15-8577-7 ISBN 978-981-15-8578-4 https://doi.org/10.1007/978-981-15-8578-4

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is devoted to convolution type equations, which occur in linear wave problems of hereditary elasticity. The main mathematical tool used below is the Fourier–Laplace transform. The possibility of making use of the Fourier–Laplace transform, when solving convolution type equations, is evident. After applying the mentioned transform, we get a simple formula for the transformed solution of the equation considered. However the essence of matter is to derive from this formula the description of behavior of the solution itself: to find its support, asymptotics, etc. We must note that, as a rule, the Fourier–Laplace transform is used only formally in papers on wave problems of hereditary elasticity, whereas profound mathematical theorems (such as the Paley–Wiener theorem and Tauberian theorems) are neglected. The purpose of this book is to construct a rigorous mathematical approach to linear hereditary problems of wave propagation theory and to demonstrate usefulness of profound mathematical theorems in hereditary mechanics. Chapter 1 is introductory, which contains some preliminary material from linear hereditary elasticity, geometry, and harmonic analysis. Chapter 2 investigates conditions of hyperbolicity for general operators with memory in case of many spatial variables. Operators of such a kind occur in problems of wave propagation in anisotropic hereditary media. It turns out that under certain geometrical restrictions of monotonicity and concavity type imposed on the functions of memory, the desired hyperbolicity condition can be formulated algebraically. Chapter 3 deals with more refined properties of wave equations with memory. This chapter discusses the one-dimensional case, which corresponds to wave propagation in linearly elastic hereditary rods. By using both real end complex Tauberian techniques, a classification of near-front asymptotics of solutions to equations considered can be given, depending on the singularity character of the memory function. In particular, it is rigorously demonstrated that in linear hereditary media with singular memory (i.e., with a memory function tending to infinity when approaching the current moment of time), strong shocks cannot propagate at all. Among other results of Chap. 3, a mathematically rigorous derivation of the formula is mentioned for wave front v

vi

Preface

velocity in an inhomogeneous hereditary rod with singular memory (Sect. 3.9) and a generalization of the well-known Cagniard–de Hoop method to the hereditary case (Sect. 3.10). The last two results demonstrate the importance of nonlinear Laplace transform in linear hereditary elasticity. Department of Mathematics and Informatics in Primary School Moscow Pedagogical University Moscow, Russia April 1994–June 2020

Alexander A. Lokshin

Notation

1. ℝn: n-dimensional real Euclidean space 2. ℂn: n-dimensional complex Euclidean space 3. The Laplace transform of a function f( y), y 2ℝ1, Z1 Ly!p 

f ðyÞepy dy, p 2 ℝ1 :

0

Here the integral is supposed to be Lebesgue convergent for p >0 large enough. 4. The Fourier-Laplace transform of a function f( y), y 2ℝn, Z1 F y!z 

Z1 ...

1

f ðyÞeizy dy1 ⋯dyn ,

1

where z 2 ℂn ,

zy  z1 y1 þ . . . þ zn yn

Here the integral is supposed to be Lebesgue convergent when Im z is contained in some open set in Im ℂn. 5. For the Laplace transform and the Fourier–Laplace transform with respect to t 2ℝ1 we use special notation: Lt!p f  f ðpÞ,

p 2 ℝ1

vii

viii

Notation

F t!λ f  ef ðλÞ,

λ 2 ℂ1

6. The convolution with respect to t: Z1 f ð t Þ  gð t Þ 

f ðt  τÞ gðτÞdτ 1

7. In Sects. 2.2 and 2.3, λ, E, Z denote the complex conjugate of λ, E, Z

Contents

1

2

The Problem of Hyperbolicity in Linear Hereditary Elasticity . . . . . 1.1 Dynamic Problems of Hereditary Elasticity and Integro-Differential Operators . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 One-Dimensional Constitutive Equations . . . . . . . . . . . . . 1.1.2 Regular and Singular Functions of Memory . . . . . . . . . . . 1.1.3 One-Dimensional Wave Operators with Memory . . . . . . . 1.1.4 Wave Operators with Memory in Cases of Two and Three Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . 1.1.5 Anisotropic Hereditary Elastic Medium: Systems of Dynamic Equations and Their Determinants . . . . . . . . . . 1.1.6 The Problem of Hyperbolicity . . . . . . . . . . . . . . . . . . . . . 1.1.7 Equivalence of Hyperbolicity of the System of Dynamic Equations to Hyperbolicity of Its Convolutional Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8 Definition of the Class of Integro-Differential Operators Under Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Homogeneous Hyperbolic Polynomials: The Propagation Cone and the Influence Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Paley–Wiener-Type Theorems . . . . . . . . . . . . . . . . . . . . . . 1.4 A Lemma About Cos-Fourier Transform . . . . . . . . . . . . . . . . . . 1.5 Lemmas About the Fourier–Laplace Transform of the Function of Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Hyperbolic Operators with Memory . . . . . . . . . . . . . . . . . 2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multiple Characteristics, Multiple Convolutions, Unbounded Normal Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Multidimensional Case: Preliminary Lemma . . . . . . . . . . . . . . .

1 1 1 2 3 4 5 8

9 12 13 15 17 18 29

.

31

.

31

. .

50 61 ix

x

Contents

2.4 2.5

The Basic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalization of the Basic Theorem to the Case of Unbounded Normal Surface . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

The Wave Equation with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Two Classical Tauberian Theorems . . . . . . . . . . . . . . . . . . . . . . 3.3 The Continuity and Monotonicity Theorem . . . . . . . . . . . . . . . . 3.4 The Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Memory Function with a Singularity, Which Is Weaker than the Logarithmic One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Memory Function with the Logarithmic Singularity . . . . . . . . . . 3.7 Memory Function with a Singularity Stronger than the Logarithmic One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 The Power Memory Function . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Application of the Nonlinear Laplace Transform to Calculating the Wave Front Velocity in an Inhomogeneous Hereditary Rod . . 3.9.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . 3.9.2 Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Line Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix: The Near Source Behaviour of Fundamental Solutions for Wave Operators with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. The One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. The Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. The Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

66 71 76 77 77 82 84 92 94 100 107 113 115 115 118 120 126 132

133 133 134 135

About the Author

Alexander A. Lokshin is Professor of Mathematics in the Faculty of Primary Education, Moscow Pedagogical University, Russia, since 1999. He completed his graduation in differential equations in 1973 in the Faculty of Mechanics and Mathematics, Moscow State University, Russia. Professor Lokshin defended his thesis on “On lacunas and weak lacunas of hyperbolic and quasi-hyperbolic equations” in 1976 at Moscow State University, Russia. Later, he defended his doctoral dissertation on “Waves in hereditarily elastic media” at the Institute of Problems of Mechanics, USSR Academy of Sciences, Russia, in 1985. He served as a junior research fellow at the Moscow Institute of Electronic Engineering and had also worked as a scientific editor for the Moscow University Press, Russia. Coauthor of The Mathematical Theory of Wave Propagation in Media with Memory and Nonlinear Waves in Inhomogeneous and Hereditary Media, Prof. Lokshin has also published several books proposing a visual and at the same time mathematically rigorous justification of the four arithmetic algorithms.

xi

Chapter 1

The Problem of Hyperbolicity in Linear Hereditary Elasticity

This chapter discusses the problem of hyperbolicity in linear hereditary elasticity. In Sect. 1.1, preliminary information about integro-differential dynamic operators of hereditary elasticity is presented. Sections 1.2, 1.3, and 1.4 are devoted to some wellknown facts from geometry and harmonic analysis. Section 1.5 contains lemmas concerning the Fourier–Laplace transform of the function of memory.

1.1 1.1.1

Dynamic Problems of Hereditary Elasticity and Integro-Differential Operators One-Dimensional Constitutive Equations

Let a homogeneous linear hereditary rod be located on the x-axis. We suppose stress σ and deformation ε of the material of the rod to be related by the following constitutive equation 2 σ ðt, xÞ ¼ A4εðt, xÞ 

Zt Rðt  τÞεðτ, xÞdτ 1

Here t is time, A (a constant) is the instantaneous module of elasticity, and the function R(t) defined for t > 0 is the relaxation kernel. Let us extend R(t) to the semiaxis t < 0 by zero. Then the previous relation can be rewritten as σ ¼ A½ε  Rðt Þ  ε

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. A. Lokshin, Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity, https://doi.org/10.1007/978-981-15-8578-4_1

ð1:1:1Þ

1

2

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

where * denotes convolution with respect to t over the whole of t-axis. Let us single out in (1.1.1) the operator applied to ε. Then Eq. (1.1.1) takes the form σ ¼ A½1  Rðt Þε

ð1:1:10 Þ

Now, we solve (1.1.10 ) for ε by multiplying both sides of this equation by the operator 1/[1– R (t)*], which is a Taylor’s series 1+ R(t)* + R(t) * R(t) * + ⋯. Now, we have ε¼

1 1 σ A 1  Rðt Þ

Now, we denote 1  1 þ Λðt Þ 1  Rðt Þ

ð1:1:2Þ

The operator equality (1.1.2) yields two equivalent relations which link the functions R(t) and Λ(t). Namely Λðt Þ ¼ Rðt Þ þ Rðt Þ  Rðt Þ þ ⋯

ð1:1:3Þ

whence, in particular, it follows that Λ(t) ¼ 0 for t < 0, and the relation dual to (1.1.3) R ðt Þ ¼ Λ ðt Þ  Λ ðt Þ  Λ ðt Þ þ ⋯

ð1:1:4Þ

The function Λ(t), defined by (1.1.2), is called the creep kernel. Using Λ(t), we can finally rewrite the expression for deformation by means of stress in the form ε¼

1.1.2

1 ½ σ þ Λ ðt Þ  σ  A

ð1:1:5Þ

Regular and Singular Functions of Memory

One can easily see that, under reasonable restrictions, (1.1.3), or (1.1.4), yields Rðt Þ  Λðt Þ as t ! þ0 Functions of memory tending to a finite limit as t ! +0 are called regular, while functions of memory tending to infinity as t ! +0 are called singular. Thus the kernels R(t) and Λ(t) are either regular or singular simultaneously.

1.1 Dynamic Problems of Hereditary Elasticity and Integro-Differential Operators

3

One may be taking singularity of memory functions as an exception to the rule. However, it is not the case in continuum mechanics. On the contrary, singular kernels appear rather often. The presence of singularity is corroborated by specific high-frequency experiments and also by theoretical considerations (see, e.g. [2]). Mathematically, the regular case (easier for investigation) is studied in detail [3, 4]. So, this case is of no interest for us. In this book, we only consider singular functions of memory. But, it should be noted that all the results concerning regular functions of memory can be easily derived from our theory.

1.1.3

One-Dimensional Wave Operators with Memory

Here, we shall try to construct a general theory of linear operators describing wave propagation in homogeneous hereditary elastic media. One of the most important operators of this theory, the wave operator with memory, appears in the following situation. Let us turn to the rod which is under consideration in Sect. (1.1.1). For simplicity, we suppose the rod to be infinite and its density is ρ (a constant). Let the rod be subjected to the longitudinal external load f(t, x) counted at a unit length. Then the equation of the rod motion will obviously take the form 2

ρ

∂ u ∂σ þ f ðt, xÞ: ¼ ∂t 2 ∂x

ð1:1:6Þ

Here u is the displacement of the element of the rod. Please note that the displacement u and deformation ε are linked by formula ε¼

∂u ∂x

ð1:1:7Þ

Substituting the expression for σ given by (1.1.1) into (1.1.6) and taking into account (1.1.7), we arrive at the one-dimensional wave equation with memory as below:  2 2 2  ∂ u ∂ u f 2 ∂ u c  Rðt Þ  2 ¼ , ρ ∂t 2 ∂x2 ∂x

rffiffiffi A c ρ

ð1:1:8Þ

Applying the operator 1/[1  R(t)*] to both sides of (1.1.8) and taking into account (1.1.2), we arrive at the following equivalent form:

4

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity 2

½1 þ Λðt Þ

2

∂ u ∂ u f  c2 2 ¼ ½1 þ Λðt Þ ρ ∂t 2 ∂x

ð1:1:9Þ

Operators from the left-hand sides of (1.1.8) and (1.1.9) are called one-dimensional wave operators with memory.

1.1.4

Wave Operators with Memory in Cases of Two and Three Spatial Dimensions

Let us consider equations of motion of an unbounded (two or three dimensional) homogeneous linear hereditary medium of density ρ: ρ

2!   ∂ u ! ¼ fλ½1  qðt Þ þ 2μ½1  hðt Þg∇ ∇  u 2 ∂t  

ð1:1:10Þ

!

!

μ½1  hðt Þ∇  ∇  u þ f ðt, xÞ

!

Here u ¼ ðu1, . . . , unÞ is the vector field of displacement; ∇ ¼



∂ ∂x1

, ...,

∂ ∂xn

 ;

λ > 0 and μ > 0 are the instantaneous elastic constants of Lamé; q(t) and h(t) are the ! corresponding relaxation kernels; f ¼ ð f 1 , . . . , f n Þ is the body force; and n ¼ 2 or 3. ! Let us represent the force f by means of the potentials of Helmholtz: !

!

f ¼ ∇Φ þ ∇  Ψ,

!

∇Ψ¼0 !

Then, by the theorem of Lamé (see 4.5), for u there exist potentials ϕ and ψ,such that !

!

1: u ¼ ∇ϕ þ ∇  ψ ; !

2: ∇  ψ ¼ 0; 2

3:

∂ ϕ λ½1  qðt Þ þ 2μ½1  hðt Þ Φ Δϕ ¼ ;  ρ ρ ∂t 2 2!

4:

ð1:1:11Þ

!

∂ ψ μ½1  hðt Þ ! Ψ Δψ ¼  ρ ρ ∂t 2

ð1:1:12Þ

Thus, the left-hand sides of (1.1.11) and (1.1.12) contain wave operators with memory in case of two or three spatial dimensions. In Chap. 3, we shall get

1.1 Dynamic Problems of Hereditary Elasticity and Integro-Differential Operators

5

acquainted with one more situation where the two-dimensional wave operator with memory appears.

1.1.5

Anisotropic Hereditary Elastic Medium: Systems of Dynamic Equations and Their Determinants

Consider an unbounded homogeneous anisotropic hereditary elastic medium. Let the dimension of the medium be equal to 2 or 3, as is well-known, for such a medium the relation between stress tensor σ ik and deformation tensor εlm is given by the formula σ ik ¼ ½λiklm þ giklm ðt Þεlm

ð1:1:13Þ

Here λiklm are the instantaneous elastic moduli, giklm(t) the corresponding memory functions which are supposed to be equal zero for t < 0. In addition, the equalities λiklm ¼ λkilm ¼ λikml ¼ λlmik



giklm ðt Þ ¼ gkilm ðt Þ ¼ gikml ðt Þ ¼ glmik ðt Þ

ð1:1:14Þ !

hold. We would like to recall here that deformation tensor εlm and displacement u are related by the equality εlm ¼

 1 ∂ul ∂um þ 2 ∂xm ∂xl

ð1:1:15Þ

Now, let us write equations of motions of the medium: 2

ρ

2

∂ ui ∂ σ ik ¼ þ f i ðt, xÞ; i ¼ 1, . . . , n ∂t 2 ∂xk

ð1:1:16Þ

!

Here f ¼ ð f 1 , . . . , f n Þ is the body force. For simplicity, the functions fi(t, x) are supposed to have a compact support. Inserting expression (1.1.13) for σ ik into the last system of equations and taking into account (1.1.14) and (1.1.15), we obtain the equations of motions of the medium in terms of displacement: 2 2 f ðt, xÞ ∂ ui 1 ∂ um λ  ½ þ g ð t Þ  ¼ i iklm iklm 2 ρ ρ ∂t ∂xk ∂xl

ð1:1:17Þ

Now suppose all the memory functions giklm (t) can be represented in the form

6

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

giklm ðt Þ ¼ ciklm ϕðt Þ

ð1:1:18Þ

where ciklm are some constants and the function ϕ(t) is equal to zero for t < 0. Then elements of a matrix G ¼ kGijk of the system (1.1.17) can be represented as follows:  Gij ¼ Gij0

 ∂ ∂ ∂ ∂ , , þ ϕðt Þ  Gij1 ∂t ∂x ∂t ∂x

where Gij0, Gij1 are linear homogeneous second-order differential operators with constant coefficients. Since differential operators with constant coefficients can be considered operators of convolution (with δ function and its derivatives), elements of the matrix G can be also considered operators of convolution (with respect to t and x). Now, let us successively eliminate all the unknown functions, except some uk, from the system (1.1.17). Evidently, all our operations will be similar to the ones in linear algebra. The only difference lies in the fact that we apply convolution operators to the equations of the system under consideration, instead of multiplying them by numbers. In doing this, we essentially make use of commutativity of convolution operators. As a result, we arrive at the following equalities corresponding to the Cramer’s formulas from linear algebra: ðdetGÞuk ¼ detBk ;

k ¼ 1, . . . , n: ð1:1:19Þ





Here Bk ¼ Bkij is a matrix, which results from the matrix G after substituting its k-th column for the column 0

f 1 =ρ

1

C B @ ⋮ A; f n =ρ det G and det Bk denote the corresponding matrix determinants taken with respect to operation of convolution; all the fi/ρ should thus be when calculating det Bk treated as convolution operators with the kernels fi(t, x)/ρ. Finally, the expression (det G) uk denotes the operator det G which is applied to the function uk. It is easy to see that in the three-dimensional case

1.1 Dynamic Problems of Hereditary Elasticity and Integro-Differential Operators

 detG ¼ V 0

  ∂ ∂ ∂ ∂ ∂ ∂ , , , þ ϕðt Þ  V 1 þ ϕð t Þ  ϕð t Þ  V 2 ∂t ∂x ∂t ∂x ∂t ∂x  ∂ ∂ , þϕðt Þ  ϕðt Þ  ϕðt Þ  V 3 ∂t ∂x

7

ð1:1:20Þ

where Vs are homogeneous sixth order linear differential operators with constant coefficients. In the two-dimensional form, the convolutional determinant of the matrix G has the following form: 

 ∂ ∂ ∂ ∂ , , detG ¼ V 0 þ ϕð t Þ  V 1 þ ϕð t Þ  ϕð t Þ ∂t ∂x ∂t ∂x  ∂ ∂ ,  V2 ∂t ∂x

ð1:1:21Þ

where Vs are homogenous fourth-order linear differential operators with constant coefficients. Note It can be demonstrated that in the isotropic case det G can be decomposed into a product of wave operators with memory. Namely, for n ¼ 3  detG ¼

 2 λð1  qðt ÞÞ þ 2μð1  hðt ÞÞ ∂ Δ  ρ ∂t 2  2 2 μð1  hðt ÞÞ ∂  Δ  ρ ∂t 2

ð1:1:22Þ

2

∂ where Δ  Σ3i¼1 ∂x 2 , and for n ¼ 2 i

 detG ¼

 2 λ½1  qðt Þ þ 2μ½1  hðt Þ ∂ Δ  ρ ∂t 2  2  μ½1  hðt Þ ∂  Δ ,  ρ ∂t 2

2

∂ where Δ  Σ2i¼1 ∂x 2 : Here, on account of the hypothesis 1.1.17 i

qðt Þ ¼ const1 ϕðt Þ,

hðt Þ ¼ const2 ϕðt Þ

ð1:1:23Þ

8

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

1.1.6

The Problem of Hyperbolicity

The question of interest (both from the mathematical and physical points of view) is whether dynamic equations of linear hereditary elasticity have solutions belonging to reasonable classes of functions and describing finite speed wave propagation. For example, let us consider Eq. (1.1.8):  2 2 2  f ðt, xÞ ∂ u ∂ u 2 ∂ u  c  R ð t Þ  ¼ 2 2 2 ρ ∂t ∂x ∂x

ð1:1:24Þ

For simplicity, we suppose f(t, x) to have a compact support in the t, x-plane. Furthermore, let E(t, x) be some fundamental solution for the operator (1.1.9). That is, a solution of the equation " # 2 2 2 ∂ E ðt, xÞ ∂ E ðt, xÞ 2 ∂ E ðt, xÞ c  Rðt Þ  ¼ δðt ÞδðxÞ ∂t 2 ∂x2 ∂x2

ð1:1:25Þ

Then the convolution Z1 Z1 E ðt  τ, x  ξÞ

uðt, xÞ ¼ 1

1

f ðτ, ξÞ dτdξ ρ

will evidently satisfy Eq. (1.1.24). One can easily see that for the convolution (1.1.25) to describe the finite speed wave propagation from the source f(t, x), the fundamental solution E(t, x) must satisfy the condition: suppE ðt, xÞ ⊆ ft, xjt constjxjg, const > 0

ð1:1:27Þ

Here supp E(t, x) denotes the support of the fundamental solution E(t, x). But does there exist such a fundamental solution for the operator (1.1.9)? Let us suppose now x 2 ℝn, n > 1, and let jx j ¼ (x21 + ⋯ + xn2)1/2. Definition Suppose an operator P has a fundamental solution E(t, x) satisfying the condition (1.1.27). Then we shall call the operator P hyperbolic in D0 (or, if it will not lead to misunderstanding, simply hyperbolic). Here D0 is the space of distributions which are continuous linear functionals on the space D of infinitely differentiable test functions with compact support [6]. Definition Suppose an operator P has a fundamental solution E(t, x) such that eMtE(t, x) 2 S0 for some M > 0 and the condition (1.1.27) holds. Then we shall call the operator P hyperbolic in S0. Here S0 is the space of temperate distributions, which are continuous linear functionals on the space S of rapidly decreasing test functions [6]. It is clear that hyperbolicity in D’ follows from hyperbolicity in S0

1.1 Dynamic Problems of Hereditary Elasticity and Integro-Differential Operators

9

(since S0 ⊂ D0). The space S0 is better adapted for deriving necessary and sufficient conditions of hyperbolicity. Therefore, our main results will concern operators hyperbolic in S0. Some results will be also established in D0 . Note As is well-known, the differential wave operator is hyperbolic (in D’). The sixth-order differential operator V0 in (1.1.20) and the fourth-order differential operator V0 in (1.1.21) are also hyperbolic. These operators correspond to the instantaneous elastic behaviour of the hereditary medium. Example Let k > 0, 0 < α < 1. Then, as we shall see in Chap. 2, the wave operator with memory  2 2 2 ∂ ∂ 2 ∂ α c  kt þ  2 ; ∂t 2 ∂x2 ∂x where t α þ ¼



t α ,

for t > 0

0,

for t < 0

is hyperbolic in S0 , while the operator  2 2 2 ∂ 2 ∂ α ∂  c þ kt t ∂t 2 ∂x2 ∂x2 is not hyperbolic in S0 (and even in D0 ). This result, evidently, is in accordance with non-negativeness of the relaxation kernel. In fact, hyperbolicity in S0 of the wave operator with singular memory is determined by the sign of the convolutional summand. Note, however, that for operators of a more general form, the algebraic criterion of hyperbolicity in S0 is more interesting.

1.1.7

Equivalence of Hyperbolicity of the System of Dynamic Equations to Hyperbolicity of Its Convolutional Determinant

Let us return to the system (1.1.17) and (1.1.18) 2 2 f ðt, xÞ ∂ ui 1 ∂ um , i ¼ 1, . . . , n; λ  ½ þ c ϕ ð t Þ  ¼ i iklm iklm ρ ∂t 2 ρ ∂xk ∂xl

for conciseness, we rewrite it in the form

ð1:1:28Þ

10

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity ! !

Gu ¼

f ρ

ð1:1:28’Þ

Here G denotes the matrix of the system (1.1.28): !

!

u ¼ ðu1, . . . , unÞ; f ¼ ð f 1 , . . . , f n Þ; n ¼ 2 or 3

We would remind to the reader that the matrix E(t, x) ¼ kEij(t, x)k is called the fundamental matrix of solutions for the system (1.1.28’), if E(t, x) satisfies the matrix equation GEðt, xÞ ¼ δðt ÞδðxÞI

ð1:1:29Þ

where I is the unity matrix. We shall call the system (1.1.28’) hyperbolic in D0 (in S0 ), if there exists a fundamental matrix of solutions E(t, x) for (1.1.28') such that supp E ij ðt, xÞ ⊆ ft, xj t const jxjg, const > 0,

ð1:1:30Þ

and Eij(t, x) 2 D' (respectively, eMtE(t, x) 2 S' for some M > 0). Here

1=2 j x j¼ x21 þ x22 þ ⋯ þ x2n The following proposition is similar to the one given in [7] for the purely differential case. Proposition The system (1.1.28') is hyperbolic in D0 (in S0 ), if and only if the operator det G is hyperbolic in D0 (respectively, in S0 ). Here det G is the convolutional determinant of the matrix G. Proof Let the system (1.1.28') be hyperbolic (e.g. in D0 ), and let E(t, x) ¼ kEij(t, x)k be the corresponding fundamental matrix of solutions describing finite speed wave propagation. Then one can take the convolutional determinant of the matrix equality (1.1.29), since all convolutions which enter det (GE) are evidently determined. Using the commutativity of convolution, we have detGdetEðt, xÞ ¼ δðt ÞδðxÞ

ð1:1:31Þ

The convolutional determinants det G and det E are, obviously, also determined. Furthermore, from (1.1.30), it follows that suppdetEðt, xÞ ⊆ ft, xjt constjxjg, const > 0: It is also clear that det E(t, x) 2 D0 . Thus the operator det G proves to be hyperbolic in D0 .

1.1 Dynamic Problems of Hereditary Elasticity and Integro-Differential Operators

11

Conversely, let the operator det G be hyperbolic in D0 , and let E (det G) be its finite speed fundamental solution. Let Q ¼ kQijk be the matrix associated with G, that is, GQ ¼ I det G. Then the desired finite speed fundamental matrix of solutions for (1.1.28') is given by formula E ðGÞ ¼ QEðdet GÞI In fact, let us apply G to the last equality. We have GEðGÞ ¼ GQEðdetGÞ  I ¼ detG  I  E ðdetGÞ  I ¼ δðt ÞδðxÞI Therefore, E(G) really is a fundamental matrix of solutions for Eq. (1.1.28'). Moreover, one can easily see that the elements Qij of the matrix Q have a structure similar to the one of det G. Therefore, the functions E ij ðGÞ ¼ Qij E ðdetGÞ satisfy Eq. (1.1.30). The fact that Eij(G) 2 D' is obvious. The case of hyperbolicity in S0 can be studied in the same manner. Note It is easy to see that our approach can be generalized to the case where the kernels giklm(t) in (1.1.13) can be represented as convergent series: giklm ðt Þ ¼ ciklm1 ϕðt Þ þ ciklm2 ϕðt Þ  ϕðt Þ þ ⋯

ð1:1:32Þ

In this case, det G also assumes the form of a series:  detG ¼ V 0

 ∂ ∂ ∂ ∂ , , þ ϕð t Þ  V 1 þ⋯ ∂t ∂x ∂t ∂x

ð1:1:33Þ

Let us explain how the operator (1.1.33) acts. Let u(t, x), f(t, x) be some distributions (e.g., from the space D0 ). Then the equality 

 V0

  ∂ ∂ ∂ ∂ , , þ ⋯ þ ϕð t Þ  . . .  ϕð t Þ  V n uðt, xÞ ¼ f ðt, xÞ ∂t ∂x ∂t ∂x

means that for each test function ψ(t, x) lim < ½V 0 þ ⋯ þ ϕ  ⋯ϕ  V N uðt, xÞ, ϕðt, xÞ >¼< f ðt, xÞ, ϕðt, xÞ >

N!1

Note If in the isotropic case the memory functions satisfy (1.1.32), then it is evident that in (1.1.22) and (1.1.23), the kernels q(t) and h(t) also have the structure of the (1.1.32) type.

12

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

Note Suppose the operator det G can be represented as a product of operator multipliers. Furthermore, let these operator multipliers be hyperbolic in D0 (or in S0 ). Then the operator det G will also be hyperbolic in D0 (respectively, in S0 ). From the theorems given below, the inverse proposition also follows. Thus in the isotropic case of 2 or 3 spatial dimensions, hyperbolicity of wave operators with memory plays the role, which is as important as the one of hyperbolicity of the one-dimensional wave operator with memory.

1.1.8

Definition of the Class of Integro-Differential Operators Under Consideration

Here we shall study hyperbolicity of operators of the (1.1.33) type, which are the natural generalization of dynamic operators of hereditary elasticity. In what follows, we consider the number of spatial dimensions equal to n (n 1). Let us introduce the following notation for operators from the class under consideration: 

  ∂ ∂ ∂ ∂ ∂ ∂ , , , þ ϕð t Þ  V 1 þ ϕðt Þ  ϕðt Þ  V 2 ∂t ∂x ∂t ∂x ∂t ∂x

W ¼ V0

þ ⋯;

ð1:1:34Þ

and  V ¼ V0

 ∂ ∂ ∂ ∂ , , þ ϕð t Þ  V 1 ∂t ∂x ∂t ∂x

ð1:1:35Þ

In Chaps. 1 and 2, we suppose that ϕðt Þ ¼ 0, for t < 0; 1. ϕ(t) is four times differentiable for t > 0. 2. ð1Þk d dtϕkðtÞ 0, for t > 0, k ¼ 0, 1, 2 and 3. ϕ(t) ! + 1 , as t ! + 0; 3. There exists r 2 (0, 1) such that function trϕ(t) is increasing for small t > 0. k

Besides that we suppose that ∂ ∂ (a) V s ¼ V s ∂t , ∂x ; s ¼ 0, 1, . . . are homogeneous mth-order differential operators with constant real coefficients; the operator V0 is supposed to be hyperbolic. (b) Maximums of moduli of coefficient of operators Vs increase not so rapidly as some geometrical progression. We shall use the following notation:

1.2 Homogeneous Hyperbolic Polynomials: The Propagation Cone and the Influence Cone

W ðλ, σ Þ ¼

1 h X

is e ðλÞ V s ðλ, σ Þ, ϕ

13

ð1:1:36Þ

s¼0

e ðλÞV 1 ðλ, σ Þ V ðλ, σ Þ ¼ V 0 ðλ, σ Þ þ ϕ

ð1:1:37Þ

Here e ðλ Þ ¼ ϕ

Zþ1

ϕðt Þeiλt dt ¼ F t!λ ϕ

1

is the Fourier–Laplace transform of the function ϕ(t); λ ¼ μ  ip; p > 0.. Expression (1.1.36) and (1.1.37) are called symbol of the corresponding operators W and V (for the classes of operators under consideration, such a definition of a symbol coincides with the usual definition of a symbol with accuracy to multiplication by im). Equations W ðλ, σ Þ ¼ 0

ð1:1:38Þ

V ðλ, σ Þ ¼ 0

ð1:1:39Þ

will be called the characteristic equations for the corresponding operators. Later on we shall see that operator V hyperbolicity in S0 , stable with respect to small (real) perturbances of coefficients of operators V0 and V1, yields operator W hyperbolicity in S0 , stable with respect to small (real) perturbances of coefficients of all operators Vs. Thus the operator V can be considered the principal part of the operator W. From our results, it will also follow that the operator V0, generally speaking, cannot be considered the principal part of the operator W. Below (see Sect. 2.1), we shall begin to study operators with memory with operators of Eq. (1.1.39) type, for which it is possible to obtain the most complete results. Then, we shall pass to more general operators of Eq. (1.1.38) type (see Sects. 2.2, 2.3, 2.4, and 2.5).

1.2

Homogeneous Hyperbolic Polynomials: The Propagation Cone and the Influence Cone

(a) Let λ 2 ℝ1, σ 2 ℝn. We would remind to the reader that a homogeneous polynomial P(λ, σ) of degree m is called hyperbolic (or, to be more precise, hyperbolic with respect to the variable λ), if for each σ, the equation P(λ, σ) ¼ 0 has m real roots λ1(σ), . . ., λm(σ). In Sect. 1.1.8, we have supposed V0 to be a hyperbolic homogeneous operator of mth order. But it is well known [7] that a homogeneous differential operator is

14

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

hyperbolic (in D0 ) if and only if its symbol is a hyperbolic polynomial. Hence, in what follows V0(λ, σ) is a hyperbolic polynomial. Furthermore, we recall that a hyperbolic polynomial V0(λ, σ) is said to be strictly hyperbolic if for σ 6¼ 0 all the roots λ1(σ), . . ., λm(σ) of the equation V0(λ, σ) ¼ 0 are different. (b) The set fλ, σjV 0 ðλ, σ Þ ¼ 0g ⊂ ℝ1  ℝn

ð1:2:1Þ

is called the normal cone for the operator V0. The connected component of ℝn\{λ, σ| V0(λ, σ) ¼ 0} containing the semi-axis λ > 0 is called the core of the normal cone for the operator V0 and will be denoted by ∘N: It is well known that the core of the normal cone for a hyperbolic operator is convex [6, 7]. (c) V0 is said to be an operator with bounded normal surface if the intersection of the normal cone (1.2.1) with the hyperplane λ ¼ 0 consists of the origin of the coordinates λ ¼ 0 and σ ¼ 0. Otherwise, we say that V0 is an operator with unbounded normal surface (see [1.8]). (d) Let g : λ, σ ! λ0, σ 0 be a linear homogeneous mapping of ℝ1  ℝn into itself, such that the semi-axis λ0 > 0 belongs to the core of the normal cone. for V0. Then the polynomial V0(λ(λ0, σ 0), σ(λ0, σ 0)) is hyperbolic with respect to λ0. If the polynomial V0(λ, σ) is strictly hyperbolic (with respect to λ), then the polynomial V0(λ(λ0, σ 0), σ(λ0, σ 0)) is also strictly hyperbolic (with respect to λ0) [6, 7]. (e) Let t 2 ℝ1, x 2 ℝn. Consider the cone ∘K dual to ∘N : ∘K ¼ ft, xjtx þ x  σ 0, forðλ, σ Þ 2 ∘N g ⊂ ℝ1  ℝn

ð1:2:2Þ

The cone ∘K is called the propagation cone for the operator V0. Let E V 0 ðt, xÞ be the operator V0 fundamental solution describing finite speed wave propagation. Then ∘K coincides with the closure of the convex hull of supp E V 0 ðt, xÞ [6, 7]. The cone ∘K can also be defined as the intersection of the half-space t 0 with the closure of the convex hull of the characteristic cone for V0). (f) Now, let K be the closure of the convex hull of the set ft, xjt > 0, x ¼ 0g [ ∘ K: It is clear that K ⊇ ∘K

ð1:2:3Þ

1.3 The Paley–Wiener-Type Theorems

15

Evidently, K ¼ ∘K if and only if the semi-axis t > 0 belongs to the cone ∘K: We shall call K the influence cone for the operator V0. It is clear that both K and ∘K are proper cones of the half-space t > 0, that is, some cone of the {t, x|t constjxj}, const >0 type contains both K and ∘K: We shall denote the bounds of K and ∘K by ∂K and ∂∘K, respectively. Evidently, the intersection ∂K \ ∂∘K contains at least one half-line. (g) Let us define a cone N ¼ fλ, σ j tλ þ x  σ > 0 for ðt, xÞ 2 K g ⊂ ℝ1  ℝn

ð1:2:4Þ

From (1.2.3), it evidently follows that N ⊆ ∘N

ð1:2:5Þ

Moreover, N ¼ ∘N if and only if K ¼ ∘K: It is easy to see that N ¼ ∘N \ fλ, σ jλ > 0g

ð1:2:6Þ

(h) Consider an example  V0 ¼

∂ ∂ þ ∂t ∂x



∂ ∂ þ2 : ∂t ∂x

Then the cones ∘N, ∘K, N, K are shown in Fig. 1.1. (i) It can be shown that in case of n > 1, the boundedness of the normal surface for the operator V0 yields N ¼ ∘N (whence K ¼ ∘K).

1.3

The Paley–Wiener-Type Theorems

The definition of the Fourier–Laplace transform for distributions can be found, for example, in [6, 7]. The symbol Fy ! z is called the Fourier–Laplace transform with respect to variables indicated in the subscript. In case where the distribution f( y), y 2 ℝn, coincides with a usual locally integrable function, the Fourier–Laplace transform reduces to the integral Z1 F y!z f ¼

Z1 ⋯

1

1

f ðyÞeizy dy1 ⋯dyn

ð1:3:1Þ

16

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

Fig. 1.1 Cones of propagation and of influence (as well as cones dual to them) for the operator from the example (h)

where zy ¼ z1y1 + ⋯ + znyn. The integral (1.3.1) is supposed to be convergent when Im z belongs to some open set in Im ℂn. Theorem 1.3.1 [9] Let t 2 ℝ1 and let f(t) be a distribution equal to zero for t < T and satisfying the condition eMtf(t) 2 S' for some M 2 ℝ1. Then the function Ft ! λf is holomorphic in the half-plane Imλ <  M and satisfies the following estimate j F t!λ f j C ð1þjλjÞν ð1 þ jM þ Imλjν Þ eTImλ

ð1:3:2Þ

for Im λ <  M. Here C ¼ const >0, ν¼ const >0. Conversely, let for Im λ < M the function g(λ) be holomorphic and satisfying (1.3.2). Then g(λ) is the Fourier–Laplace transform of some distribution equal to zero for t < T and such that eMtf(t) 2 S'. Theorem 1.3.2 [7, 10] Let t 2 ℝ1, x 2 ℝn and suppose K is a closed convex cone in ℝ1  ℝn satisfying the condition K ⊆ {t, x| t const| x| }, const > 0. Furthermore, let f(t, x) be a distribution such that supp f(t, x) ⊆ K and eMtf(t, x) 2 S' for some M 2 ℝ1. Then the function Ft, x ! λ, σ f is holomorphic in the complex domain

1.4 A Lemma About Cos-Fourier Transform

17

fλ, σjt ðM þ ImλÞ þ x  Imσ < 0for ðt, xÞ 2 K g

ð1:3:3Þ

(λ 2 ℂ1, σ 2 ℂn) and satisfies the estimate j F t,x!λ,σ f j CðM þ Imλ, Imσ Þ  ð1þjλj þ jσjÞν ½1 þ ðjM þ Imλj þ jImσjÞν 

ð1:3:4Þ

in the domain (1.3.3). Here C(Imλ, Imσ) is a function locally bounded on the open set {Imλ, Imσ | t Im λ + x  Im σ < 0 for (t, x) 2 K} and satisfying the condition C(s Im λ, s Im σ) ¼ C(Imλ, Imσ), for s > 0; ν ¼ const > 0: Conversely, suppose a function g(λ, σ) is holomorphic in the domain (1.3.3) and satisfies the estimate (1.3.4) in the mentioned domain. Then g(λ, σ) is the Fourier– Laplace transform of some distribution f(t, x) such that supp f(t, x) ⊆ K and eMtf(t, x) 2 S'. Theorem 1.3.3 [6, 11] Let x 2 ℝn and suppose f(x) 2 D' is a distribution such that supp f(x) ⊆ Q, where Q is a bounded open convex set. Then Fx ! σ f is an entire function satisfying the estimate j F x!σ f j Cð1þjσjÞν ehðImσÞ

ð1:3:5Þ

where C ¼ const > 0, ν ¼ const > 0; hðηÞ ¼ sup ðx  ηÞ, η 2 ℝn x2Q

Conversely, suppose an entire function g(σ) satisfies the estimate (1.3.5). Then g (σ) is the Fourier–Laplace transform of some distribution f(x) 2 D' such that supp f (x) ⊆ Q.

1.4

A Lemma About Cos-Fourier Transform

Lemma 1.4.1 [12] Let a function ϕ(t) be continuous and concave for t > 0, locally integrable on [0, 1] and tending to zero for t ! + 1 . Then for μ 6¼ 0 Z1 ϕðt Þ cos μt dt 0: 0

ð1:4:1Þ

18

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

Note The improper integral in (1.4.1) should be understood as ZA ϕðt Þ cos μt dt:

lim

A!1 0

Proof of the Lemma From conditions of the lemma, it easily follows that ϕ(t) is monotone decreasing for t > 0. Therefore, the existence of the integral (1.4.1) is obvious. Furthermore, we rewrite the integral (1.4.1) in the form π=2   1 Z   X t þ 2nπ π  t þ 2nπ ϕ ϕ jμj jμj n¼0 0     π þ t þ 2nπ 2π  t þ 2nπ cos t  ϕ ϕ dt jμj jμj jμj

ð1:4:2Þ

Since φ(t) is concave, we have 

   t þ 2nπ π  t þ 2πn π þ t þ 2nπ 2π  t þ 2nπ φ φ φ φ jμj jμj jμj jμj Thus the expression in braces in (1.4.2) is non-negative, whence follows the required result.

1.5

Lemmas About the Fourier–Laplace Transform of the Function of Memory

Before proceeding further, we recall that everywhere in this chapter we supposed the function of memory ϕ(t) to possess properties 1–5 given in Sect. 1.1.8. As before, we denote the Fourier–Laplace transform e ðλÞ, λ ¼ μ  ip F t!λ ϕ by ϕ Lemma 1.5.1 Prove that e ðλÞ ! 0 as p ! þ1 ϕ uniformly with respect to μ.

ð1:5:1Þ

1.5 Lemmas About the Fourier–Laplace Transform of the Function of Memory

19

Proof Let us represent ϕ(t) as a sum ϕ1 ðt Þ þ ½ϕðt Þ  ϕ1 ðt Þ where the function ϕ1(t), equals zero for t < 0 and for t > 0 large enough, is continuous for t > 0, coincides with ϕ(t) for small t > 0, and satisfies the condition Z1 j ϕ1 ðt Þ j dt ε;

ε>0

ð1:5:2Þ

0

Note that, first of all, Eq. (1.5.2) yields the inequality e 1 ðλÞ j

jϕ

Z1

j ϕ1 ðt Þ j ept dt ε; p > 0

ð1:5:3Þ

0

Furthermore, one can easily see that under the above assumptions, the following estimate is true j ϕðt Þ  ϕ1 ðt Þ j C where C is some constant depending on the choice of ϕ1. Therefore e ðλÞ  ϕ e 1 ðλÞ j

jϕ

Z1 j ϕð t Þ  ϕ1 ð t Þ j e 0

pt

Z1 dt C

ept dt ¼

C ; p

p

0

>0

ð1:5:4Þ

Thus (1.5.3 and 1.5.4) yield e ðλÞ j j ϕ e 1 ðλÞ j þ j ϕ e ðλ Þ  ϕ e 1 ðλÞ j ε þ jϕ

C ; p>0 p

Since ε> 0 can be taken however small, the required result follows. Lemma 1.5.2 Let p > 0. Then Z1 μ 0

ϕðt Þ ept

 8  π pπ 0. We have Z1 μ

ϕð t Þ e

pt

π= Zð2μÞ

sin μt dt ¼μ

ϕðt Þ ept sin μt dt

0

0

Z1  π π þμ ϕ t þ epðtþ2μÞ cos μt dt 2μ

ð1:5:6Þ

0

From Properties 1–5 of the function ϕ(t), it follows that the function  π π φ ðt Þ ¼ ϕ t þ epðtþ2μÞ 2μ satisfies the conditions of Lemma 1.4.1. Therefore, by virtue of Lemma 1.4.1, the second summand on the right-hand side of (1.5.6) is non negative. Thus Z1 ϕð t Þ e

μ

pt

π= Zð2μÞ

sin μt dt μ

ϕðt Þ ept sin μt dt

0

0

Zπ=2  t pt=μ ¼ e ϕ sin t dt μ 0



Zπ=2 π pπ=ð2μÞ ϕ e sin μt dt, 2μ 0

which gives the result required. Lemma 1.5.3 Let p > 0, jμ j M, where M > 0 is sufficiently large. Then Z1 0 μ

ϕð t Þ e

pt



π sin μt dt π ϕ jμj 2

ð1:5:7Þ

0

Proof To be specific, suppose μ > 0. Then by virtue of the decrease of the function ϕ(t), for t > 0

1.5 Lemmas About the Fourier–Laplace Transform of the Function of Memory

Z1 0 μ

ϕð t Þ e

pt

Zπ=μ sin μt dt μ

21

ϕðt Þ ept sin μt dt

0

0

Zπ=μ

μ

ϕðt Þμt dt 0

By virtue of fifth property of ϕ(t), the function tϕ(t) is increasing for small t > 0. Therefore, provided μ is sufficiently large, we have

μ



Zπ=μ

Zπ=μ ϕðt Þμt dt ¼ μ

tϕðt Þdt μ2

2

0

   π π π π , ϕ ¼ π2 ϕ μ μ μ μ

0

which proves the lemma. Lemma 1.5.4 Let p > 0. Then   8 π pπ < ϕ exp , e ðλÞ

2jμj 2jμj Im λϕ : 0, h

i

for μ 6¼ 0

ð1:5:8Þ

forμ ¼ 0

Proof Using Lemma 1.4.1, we have Z1 Z1 h i e ðλ Þ ¼  p Im λϕ ϕðt Þept cos μt dt  μ ϕðt Þept sin μt dt 0

Z1

μ

0

ϕðt Þ ept sin μt dt

0

ð1:5:9Þ Now, the required result follows from Lemma 1.5.2. Lemma 1.5.5 Let p M, where M > 0 is large enough. Then the following estimate holds   h h i i  e ðλÞ  constIm λϕ e ðλÞ ,  Re λϕ

const > 0

ð1:5:10Þ

22

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

Proof We have 0 e ðλÞ ¼@μ λϕ 0

Z1

ϕðt Þ ept cos μt dt  p

0

Z1

1 ϕðt Þ ept sin μt dt A

0

þi@μ

Z1

ϕðt Þ ept sin μt dt  p

0

Z1

1

ð1:5:11Þ

ϕðt Þ ept cos μt dt A

0

Clearly, both summands inside of the extreme right parentheses in (1.5.11) are negative for all μ. To be specific, we restrict ourselves to the case of μ > 0. At first, we suppose μ ν > 0, p M > 0 p Then the first summand of the imaginary part of (1.5.11), after being multiplied by an appropriate constant, dominates the second summand of the real part of (1.5.11).   1   1   Z   Z     constμ ϕðt Þept sin μt dt  p ϕðt Þ ept sin μt dt ;     0

0

where const ¼ const (ν) > 0. Let us demonstrate that the first summand of the imaginary part of (1.5.11) dominates also the first summand of the real part of (1.5.11). For p and μ satisfying the above conditions Lemma 1.5.2 yields the following inequality Z1 ϕð t Þ e

pt

 const π ; ϕ sin μt dt μ 2μ

const ¼ const ðνÞ > 0

ð1:5:12Þ

0

Furthermore, without loss of generality, we can consider μ > 0 sufficiently large (since M > 0 can be taken large enough). Now, taking into account both the decrease of ϕ(t) and its fifth property,

1.5 Lemmas About the Fourier–Laplace Transform of the Function of Memory

Z1 0

ϕð t Þ e

pt

1 cos μt dt ¼ μ

23

Zπ=2  t pt=μ e ϕ cos t dt μ 0

0

Z1  1 t pt=μ þ e ϕ cos t dt μ μ π=2

1

μ

Zπ=2  t pt=μ ϕ dt e μ 0



1 μ

Z1

 2  π 3 π1 p tþ 2 5 sin t dt 2A exp 4 ϕ@ μ μ 0

tþ

0

ð1:5:13Þ

Zπ=2  1 t pt=μ

e ϕ dt μ μ 0

1 ¼ μ

Zπ=2   γ  γ t t t ϕ ept=μ dt μ μ μ 0

  γ Zπ=2  γ 1 π π t

ϕ et=ν dt μ 2μ 2μ μ 0  const ðνÞ π ϕ ; const ðνÞ > 0

μ 2μ Formulas (1.5.12 and 1.5.13) yield   1   1   Z   Z     const1 ðνÞμ ϕðt Þ ept sin μt dt  μ ϕðt Þ ept cos μt dt ;     0

0

where const1(ν) > 0. Thus for μ/p ν > 0; p M > 0, the required result is proved. Now let 0 μ/p ν, p M > 0. Then the second summand of the imaginary part of (1.5.11), after being multiplied by a constant, dominates the first summand of the real part of (1.5.11). Let us demonstrate that it dominates also the second summand of the real part of (1.5.11). Making use of concavity of the function 0

½ϕðt Þ ept t ,

for t > 0

from Property 3 of the function ϕ(t), and taking into account Lemma 1.5.1, we have

24

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

Z1 ϕð t Þ e

pt

1 cos μt dt ¼  μ

0



Z1

0

½ϕðt Þ ept t sin μt dt

0 π= Zð2μÞ

1 μ

0

½ϕðt Þept t sin μt dt

0

2  πμ

π= Zð2μÞ

ð1:5:14Þ

μt d½ϕðt Þept 

0

  1 π pπ exp  ¼ ϕ μ 2μ 2μ 2 þ π

π=2μ Z

ϕðt Þ ept dt

0

On the other hand, from the decrease of ϕ(t), for t > 0, it follows that Z1 0

ϕð t Þ e

pt

Zπ=μ sin μt dt

ϕð t Þ e 0

0

pt

π= Zð2μÞ

dt 2

ϕðt Þ ept dt

ð1:5:15Þ

0

We want to demonstrate that the quantity on the extreme right-hand side of (1.5.14) dominates the quantity on the extreme right-hand side of (1.5.15). In other words, we want to find a small constant κ > 0 such that π= π=   Zð2μÞ Zð2μÞ 1 π pπ 2 pt  ϕ ϕðt Þ e dt 2κ ϕðt Þ ept dt exp  þ μ 2μ 2μ π 0

ð1:5:16Þ

0

That is 2 ð1  πκÞ π

π= Zð2μÞ

  1 π pπ exp  ϕðt Þ ept dt ϕ μ 2μ 2μ

ð1:5:17Þ

0

for values of p and μ under consideration. But, by virtue of concavity of the function, ϕ(t) ept, for t > 0, we have

1.5 Lemmas About the Fourier–Laplace Transform of the Function of Memory π= Zð2μÞ

ϕð t Þ e

pt

25

  π π pπ exp  ϕ dt 2μ 4μ 4μ

0

for each p > 0. Therefore, Eq. (1.5.17) will be established if we demonstrate the validity of the inequality     pπ π pπ π ð1  πκÞ exp  ϕ exp  ϕ 4μ 4μ 2μ 2μ which, in its turn, can be deduced from the inequality ð1  πκ Þ exp

    π π π ϕ ϕ 4ν 4μ 2μ

However, for κ > 0 small enough, the last inequality follows from the decrease of ϕ(t) for t > 0. The lemma is proved. Lemma 1.5.6 For p ¼ const >0, jμ j ! 1    π  e  const ϕ  ϕð λ Þ 

jμj 2j μ j

ð1:5:18Þ

Proof From formulas (1.5.7 and 1.5.13) Lemma 1.5.7 For μ=p ν > 0, p M > 0 π e ðλÞ θ0 < 0;  arg ϕ 2

ð1:5:19Þ

μ=p ν < 0, p M > 0 π e ðλ Þ θ 0 > 0 argϕ 2

ð1:5:20Þ

Here M is supposed to be sufficiently large. Proof Let us prove, for example, the proposition (a) of the lemma. From the proof of Lemma 1.5.5, it follows that for μ/p ν, p M, the inequality Z1 0

ϕð t Þ e

pt

Z1 cos μt dt const

0

holds for const ¼ const (ν) > 0. That is

0

ϕðt Þ ept sin μt dt

ð1:5:21Þ

26

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

h i e ðλÞ const Im ϕ e ðλ Þ , 0 Re ϕ which gives the required result. Lemma 1.5.8

h i e ðλÞ ! 1, as p ! þ1 Im λϕ

ð1:5:22Þ

uniformly with respect to μ. Proof To be specific, we consider μ > 0. At first, let μ/p ν > 0, p M > 0. Then (1.5.9 and 1.5.12) yield  Z1 i π pt e Im λϕðλÞ μ ϕðt Þ e sin μt dt const ϕ 2μ 0  π

const ϕ ! 1, as p ! þ1; 2νp h

ð1:5:23Þ

for const ¼ const (ν) > 0.. Now, let μp ν, p > M > 0: Then (1.5.5) yields Z1 h i e Im λϕðλÞ p ϕðt Þept cos μt dt

ð1:5:24Þ

0

Furthermore, it follows from the proof of Lemma 1.5.5 that, for p and μ under consideration, the inequality Z1 ϕðt Þe

pt

π= Zð2μÞ

cos μt dt const

ϕðt Þept dt;

ð1:5:25Þ

0

0

holds for const ¼ const (ν). See (1.5.14 and 1.5.16)). Now, (1.5.24 and 1.5.25) give π= πp= Zð2μÞ Z ð2μÞ  i t t pt e Im λϕðλÞ const p ϕðt Þe dt ¼ const ϕ e dt p

h

0

0

ð1:5:26Þ

1.5 Lemmas About the Fourier–Laplace Transform of the Function of Memory π= Zð2νÞ

const

27

 t t e dt ! 1, as p ! þ1 ϕ p

0

From (1.5.23 and 1.5.26), the required result follows. Lemma 1.5.9 Let   1 ϕðt Þ const ln , t

const > 0,

ð1:5:27Þ

for small t > 0. Then    e  λϕðλÞ const ½ ln ðjλj þ 1Þ þ 1,

const > 0

ð1:5:28Þ

Proof We have Z1 Z1 h i pt e ðλÞ ¼ μ ϕðt Þe sin μt þ p ϕðt Þept cos μt dt Im λϕ 0

0

 I1 þ I2 Consider at first the summand Z1 I1 ¼ μ

ϕðt Þept sin μt dt

0

For |μ| large enough, we have by virtue of Lemma 1.5.3: 

π 0 I1 π ϕ jμj 2



const ½ ln ðjμj þ 1Þ,

const > 0

Now, let |μ| const. Then evidently 0 I1 const. Therefore 0 I 1 const ½ ln ðjμj þ 1Þ þ 1, for all μ. Now consider the summand

const > 0,

ð1:5:29Þ

28

1 The Problem of Hyperbolicity in Linear Hereditary Elasticity

Z1 I2 ¼ p

ϕðt Þ ept cos μt dt

0

For all μ, we have Z1 0 I2 p

ϕðt Þ ept dt

0

By virtue of the conditions of the lemma, one can construct a function ϕ1(t), which coincides with ϕ(t), for small t > 0, satisfies the inequality 1 ϕ1 ðt Þ α ln , t

α > 0,

ð1:5:30Þ

on the whole of semi-axis t > 0, coincides with β ln (1/t), β >0, for large t > 0, and is continuous for t > 0. Then, on the one hand, (1.5.30) yields Z1 ϕ1 ð t Þ e

p

pt

dt αp

0

Z1  ln

 1 pt e dt ¼ αðC þ ln pÞ t

ð1:5:31Þ

0

where C ¼ 0.57... is the Euler’s constant. On the other hand Z1 ½ ϕð t Þ  ϕ1 ð t Þ  e

p 0

pt

Z1 dt ¼ p

½ϕðt Þ  ϕ1 ðt Þ eεt eðpεÞt dt,

ð1:5:32Þ

0

for ε > 0. By virtue of the above assumptions, it is evident that j½ϕðt Þ  ϕ1 ðt Þ eεt j const,

t 0:

Therefore, it follows from (1.5.32) that   1 2 1 3   Z Z   p ½ϕðt Þ  ϕ1 ðt Þ ept dt  const4 p eðpεÞt dt 5     0

0

p

const for p M > ε

const pε Thus, on account of (1.5.31 and 1.5.33), we have for p M,

ð1:5:33Þ

References

29

Z1 0 I2 p

ϕðt Þept dt

0

  1   1   Z   Z     pt  pt   

p ϕ1 ðt Þe dt  þ p ½ϕðt Þ  ϕ1 ðt Þe dt      0 0   1   Z   þp ½ϕðt Þ  ϕ1 ðt Þept dt ,  

ð1:5:34Þ

0

const½ ln ðp þ 1Þ þ 1 for const >0. Now, (1.5.29 and 1.5.34) and Lemma 1.5.5 give the result required. Lemma 1.5.10 Let ϕðt Þ const ðt α Þ, 0 < α < 1,

const > 0

ð1:5:35Þ

for small t > 0. Then    e  α λϕðλÞ const jλj ,

const > 0,

ð1:5:36Þ

for p M > 0. The proof of this lemma is similar to the one of Lemma 1.5.9.

References 1. Lokshin, A.A., Suvorova, J.V.: Mathematical Theory of Wave Propagation in Media with Memory, pp. 1–151. Moskow University Press, Moscow (1982) 2. Kelbert, M. J. and Chaban, I. A. (1986). Izv. Akad. Nauk. SSSR MZhG 4, 164. 3. Rabotnovy, J.N.: Elements of Hereditary Mechanics of Solids, pp. 1–383. Nauka, Moscow (1977) 4. Christensen, R.M.: Theory of Viscoelasticity. Academic Press, New York (1971) 5. Aki, K., Richards, P.: Quantitative Seismology, vol. 1. WH. Freeman and Company, San Francisco (1980) 6. Hormander, L.: Linear Partial Differential Operators. Springer, Berlin (1963) 7. Atiah, M.F., Bott, R., Görding, L.: Acta Math. 24, 109 (1970) 8. John, F.: Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience Publishers, New York (1955) 9. Volevitch, L.R., Gindikin, S.G.: Usp. Mat. Nauk. 2, 65 (1972) 10. Vladimirov, V.S.: Distributions in Mathematical Physics, pp. 1–318. Nauka, Moscow (1979) 11. Plancherel, M., Polya, G.: Comm. Math. Helv. 9, 224 (1937) 12. Ahieser, N.I.: Lectures on Approximation Theory, pp. 1–407. Nauka, Moscow (1965)

Chapter 2

General Hyperbolic Operators with Memory

This chapter is devoted to the study of conditions of hyperbolicity for intergrodifferential operators of (1.1.34) and (1.1.35) types, that is, conditions under which the operators in question describe finite speed wave propagation. In Sects. 2.1 and 2.2, we deal with the one-dimensional case; Sects. 2.3, 2.4, and 2.5 are devoted to the case of n spatial dimensions.

2.1

One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

Let V be an operator with the symbol e ðλÞV 1 ðλ, σ Þ; λ 2 ℂ1 , σ 2 ℂ1 V ðλ, σ Þ ¼ V 0 ðλ, σ Þ þ ϕ

ð2:1:1Þ

satisfying the conditions formulated in Sect. 1.1.8 in the one-dimensional case. The symbol of a homogeneous hyperbolic operator V0 of order m, clearly, can be decomposed into a product of first-order factors V 0 ðλ, σ Þ ¼ V 0 ð1, 0Þ

m  Y

λ  c jσ



ð2:1:2Þ

j¼1

If the operator V0 is strictly hyperbolic with a bounded normal surface, then all the cj, j ¼ 1,. . ., m are different and distinct from zero and hence the coefficients

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. A. Lokshin, Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity, https://doi.org/10.1007/978-981-15-8578-4_2

31

32

2 General Hyperbolic Operators with Memory

  V 1 ðλ, σ Þ  kj  ∂  λ ∂λ V 0 ðλ, σ Þλ¼c

; j ¼ 1, . . . , m

ð2:1:3Þ

j σ,σ6¼0

are defined correctly. Theorem 2.1.1 Let V0 be a strictly hyperbolic operator with a bounded normal surface. Then the operator V is hyperbolic in S0 if and only if the inequalities k j  0; j ¼ 1, . . . , m

ð2:1:4Þ 

hold. The condition (2.1.4) if satisfied, supp E(t, x) ⊆ K and Eðt, xÞ  0 in a however small neighbourhood of an arbitrary point P 2 ∂K \ ∂∘K. (Here E(t, x) is the operator V fundamental solution describing finite speed wave propagation; K and ∘K are the influence cone and the propagation cone for the operator V0, respectively). Note About Uniqueness From the uniqueness theorem for equations in convolutional algebras [1], it follows that the operator V has no more than one fundamental solution E(t, x) 2 D' with a support contained in a proper cone of the half-space t  0. Really, let E(t, x) and E1(t, x) 2 D' be the operator V fundamental solutions vanishing outside a proper cone of the half-space t  0. Then V ðEðt, xÞ  E 1 ðt, xÞÞ ¼ 0 We shall denote the convolution with respect to t, x by  : Then t, x

E ðt, xÞ  E 1 ðt, xÞ ¼ ðE ðt, xÞ  E 1 ðt, xÞÞ  δðt ÞδðxÞ t, x

¼ ðE ðt, xÞ  E 1 ðt, xÞÞ  VE 1 ðt, xÞ: t, x

Since V is a convolution operator, the previous expression, by commutativity of convolution, equals V ðEðt, xÞ  E 1 ðt, xÞÞ  E1 ðt, xÞ ¼ 0  E1 ðt, xÞ ¼ 0: t, x

t, x

Thus E(t, x) ¼ E1(t, x).. In what follows, the reader must have in mind the above note. Before proceeding, we have to establish the following algebraic result. Lemma 2.1.1 Let σ j, j ¼ 1, . . ., m, be the roots of the characteristic equation e ðλÞV 1 ðλ, σ Þ ¼ 0 V ðλ, σ Þ  V 0 ðλ, σ Þ þ ϕ

ð2:1:5Þ

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

33

where V0(λ, σ) is the symbol of a strictly hyperbolic operator with bounded normal surface. Then for large  Im λ, the roots σ j(λ) are continuous1 and satisfy the relations: (a) Uniformly with respect to Reλ,

σ j ðλÞ ¼

λ k j ½1 þ oð1Þ e þ λϕðλÞ; cj cj

ð2:1:6Þ

for j ¼ 1, . . ., m; Im λ !  1 . (b) Uniformly with respect to Reλ,

Imσ j ðλÞ ¼

h i Imλ k j ½1 þ oð1Þ e ðλÞ ; þ Im λϕ cj cj

ð2:1:7Þ

for j ¼ 1, . . ., m; Im λ !  1 . (c) For real p, Imσ j ðipÞ ¼ 

p ½1 þ oð1Þ; cj

ð2:1:8Þ

for j ¼ 1, . . ., m; p ! + 1 . Proof (a) Let us write out expansions of the symbols V0(λ, σ) and V1(λ, σ) in powers of λ  c1σ. For a0 6¼ 0, we have h i V 0 ðλ, σ Þ ¼ ðλ  c1 σ Þ am1 ðλ  c1 Þm1 þ am2 ðλ  c1 Þm2 σ þ ⋯ þ a0 σ m1 , V 1 ðλ, σ Þ ¼ bm ðλ  c1 σ Þm þ bm1 ðλ  c1 σ Þm1 σ þ ⋯ þ b0 σ m Now, the characteristic Eq. (2.1.5) assumes the form h i ðλ  c1 σ Þ am1 ðλ  c1 σ Þm1 þ am2 ðλ  c1 σ Þm2 σ þ ⋯ þ a0 σ n1 h i e ðλÞ bm ðλ  c1 σ Þm þ bm1 ðλ  c1 σ Þm1 σ þ ⋯ þ b0 σ m ¼ 0 þϕ Let us introduce a change:

1

Moreover, σ j(λ) prove to be holomorphic [2].

ð2:1:9Þ

34

2 General Hyperbolic Operators with Memory

z¼

λ  c1 σ e ðλ Þ σϕ

ð2:1:10Þ

e ðλ Þ : After substituting (2.1.10) into (2.1.9), we obtain by cancellation by σ m ϕ  z am1

h

e ðλÞ ϕ



im1 z

m1

þ ⋯ þ a0

h im e ðλÞ zm þ ⋯ þ b0 ¼ 0: þ bm ϕ

ð2:1:11Þ

e ðλÞ ! 0 uniformly with respect to Reλ as Imλ !  1 (see Lemma 1. Since ϕ 5.1), it follows that the limiting equation for (2.1.11) (as Imλ !  1) has the form a0 z þ b0 ¼ 0,

a0 6¼ 0:

Now, from the theorem about implicit functions, it easily follows that for  Im λ large enough, (2.1.11) has a root z1(λ) which is continuous in λ and can be represented in the form z1 ðλÞ ¼ 

b0 þ oð 1Þ a0

Here the quantity o(1) tends to zero uniformly with respect to Reλ as Imλ !  1 . Furthermore, it is easy to see that b0 ¼ c1 k 1 a0 where k1 is defined by (2.1.3) with j ¼ 1. Therefore, the characteristic Eq. (2.1.5) has for large  Im λ a continuoues root: σ 1 ðλÞ ¼

λ λ k þ oð1Þ e ¼ þ 1 λϕðλÞ c1 c e 1 c1 þ z1 ðλÞϕðλÞ

It is easy to see that in case of k1 ¼ 0, the equality σ 1(λ) ¼ λ1/c holds. Therefore, it is possible to rewrite the previous expression for σ 1(λ) in the form σ 1 ðλÞ ¼

λ k 1 ½1 þ oð1Þ e þ λ ϕð λ Þ c1 c1

The rest of the roots of the characteristic equation can be calculated in a similar way. (b) The assertion of this point of the lemma follows from the assertion of the point (a) by virtue of Lemma 1.5.5.

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

35

(c) The assertion of this point follows from the one of the point (b) by virtue of Lemma 1.5.1. Proof of the Theorem: Necessity Let, for example, k1< 0 and suppose the operator V has a fundamental solution E(t, x) describing finite speed wave propagation and satisfying the condition eM 0 t Eðt, xÞ 2 S0 for some M0 > 0. Applying the Fourier–Laplace transform to the equality VE ðt, xÞ ¼ δðt Þδðt Þ we have F t,x!λ,σ E ¼ ½im V ðλ, σ Þ1 iYm  n h o1 e ðλÞV 1 ð0, 1Þ ð Þ ¼ im V 0 ð0, 1Þ þ ϕ σ  σ λ : j j¼1

ð2:1:12Þ

Our purpose is to obtain a contradiction by finding for the function (2.1.12) singularities of the sort which (2.1.12) cannot have by virtue of Theorem 1.3.2. Let us study the sign of c1 Im σ 1(λ) along the straight line Imλ ¼  M where M > 0 is large enough. By virtue of Lemma 2.1.1(c), we have for Reλ ¼ 0 c1 Imσ 1 ðiM Þ < 0

ð2:1:13Þ

Now, let Reλ ! + 1 . Then it follows from Lemma 2.1.1 (b) that h i e ðλ Þ c1 Imσ 1 ðλÞ ¼ Imλ þ ½k1 þ εM ðλÞ Im λϕ where one can consider |εM(λ)| < k1/2 if M is chosen sufficiently large. Lemma 1.5.4 yields h i e ðλÞ ! þ1, for Imλ ¼ M, Re λ ! þ1 Im λϕ Therefore c1 Imσ 1 ðλÞ > 0, for Imλ ¼ M, Re λ ! þ1

ð2:1:14Þ

(since k1 < 0). By virtue of continuity of the root σ 1(λ), it follows from (2.1.13, 2.1.14) that on the straight line Imλ ¼  M there exists a point λ0 such that Imσ 1 ðλ0 Þ ¼ 0:

ð2:1:15Þ

Now, from (2.1.15), we obtain that the function (2.1.12) has a singularity for Imλ ¼  M, σ 2 ℝ1. Since M > 0 can be chosen however large, it follows from the

36

2 General Hyperbolic Operators with Memory

above-mentioned that the function (2.1.12) cannot be the Fourier–Laplace transform of a function E(t, x) which describes finite speed wave propagation and satisfies the condition eM 0 t E ðt, xÞ 2 S0 . Thus we have arrived at the desired contradiction. Sufficiency The function [inV(λ, σ)]1 being the Fourier–Laplace transform of a distribution E(t, x) such that supp E(t, x) ⊆ K and eMtE(t, x) 2 S' for some M > 0 if demonstrated, then E(t, x) being the desired fundamental solution for the operator V will be also established. By virtue of Theorem 1.3.2, it suffices to establish the inequality 1  const jV ðλ, σ Þj

ð2:1:16Þ

on the set fλ, σj Imλ  min ½ð1 þ δÞc1 Imσ, . . . , ð1 þ δÞcm Imσ, M g

ð2:1:17Þ

where δ > 0 is however small and M > 0 is sufficiently large. If we demonstrate that, on the set (2.1.17), moduli of all factors of the product m  h iY  e ðλÞV 1 ð0, 1Þ V ðλ, σ Þ ¼ V 0 ð0, 1Þ þ ϕ σ  σ j ðλÞ j¼1

are greater than a positive constant, then we shall get the estimate (2.1.16). Since V0(0, 1) 6¼ 0, by virtue of boundedness of the normal surface for the operator V0,   e ðλÞV 1 ð0, 1Þj  const > 0, for Imλ  M V 0 ð0, 1Þ þ ϕ Furthermore, it follows from Lemma 2.1.1(b) that the inequalities c j Imσ j ðλÞ  Imλ; for Imλ  M, j ¼ 1, . . . , m

ð2:1:18Þ

hold. We have taken intohaccount i non-negativeness of the coefficients kj, j ¼ 1, . . ., e ðλÞ  0, which follows from Lemma 1.5.4. From the m and the inequality Im λϕ inequalities (2.1.18), we obtain that, on the set (2.1.17)

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

37

    σ  σ j ðλÞ   1  c j Imσ  c j Imσ j ðλÞ c j   1  ¼   c j Imσ  Im λ þ Imλ  c j Imσ j ðλÞ cj  1     c j Imσ  Imλ  const > 0; 2 j ¼ 1, . . . , m cj Thus the desired estimate (2.1.16) is obtained. A more precise description of the support of the fundamental solution E(t, x) can be given with the help of Theorem 1. 3.1. To make use of Theorem 1.3.1, we have to carry out some preliminary calculations. Let us represent the function [imV(λ, σ)]1 as a sum of elementary fractions m X H j ðλÞ 1 ¼ m i V ðλ, σ Þ σ  σ j ðλÞ j¼1

ð2:1:19Þ

where ( H j ðλÞ 

im

h

)1 i Y  e ðλÞV 1 ð0, 1Þ V 0 ð0, 1Þ þ ϕ σ j ðλÞ  σ s ðλÞ s, s6¼j



1 ∂  V λ, σ j ðλÞÞ ¼ im ∂σ

ð2:1:20Þ

Note that by virtue of (2.1.18), the inequalities Imσ j ðλÞ  const > 0, if c j < 0

ð2:1:21Þ

Imσ j ðλÞ  const < 0, if c j > 0 hold for  Im λ large enough. Hence e ðλ, σ Þ ¼ F 1 E σ!x F t,x!λ,σ E 1 ¼ F 1 σ!x m i V ðλ, σ Þ Z1 X m H j ðλÞ ixσ 1 ¼ e dσ 2π σ  σ j ðλÞ j¼1 1 X X H j ðλÞeiσ j ðλÞx ΘðxÞ  i H j ðλÞeiσ j ðλÞx ΘðxÞ ¼i j, c j 0

ð2:1:22Þ

38

2 General Hyperbolic Operators with Memory

pffiffiffiffiffiffiffi i ¼ 1, ΘðxÞ ¼



x>0 x 0, some ν > 0 and sufficiently large  Im λ, we must have   E e ðλ, xÞ  constð1 þ jλjÞν exp



x Imλ , const > 0, j c1 j  δ

whence by virtue of (2.1.22), it follows that for p large enough, the inequality  exp ½xImσ 1 ðipÞ  const ð1 þ pÞν exp 

x p , const > 0, j c1 j  δ

must hold. Therefore, we must have Imσ 1 ðipÞ 

p ½1 þ oð1Þ, jc1 j  δ

p ! þ1,

which evidently contradicts Lemma 2.1.1 (c). Finally, it is clear that the point P ¼ (t(x), x) belongs to ∂K \ ∂∘K: The theorem is proved. Notes About Supp E(t, x) Supposing the hypotheses of the above theorem, let us give some supplements to the proof of sufficiency: 1. If the cone K coincides with the cone ∘K, then K evidently is the closure of the convex hull of supp E(t, x). 2. Now suppose K ⊃ ∘K (that is, K is a proper subset of ∘K); this means that all cj, j ¼ 1, . . ., m have the same sign. To be specific, we suppose c1 < ⋯ < cm < 0. Let us demonstrate that provided V1(0, 1) 6¼ 0 and the function ψ ðt Þ ¼ F 1 λ!t

e ðλ Þ ϕ e ðλÞ V 1 ð0, 1Þ V 0 ð0, 1Þ þ ϕ

has a non-compact support, the cone K is the closure of the convex hull of supp E (t, x). Suppose the contrary. Then there must exist δ > 0, ν > 0 such that

40

2 General Hyperbolic Operators with Memory

E ðt, xÞ ¼ 0,

for x 2 ½0, δ, t 2 ð1, 0Þ [ ðν, þ1Þ,

whence it follows that lim

∂

m1

x!þ0

E ðt, xÞ ¼ 0, ∂xm1

t 2 ð1, 0Þ [ ðν, þ1Þ:

Furthermore, (2.1.22) gives F t!λ lim

∂

e ðλ, xÞ E ðt, xÞ ∂ E ¼ lim x!þ0 ∂xm1 ∂xm1  m1 m X iσ j ðλÞ ¼i  ∂  j¼1 im V λ, σ j ðλÞ Z ∂σ 1 σ m1 ¼ dσ 2πi V ðλ, σ Þ

m1

x!þ0

m1

ð2:1:23Þ

γ

where γ is a closed counter-clockwisely oriented contour (in the complex σ‐plane) containing inside of itself all the points σ ¼ σ j(λ). The expression on the right-hand side of (2.1.23) evidently equals the coefficient to σ 1 in the Laurent’s expansion of the function σ m  1/V(λ, σ) about infinity; as it is easy to check, the mentioned coefficient equals

V 1 ð0, 1Þ 1 1 e ðλÞ : ¼ ψ e ðλÞV 1 ð0, 1Þ V 0 ð0, 1Þ V 1 ð0, 1Þ V 0 ð0, 1Þ þ ϕ Therefore m1

lim

x!þ0

∂

E ðt, xÞ δðt Þ ψ ðt ÞV 1 ð0, 1Þ ¼  V 0 ð0, 1Þ V 0 ð0, 1Þ ∂xm1

whence it follows that the function ψ(t) has a compact support, which gives the contradiction required. 3. Let c1 < ⋯ < cm < 0 and suppose two following conditions to be valid: (a) The closure of the convex hull of supp ψ(t) coincides with the segment [0, ν]; (b) V1(λ, σ) ¼ const V0(λ, σ). Then supp E(t, x) ⊆ Q (see Fig. 2.1). Moreover, by theorem of Lions [3], the closure of the convex hull of supp E(t, x) coincides with Q.

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

41

Fig. 2.1 Figure representing supp E(t, x) ⊆ Q

Note About Inhomogeneous Operators Consider an operator with the symbol e ðλÞ½V 1 ðλ, σ Þ þ l1 ðλ, σ Þ V 0 ðλ, σ Þ þ l0 ðλ, σ Þ þ ϕ

ð2:1:24Þ

where Vi(λ, σ) are homogenous polynomials of degree m with real coefficient li(λ, σ) are arbitrary polynomials of degree m  1; i ¼ 0, 1. As before, we consider V0(λ, σ) a symbol of a strictly hyperbolic operator with a bounded normal surface (that is, in the decomposition (2.1.2), all cj are different and distinct from zero). Let us write expansions of the polynomials Vi(λ, σ) and li(λ, σ) in powers of (λ  c1σ). Then the characteristic equation V(λ, σ) ¼ 0 will assume the form h i ðλ  c1 σ Þ am1 ðλ  c1 σ Þm1 þ ⋯ þ a0 σ m1 þam1,1 ðλ  c1 σ Þm1 þ ⋯ þ a0,1 σ m1 þ ⋯ h e ðλÞ bm ðλ  c1 σ Þm þ ⋯ þ b0 σ m þbm,1 ðλ  c1σ Þm1 þ ⋯ þ b0,1 σ m1 þ ⋯ ¼ 0 þϕ ð2:1:25Þ The change (2.1.10) reduces the last equation to the form

42

2 General Hyperbolic Operators with Memory



 h im1 m1 e z am1 ϕðλÞ z þ ⋯ þ a0   h im1 h im 1 m1 e e ð λ Þ z m þ ⋯ þ b0 þ z þ ⋯ þ a0,1 þ ⋯ þ bm ϕ am1, ϕðλÞ e ðλÞ λϕ   h im1 1 m1 e þ z þ ⋯ þ b0,1 þ ⋯ ¼ 0 b ϕð λ Þ λ m1,1 ð2:1:26Þ e ðλÞ and Taking into account the fact that ϕ

1

ϕðλÞ λe

tend to zero uniformly with respect

to Reλ as Im λ !  1 (see Lemmas 1.5.1 and 1.5.8), we obtain that the Eq. (2.1.26) has a root z1 ¼ 

b0 þ oð1Þ a0

where, as we know, b0/a0 ¼ c1k1 and the quantity o(1) is uniform with respect to Reλ. The corresponding root of Eq. (2.1.25) has the form σ 1 ðλÞ ¼

λ k 1 þ oð 1Þ e þ λϕðλÞ c1 c1

σ j ðλÞ ¼

λ k j þ oð 1Þ e þ λ ϕð λ Þ cj cj

Similarly, we obtain

for the rest of j. All of these roots are continuous for Imλ   M (provided M > 0 is sufficiently large). Now, in the manner of Theorem 2.1.1, we obtain the following result. The operator (2.1.24) is hyperbolic in S0 and preserves this property under small real perturbances of the coefficients of operators Vi, i ¼ 0, 1 and for small complex arbitrary perturbances of the coefficients of operators li, i ¼ 0, 1 if and only if the inequalities k j > 0; j ¼ 1, 2, . . . , m hold. Further theorems of this chapter can also easily be generalized to the case of inhomogeneous operators. We leave it for the reader to do. Now, we pass to the case where the function of memory has (as t ! + 0) a singularity, which is not stronger than a logarithmic one.

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

43

Theorem 2.1.2 Let V0 be a strictly hyperbolic operator with a bounded normal surface, and let ϕðt Þ  const ln

1 , const > 0 t

ð2:1:27Þ

for small t > 0. Then the operator V is hyperbolic in D’, supp E(t, x) ⊆ K. And E ðt, xÞ does not identically wanish in a however small neighbourhood of an arbitrary point P 2 ∂K \ ∂∘K: Here E(t, x) is the operator V fundamental solution describing finite speed wave propagation; K and ∘K are the influence cone and the propagation cone for the operator V0, respectively. Proof Consider a functional E(t, x) defined on the space D of test functions (see Sect. 1.1.6) by the following integral: 1 hEðt, wÞ, ψðtxÞi ¼ ð2π Þ2

Z

Z1 dλ

 m  1 i V λ, σ F t,x!λ,σ ψ dσ

ð2:1:28Þ

  Hj λ   F t,x!λ,σ ψdσ σ σj λ

ð2:1:29Þ

1

Γ

or, which is the same 1 hEðt, xÞ, ψðtxÞi ¼ ð2π Þ2

Z dλ Γ

Z1 X m 1

j¼1

Here the bar denotes the complex conjugation, functions Hj are defined by (2.1.20), and the contour Γ is defined by the equation Imλ ¼ M ½ ln ðj Re λj þ 1Þ þ 1, M > 0

ð2:1:30Þ

Here we consider M > 0 sufficiently large. The contour orientation is given by the condition of coincidence of its projection on the Imλ‐ axis with the orientation of Imλ‐ axis. First of all, let us prove the continuity of the functional E(t, x). To do this, we have to study the behaviour of the quantities Imσ j(λ), j ¼ 1, . . ., m, when λ belongs to the contour Γ, which is defined by the equation Imλ ¼ M ½ ln ðj Re λj þ 1Þ þ 1

ð2:1:31Þ

As usual, we denote σ j(λ) by the roots of the characteristic equation V(λ, σ) ¼ 0. Consider, for example, the root σ 1(λ). As we know from Lemma 2.1.1(b)

44

2 General Hyperbolic Operators with Memory

h i e ðλ Þ c1 Imσ 1 ðλÞ ¼ Imλ þ k 1 ½1 þ oð1Þ Im λϕ where the quantity o(1) goes to zero uniformly with respect to Reλ as Imλ !  1 . Let us demonstrate that c1 Imσ ðλÞjΓ  const < 0

ð2:1:32Þ

By Lemma 1.5.9, we have for Imλ   ε < 0:   e ðλÞ  a½ ln ðj Re λj þ 1Þ þ ln ðjImλj þ 1Þ þ 1; a > 0; 0  Im λϕ

ð2:1:33Þ

From Eq. (2.1.33), it follows that to prove the inequality (2.1.32), it suffices to demonstrate, for λ 2 Γ, the inequality jImλj  j2k 1 a ½ ln ðj Re λj þ 1Þ þ ln ðjImλj þ 1Þ þ 1 j: But since by virtue of (2.1.30) ln ðj Re λj þ 1Þ ¼

Imλ  1, M

it suffices to demonstrate, for λ 2 Γ, that the inequality 

   jImλj  þ ln ðjImλj þ 1Þ  jImλj  2k1 a M

ð2:1:34Þ

holds. However, it is clear that for M > 0 large enough, (2.1.34) will hold true (for arbitrary sign of k1) in the half-plane. Imλ   M. But the contour Γ is located just in this half-plane. Thus the inequality (2.1.32) is established. Similar constructions can be carried out for the rest of the roots σ j (λ). Therefore, one can choose M > 0 so large that the inequalities c j Imσ j ðλÞjΓ  const < 0; j ¼ 1, . . . , m

ð2:1:35Þ

will hold true. But these inequalities yield the functional continuity on the space D of infinitely differentiable test functions with compact support. That is E(t, x) 2 D0. Now let us demonstrate that E(t, x) is a fundamental solution for the operator V. In fact

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory







45



 ∂ ∂ ∂ ∂ m , , VE ðt, xÞ, φðt, xÞi ¼ E ðt, xÞ, ð1Þ V 0 þϕðt Þ  ð1Þ V 1 ∂t ∂x ∂t ∂x Z Z1  m  1   1 dλ i V λ, σ  im V λ, σ F t,x!λ,σ φ dσ φðt, xÞi ¼ 2 ð2π Þ m

1 ¼ ð2π Þ2

Γ

1

Z

Z1 dλ 1

Γ

1 F t,x!λ,σ φ dσ ¼ ð2π Þ2

Z1

Z1 F t,x!λ,σ φ dσ

dλ 1

1

¼ φð0, 0Þ Here we use the well-known properties of convolution and the Cauchy theorem. But the last equality means that VE ðt, xÞ ¼ δðt Þ δðxÞ: Now, let us study the support of the fundamental solution E(t, x). To do this, we employ the Parseval’s identity in the inner integral in (2.1.29) * 1 Eðt, xÞ, φðt, xÞi ¼ 2π

Z

Z1 dλ

Γ

1

F 1 σ!x

m X j¼1

  Hj λ   F 1 F 1 φdx ð2:1:36Þ σ  σ j λ σ!x t,x!λ,σ

By inequalities (2.1.35), the following formula F 1 σ!x

m X X H j ðλÞ H j ðλÞ eiσ j ðλÞx ΘðxÞ ¼i σ  σ j ðλÞ j¼1 j, c j 0

e x ðλÞ E   e x ðλÞ can be holds true for λ 2 Γ that is, λ 2 Γ [see (2.1.22)]. Clearly, the function E analytically extended to some half-plane  Im λ  const > 0. To be specific, suppose x > 0, c1 < 0 and c1 < ⋯ < cm. Since for large  Im λ, (2.1.7 and 2.1.33) yield the estimates   c j Imσ j ðλÞ  Imλ  υ½ ln ðjλj þ 1Þ þ 1; for υ ¼ const > 0; j ¼ 1, . . ., m, we obtain that for large  Im λ,the following inequality

46

2 General Hyperbolic Operators with Memory

  E e x ðλÞ  const ð1 þ jλjÞυ exp



x Imλ ; const > 0 j c1 j

holds true, while none inequality of the type   E e x ðλÞ  const ð1 þ jλjÞυ1 exp



x Imλ ; jc1 j  δ

δ > 0, const > 0

is valid. Hence by Theorem 1.3.1, it follows that the function e Ex ðt Þ ¼ F 1 λ!t E x ðλÞ vanishes for t < t(x) ¼ x/|c1| and is not identically equal to zero in, however, small neighbourhood of the point t ¼ t(x). Finally, interchanging integrals in (2.1.36) and using the Cauchy’s theorem and Parseval’s identity, we obtain for M > 0 large enough 1 hE ðt, xÞ, φðt, xÞi ¼ 2π iMþ1 Z

iM1

Z1 dx 1

Zþ1

  e x λ F t!λ φðt, xÞdλ ¼ E

Z

  e x λ F t!λ φðt, xÞdλ ¼ 1 E 2π

Γ

heMt E x ðt Þ, eMt φðt, xÞidx ¼

1

Z1 dx 1

Zþ1 hE x ðt Þ, φðt, xÞidx 1

Whence it follows that the fundamental solution E(t, x) possesses all the required properties. The theorem is proved. The following lemma presents a modification of a result from [4]. Lemma 2.1.2 Suppose for each M  const > 0, there exists a point (λ0(M), σ 0(M )) 2 ℂ1  ℂ1 whose coordinates satisfy the following conditions: Imλ0 ðM Þ ¼ ½ ln ðj Re λ0 ðM Þj þ 1Þ þ 1,

ð2:1:38Þ

Im σ 0 ðM Þ ¼ 0,

ð2:1:39Þ

γ 1 ¼ const > 0, γ 2 ¼ const > 0,

ð2:1:40Þ

jσ 0 ðM Þj  γ 1 jλ0 ðM Þjγ2 ;

V ðλ0 ðM Þ, σ 0 ðM ÞÞ ¼ 0 Then the operator V cannot be hyperbolic in D’.

ð2:1:41Þ

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

47

Proof: Suppose the Contrary Let E(t, x) 2 D' be the operator V fundamental solution such that supp E(t, x) ⊆ Q, where Q is a proper cone of the half-plane t  0.. Following the method of [4], we consider an infinitely differentiable function ψ(t, x), which has a compact support and equals unity in a neighbourhood of the origin of coordinates. Then V ½ψ ðt, xÞ E ðt, xÞ ¼ δðt Þ δðxÞ þ gðt, xÞ

ð2:1:42Þ

where gðt, xÞ  V ½ðψ ðt, xÞ  1ÞE ðt, xÞ: It is easy to see that distributions ψ(t, x)E(t, x) and g(t, x) have compact supports. Let hðη1 , η2 Þ ¼

sup t, x2supp gðt, xÞ

  ðtη1 þ xη2 Þ, η1 2 ℝ1 , η2 2 ℝ1 :

It is clear that hðsη1 , sη2 Þ ¼ shðη1 , η2 Þ,

s > 0,

and hð1, 0Þ ¼ A1 < 0, since supp g(t, x) is a compact subset of Q\{0}. Now, let us apply the Fourier– Laplace transform to the equality (2.1.42). We have ðiÞm V ðλ, σ Þ H ðλ, σ Þ ¼ 1 þ Gðλ, σ Þ

ð2:1:43Þ

where H ðλ, σ Þ ¼ F t,x!λ,σ ðψEÞ and Gðλ, σ Þ ¼ F t,x!λ,σ g are entire functions of λ, σ, as ψ(t, x) E(t, x) and g(t, x) have compact supports). Moreover, it follows from Theorem 1.3.3 that jGðλ, σ Þj  Cð1 þ jλj þ jσ jÞυ ehðImλ,ImσÞ

ð2:1:44Þ

where C ¼ const > 0, υ ¼ const > 0. Furthermore, (2.1.41, 2.1.43, and 2.1.44) yield

48

2 General Hyperbolic Operators with Memory

hðIm λ0 ðM Þ, Im σ 0 ðM ÞÞ  A2 ½ ln ðjλ0 ðM Þj þ jσ 0 ðM Þj þ 1Þ þ 1

ð2:1:45Þ

where A2 ¼ const > 0. But from the condition (2.1.39), it follows that hðImλ0 ðM Þ, Imσ 0 ðM ÞÞ ¼ hðImλ0 ðM Þ, 0Þ ¼ hðjImλ0 ðM Þj, 0Þ ¼ A1 jImλ0 ðM Þj Hence (2.1.45) assumes the form jImλ0 ðM Þj 

A2 ½ ln ð jλ0 ðM Þj þ jσ 0 ðM Þj þ 1Þ þ 1 A1

whence on account of the condition (2.1.40), it follows that we must have the inequality jIm λ0 ðM Þj  A ½ ln ðjImλ0 ðM Þj þ j Re λ0 ðM Þj þ 1Þ þ 1

ð2:1:46Þ

where A > 0 is some constant, which does not depend on M. Finally, (2.1.46) yields the inequality  jIm λ0 ðM Þj  1  jIm λ0 ðM Þj  1 j Re λ0 ðM Þj  exp A which evidently contradicts the condition (2.1.38) for M > 0 sufficiently large. The lemma is proved. Note In proving Lemma 2.1.2, we did not employ the special form of the operator V (in fact, we employed only the symbol V(λ, σ) being holomorphic for Imλ  const). Therefore, Lemma 2.1.2 will hold true if we replace the operator V by W with the symbol (1.1.34). Theorem 2.1.3 Let V0 be a strictly hyperbolic operator with a bounded normal surface and let lim

t!þ0

ϕð t Þ ¼ þ1 ln ð1=t Þ

ð2:1:47Þ

Then all the assertions of Theorem 2.1.1 hold true for the space D0 . Proof In essence, we have to prove only the fact that the operator V cannot be hyperbolic in D0 if among kj, j ¼ 1, . . ., m, there exists even though one negative. Let, for example, k1 < 0. Consider the root σ 1(λ) of the characteristic equation V (λ, σ) ¼ 0. As we know from Lemma 2.1.1(b)

2.1 One-Dimensional Case: The Simplest Hyperbolic Operators with Memory

49

h i e ðλ Þ c1 Imσ 1 ðλÞ ¼ Imλ þ k1 ½1 þ oð1Þ Im λϕ where the quantity o(1) goes to zero uniformly with respect to Reλ, as Im λ !  1 . Let us study the sign of c1 Im σ 1 (λ) on the contour Γ given by the equation Im λ ¼ M ½ ln ðj Re λj þ 1Þ þ 1 where M > 0 is sufficiently large. At first let Reλ ¼ 0, λ 2 Γ, then λ ¼  iMand, as we know c1 Im σ 1 ðiM Þ < 0

ð2:1:48Þ

for M > 0 large enough. Now let Re λ ! + 1, then we evidently have Re λ  const < 0 Im λ on the contour Γ. Therefore, on account of Lemma 1.5.4, for Re λ ! + 1 , λ 2 Γ, the following inequality holds true: h i   π e ðλÞ  const ϕ Im λϕ , const > 0 2 Re λ Hence taking into account the condition (2.1.47) and the fact that k1 < 0, we obtain that, for Reλ large enough, on the contour Γ, there do exist points λ such that c1 Imσ 1 ðλÞ < 0

ð2:1:49Þ

Thus by virtue of continuity of the root σ 1 (λ), it follows from (2.1.48, 2.1.49) that on the contour Γ, there exists a point λ0 ¼ λ0(M) for which Im σ 1 ½λ0 ðM Þ ¼ 0:

ð2:1:50Þ

Besides that, it is evident from Lemma 2.1.1(a) that for M > 0 large enough jσ 1 ðλ0 ðM ÞÞj 

2jλ0 ðM Þj : c1

ð2:1:51Þ

Letting σ 0(M)¼σ 1(λ0(M )), we obtain from (2.1.50, 2.1.51) that the coordinates of the point (λ0(M), σ 0(M)) satisfy all the conditions of Lemma 2.1.2 for M > 0 large enough, which gives the result required.

50

2 General Hyperbolic Operators with Memory

2.2

Multiple Characteristics, Multiple Convolutions, Unbounded Normal Surface

This section is devoted to some generalizations of Theorems 2.1.1, 2.1.2, and 2.1.3. Theorem 2.2.1 Let W be an operator with the symbol W ðλ, σ Þ ¼

1 h is X e ðλÞ V s ðλ, σ Þ; ϕ

λ 2 ℂ1 , σ 2 ℂ1 :

ð2:2:1Þ

s¼0

For the operator W to be hyperbolic in D’, it is necessary that two following conditions hold: (a) If λ  cjσ, cj 6¼ 0 is a multiplier of multiplicity qj of V0(λ, σ), then λ  cjσ is also a multiplier of multiplicity  max (qj  s, 0) of Vs (λ, σ); s ¼ 1, 2, . . ., (b) If λ is a multiplier of multiplicity q of V0 (λ, σ), then λ is also a multiplier of multiplicity q of Vs (λ, σ); s ¼ 1,2,. . . Proof For simplicity, we suppose Vs (λ, σ) ¼ 0 for s  3. (The general case can be treated similarly.) (a) Let λ  cjσ, c 6¼ 0, be a multiplier of multiplicity qj of the polynomial V0(λ, σ) (for ease of notation, we shall consider j ¼ 1). Then polynomials Vs(λ, σ) can be represented (in a unique way) in the following form:  V 0 ðλ, σ Þ ¼ ðλ  c1 σ Þq1 amq1 ðλ  c1 σ Þmq1 þ þ⋯ þ a0 σ mq1 ,

a0 6¼ 0, V 1 ðλ, σ Þ ¼ bm ðλ  c1 σ Þm þ bm1 ðλ  c1 σ Þm1 σ þ ⋯ þ b0 σ m , V 2 ðλ, σ Þ ¼ dm ðλ  c1 σ Þm þ dm1 ðλ  c1 σ Þm1 σ þ ⋯ þ d 0 σ m ,

so the characteristic equation assumes the form   W ðλ, σ Þ  ðλ  c1 σ Þq1 amq1 ðλ  c1 σ Þmq1 þ ⋯ þ a0 σ mq1 e ðλÞ½bm ðλ  c1 σ Þm þ ⋯ þ b0 σ m  þϕ h i2 e ðλÞ ½d m ðλ  c1 σ Þm þ . . . þ d0 σ m  ¼ 0 þ ϕ

ð2:2:2Þ

First of all, let us introduce a change y¼ Then (2.2.2) assumes the form

λ  c1 σ , σ

ð2:2:3Þ

2.2 Multiple Characteristics, Multiple Convolutions, Unbounded Normal Surface

  e ðλÞðbm ym þ ⋯ þ b0 Þ y1 q1 amq1 ymq1 þ ⋯ þ a0 þ ϕ h i2 e ðλÞ ðd m ym þ ⋯ þ d 0 Þ ¼ 0: þ ϕ

51

ð2:2:4Þ

Let bx and dk be the lowest-order nonzero coefficients in the corresponding brackets in (2.2.4) (that is, 0 ¼ bx  1 ¼ bx  2 ¼ ⋯, 0 ¼ dk  1 ¼ dk  2 ¼ ⋯). Let us show that a necessary condition for the operator W to be hyperbolic is x  q1  1;

k  q1  2

Suppose the contrary. Namely, let even if one of two inequalities x < q1  1; k < q1  2

ð2:2:5Þ

be a valid. To be specific, we consider both inequalities (2.2.5) to be performed. Let us define the number w by the condition q1 w ¼ min f1 þ xw, 2 þ kwg: It is clear that under the conditions (2.2.5), the solution of the last equation is given by formula  w ¼ min

1 2 , q 1  x q1  k



whence it follows that w is a rational number which can be represented as an irreducible fraction N1/N2 and 0 < w < 1. Let us introduce one more change: z¼h

y e ðλÞ ϕ

iw ,

ð2:2:6Þ

then Eq. (2.2.4) will take the form h iq w h ixw o n e ðλÞ 1 zq1 ½⋯ þ a0  þ ϕ e ðλÞ ⋯ þ bx ϕ e ðλ Þ ϕ zx h i2  h ikw  e ðλ Þ e ðλÞ zk ¼ 0: þ ϕ ⋯ þ dk ϕ

ð2:2:7Þ

h iq w e ðλÞ 1 , we have, as ϕ e ðλÞ ! 0 (that is, as Imλ !  1 ), the By cancelling ϕ limiting equation for z ¼ z(λ) has one of three following forms:

52

2 General Hyperbolic Operators with Memory

(a) a0 zq1 þ bx zx ¼ 0; (b) a0 zq1 þ dk zk ¼ 0; (c) a0 zq1 þ bx zx þ dk zk ¼ 0, x > k where a0, bx, dk 6¼ 0.. Suppose, for example, the case (c) takes place. To be specific, consider the case where the limiting equation has two complex conjugated roots Z 1 and Z 2 ¼ Z 1 : Therefore, for  Im λ large enough, Eq. (2.2.7) has continuous solutions of the form z1 ðλÞ ¼ Z 1 þ oð1Þ, z2 ðλÞ ¼ Z 2 þ oð1Þ whence it follows that the original Eq. (2.2.2) has continuous solutions of the form σ1 ¼

h iw λ Z þ oð 1Þ e h iw ¼  1 λ ϕðλÞ c1 c1 e ðλ Þ c 1 þ z1 ðλÞ ϕ

ð2:2:8Þ

σ2 ¼

h iw λ Z þ oð 1Þ e h iw ¼  2 λ ϕðλÞ c1 c1 e ðλ Þ c 1 þ z2 ðλÞ ϕ

ð2:2:9Þ

λ

λ

Here the quantity o(1) goes to zero uniformly with respect to Reλ, as Imλ !  1 ). Let us begin with the case where in the representation of the number was an irreducible fraction, N1/N2 the denominator N2  3. Let us study the sign of c1 Im σ 1(λ) on the contour Γ given by the equation Imλ ¼ M ½ ln ðj Re λj þ 1Þ þ 1, M > 0:

ð2:2:10Þ

At first, suppose Reλ ¼ 0 (λ 2 Γ), then λ ¼  iM, and as we know, c1 Imσ 1 ðiM Þ < 0

ð2:2:11Þ

provided M is sufficiently large. Furthermore, from Lemma 1.5.7, it follows that for Reλ !  1  h i e ðλÞ 2 θ0 , π arg ϕ 2 Γ

ð2:2:12Þ

where 0 h< θ0 i< π/2). It is geometrically evident that one can always choose the w e ðλÞ branch such that two following conditions function ϕ

2.2 Multiple Characteristics, Multiple Convolutions, Unbounded Normal Surface

iw o   n h   Im Z λ ϕ e ðλÞ 1     n o h i w   const > 0; λ 2 Γ, Re λ ! 1,  e ðλ Þ   Re Z 1 λ ϕ n h iw o e ðλ Þ Im Z 1 λ ϕ > 0; λ 2 Γ, Re λ ! 1

53

ð2:2:13Þ ð2:2:14Þ

hold. The condition (2.2.13) yields h iw o  n e ðλ Þ Imλ ¼ o Im ½1 þ oð1ÞZ 1 λ ϕ ; λ 2 Γ, Re ! 1

ð2:2:15Þ

Really, on the one hand jImλj  const j ln ðjλj þ 1Þ þ 1j; λ 2 Γ, Re λ ! 1 by virtue of Eq. (2.2.10). On the other hand, by Eq. (2.2.13), for λ 2 Γ, Re λ !  1 ,   n h iw o h iw     e ðλÞ e ðλÞ  Im ½1 þ oð1ÞZ 1 λ ϕ   const λ1w λϕ  const jλj1w ,

const > 0,

since by Lemma 1.5.8    e  λϕðλÞ ! 1, for λ 2 Γ, Re λ ! 1: Thus the relation (2.2.15) is proved. Now, from (2.2.8, 2.2.14, and 2.2.15), it is clear that c1 Imσ 1 ðλÞjΓ > 0,

as Re λ ! 1

ð2:2:16Þ

Thus from (2.2.11, 2.2.16) by virtue of continuity of σ 1(λ), it follows that there exists a point λ0 ¼ λ0(M) 2 Γ such that Imσ 1 ½λ0 ðM Þ ¼ 0

ð2:2:17Þ

Besides that, from (2.2.8), it is evident that jσ 1 ½λ0 ðM Þj 

2j λ 0 ð M Þ j j c1 j

ð2:2:18Þ

for M large enough. Letting σ 0(M ) ¼ σ 1[λ0(M )], we obtain from (2.2.17, 2.2.18) that for sufficiently large M, the coordinates of the point (λ0(M ), σ 0(M)) satisfy all the

54

2 General Hyperbolic Operators with Memory

conditions of Lemma 2.1.2 (where one should replace the operator V by W ), whence it follows that the operator W cannot be hyperbolic in D'. Consider now the case where in the representation w ¼ N1/N2, the denominator N2 ¼ 2. Then w ¼ 1/2 (since, as we know, 0 < w < 1). Without loss of generality, we consider h i h i π π arg Z 1 2 π,  [ 0, 2 2

ð2:2:19Þ

If this condition is not performed, one should replace Z 1 by Z 2 ¼ Z 1 : From (2.2.12, 2.2.19) and the relation arg λjΓ ! π, as Re λ ! 1 h i1=2 e ðλ Þ it is geometrically evident that one of two branches of the function ϕ will satisfy the conditions (2.2.13, 2.2.14). Hence, as before, it follows that the operator W cannot be hyperbolic in D'. (b) Let λ be a multiplier of multiplicity q of the polynomial V0(λ, σ). Then the characteristic equation W(λ, σ) ¼ 0 can be written in the form analogous to (2.2.2)   λq amq λmq þ amq1 λmq1 σ þ ⋯ þ a0 σ mq   e ðλÞ bm λm þ bm1 λm1 σ þ ⋯ þ b0 σ m þϕ h i2   e ðλÞ dm λm þ dm1 λm1 σ þ ⋯ þ d 0 σ m þ ϕ ¼ 0:

ð2:2:20Þ

Let bx and dk be the lowest-order nonzero coefficients in the corresponding brackets in (2.2.20). Let us show that for the operator W to be hyperbolic, it is necessary that x  q; k  q. Suppose the Contrary Namely, let even if one of two inequalities x < q;

k0

h iw e ðλÞ and making use of Now, choosing the principal branch of the function ϕ the changes (2.2.3, 2.2.6), we easily obtain that for  Im λ large enough, the Eq. (2.2.20) has a continuous solution of the form λ iw , σ ðλÞ ¼ ½const þ oð1Þ h e ðλ Þ ϕ

const 6¼ 0

ð2:2:22Þ

where o(1) goes to zero uniformly with respect to Reλ, as Im λ !  1 . Note that by virtue of Lemma 1.5.8,      λ  jλj1þw w  constjλj1þw iw  ¼ const  jσ ðλÞj  const h  e ðλÞ  e ðλÞ  ϕ λϕ

ð2:2:23Þ

for  Im λ sufficiently large. Furthermore, from Lemma 1.5.7, it follows that h iw  h i e ðλÞ  2  πw, θ0 w , as Re λ ! þ1, arg ϕ 2 Γ or h iw  h i e ðλÞ  2 θ0 w, πw , as Re λ ! 1, arg ϕ 2 Γ

ð2:2:24Þ

where 0 < θ0 < π/2. Taking into account the relations arg λjΓ ! 0, as Re λ ! þ1, arg λjΓ ! π, as Re λ ! 1, one can easily obtain from (2.2.24) that λ iw arg h e ϕð λ Þ assumes, by continuity, each interim value from the interval (π  θw, θ0w).Hence arg σ(λ)|Γ also assumes each interim value from some interval of length π + 2θ0w > π. Therefore, there exists a point λ0 ¼ λ0 (M ) 2 Γ such that

56

2 General Hyperbolic Operators with Memory

Imσ ½λ0 ðM Þ ¼ 0

ð2:2:25Þ

Letting σ 0(M) ¼ σ[λ0(M )], we obtain from (2.2.23, 2.2.25) that for M large enough, the coordinates of the point (λ0(M ), σ 0(M)) satisfy all conditions of Lemma 2.1.2 (where one should replace the operator V by W), whence it follows that the operator W cannot be hyperbolic in D’. The theorem is proved. The following theorem presents a natural generalization of a result known for differential hyperbolic operators (see [5, 6]). Theorem 2.2.2 Let ϕ  const ln

1 t

for small t > 0. Then the conditions of the previous theorem are also sufficient for the operator W to be hyperbolic (in D’). These conditions, if satisfied, supp E(t, x) ⊆ K. Here E(t, x) is the operator W fundamental solution describing finite speed wave propagation; K is the influence cone for the operator V0. Proof The proof of this theorem is similar, in the main, to the one of Theorem 2.1.2. At first, let V0 be an operator with a bounded normal surface. It is easy to show that in the case considered, the roots of the characteristic equation W(λ, σ) ¼ 0 have the following form as Imλ !  1 σ j ðλÞ ¼

λ d j þ oð 1Þ e þ λϕðλÞ; cj cj

j ¼ 1, . . . , m:

Here the quantity o(1) is uniformly small with respect to Reλ. For simplicity we suppose c1 < c2 < ⋯ < cm < 0. Then one can write the following formula, analogous to (2.1.37):   1 ⋯ 1      σ 1 ðλÞ ⋯ σ m ðλÞ     ⋮  det ⋮    σ m2 ðλÞ ⋯ σ m2 ðλÞ  m   1  iσ1 ðλÞx iσ m ðλÞx  pffiffiffiffiffiffiffi e ⋯ e e x ðλÞ ¼   E Θ ð x Þ; i ¼ 1 h i h i 1 s Q P e ðλ Þ im V s ð0, 1Þ ϕ σ l ðλÞ  σ ðλÞ s¼0

l, τ, l 0 the function E

2

The expression Δð1Þ ð f ; σ 1 , σ 2 Þ ¼

f ðσ 2 Þ  f ðσ 1 Þ σ2  σ1

is called the first-order divided difference for the function f(σ) with respect to points σ 1 and σ 2; the expression Δð2Þ ð f ; σ 1 , σ 2 , σ 3 Þ ¼

Δð1Þ ð f ; σ 2 , σ 3 Þ  Δð1Þ ð f ; σ 1 , σ 2 Þ σ3  σ1

is called the second-order divided difference for the function f(σ)with respect to points σ 1, σ 2, σ 3, etc.

58

2 General Hyperbolic Operators with Memory

  x E e x ðλÞ  const ð1þjλjÞυ ejcl jlmλ ; const > 0, υ ¼ const > 0 Now the assertion of the theorem follows from Theorem 1.3.1. Finally, the case of an unbounded normal surface reduces to the case of a bounded normal surface by virtue of Theorem 2.2.1. The theorem is proved. Let us pass to the case where lim

t!þ0

ϕð t Þ ¼ þ1 ln ð1=t Þ

We shall demonstrate that in this case, even under necessary conditions of Theorem 2.2.1, the operator W hyperbolicity is not determined by operators V0 and V1, provided the characteristics of V0 are multiple. Let us consider, as an example, the operator whose symbol has the form h i W ðλ, σ Þ ¼ ðλ  c1 σ Þ2 am2 ðλ  c1 σ Þm2 þ am3 ðλ  c1 σ Þm3 σ þ ⋯ þ a0 σ m2 h i e ðλÞ ðλ  c1 σ Þ bm1 ðλ  c1 σ Þm1 þ bm2 ðλ  c1 σ Þm2 σ þ . . . þ b0 σ m1 þϕ  2 h i e ð λÞ þ ϕ dm ðλ  c1 σ Þm þ dm1 ðλ  c1 σ Þm1 σ þ ⋯ þ d 0 σ m ð2:2:29Þ

where c1 6¼ 0, a0 6¼ 0. Letting Z¼

λ  c1 σ e ðλ Þ σϕ

we easily obtain that the characteristic equation W(λ, σ) ¼ 0 assumes the form



 m2  m1 m2 m1 e e Z þ ⋯ þ a0 þ Z bm1 ϕðλÞ Z þ ⋯ þ b0 Z am2 ϕðλÞ  m e ðλ Þ Z m þ ⋯ þ d 0 ¼ 0 þ dm ϕ 2

For  Im λ large enough, the last equation has, in particular, two roots

Z 1,2 ðλÞ ¼

b0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b20  4a0 d 0 2a0

þ oð1Þ,

therefore, the characteristic equation W(λ, σ) ¼ 0 has two continuous roots

2.2 Multiple Characteristics, Multiple Convolutions, Unbounded Normal Surface

2 λ σ 1,2 ðλÞ ¼ þ 4 c1

b0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b20  4a0 d0 2a0 c1

59

3 e ðλÞ þ oð 1Þ 5 λ ϕ

Here the quantity o(1) tends to zero uniformly with respect to Reλ, as Im λ !  1 . Hence conditions of hyperbolicity for operators of the (2.2.29) type present concordance conditions for coefficients of three operators V0, V1 and V2. The complexity of these conditions evidently grows with the multiplicity of roots and convolutions. Now let us consider the case where the polynomial V0 has a root of multiplicity 1. Then, as is easy to see, the following result generalizing Lemma 2.1.1 holds. Lemma 2.2.1 Suppose in the decomposition V 0 ðλ, σ Þ ¼ V 0 ð1, 0Þ

m  Y

λ  c jσ



j¼1

the coefficient c j0 is distinct from zero and from all cj, j 6¼ j0. Then for  Im λ large enough, the characteristic equation W(λ, σ) ¼ 0 has a continuous root σ j0 ðλÞ such that (a) Uniformly with respect to Reλ k λ þ σ j0 ðλÞ ¼ c j0

j0

þ oð 1Þ e λϕðλÞ; k c j0

j0

  V 1 ðλ, σ Þ   ∂  λ ∂λ V 0 ðλ, σ Þλ¼c

; j0 σ,σ6¼0

where Imλ !  1 . (b) Uniformly with respect to Reλ Imσ j0 ðλÞ ¼

Imλ k þ c j0

j0

þ oð 1Þ h e i Im λϕðλÞ ; Imλ ! 1 c j0

(c) For real p, Imσ j0 ðipÞ ¼ 

p ð1 þ oð1ÞÞ; p ! þ1 c j0

From Lemma 2.2.1, it evidently follows that for the operator W to be hyperbolic in S0 , it is necessary that k

j0

 0:

60

2 General Hyperbolic Operators with Memory

Now, if V0(λ, σ) is strictly hyperbolic, then Lemma 2.2.1 and Theorem 2.2.1 yield the following description of hyperbolic operators with memory. Theorem 2.2.3 Let V0 be a strictly hyperbolic operator. 1. For the operator W to be hyperbolic in S0 , it is necessary that the following conditions hold: (a) If λ is a multiplier of V0(λ, σ), then λ is also a multiplier of no less multiplicity of all the polynomials Vs(λ, σ); s ¼ 1, 2. . ., (b) kj0 for all j such that cj 6¼ 0. 2. Let the condition (a) be valid, and suppose that 

(b’) kj > 0 for all j such that cj 6¼ 0. Then the operator W is hyperbolic in S0 , supp E ðt, xÞ ⊆ K and E ðt, xÞ  0 in a however small neighbourhood of an arbitrary point P 2 ∂K \ ∂∘K: Here E(t, x) is the operator W fundamental solution describing finite speed wave propagation, K and ∘K are the influence cone and the propagation cone for the operator V0, respectively. Corollary Suppose the relaxation kernel R(t) can be represented in the form R ð t Þ ¼ ϕð t Þ þ a1 ϕð t Þ  ϕð t Þ þ ⋯ where jaij are increasing not so rapidly as a geometrical progression. Then the wave operator with memory  2 2 2 ∂ ∂ ∂ 2  c  R ð t Þ  ∂t 2 ∂x2 ∂x2 is hyperbolic in S'. In fact, for this operator k1,2

  c2 σ 1   ¼ ∂ 2 ¼ > 0: λ ∂λ λ  c2 σ 2 λ¼ cσ,σ6¼0 2

Theorem 2.2.4 Let V0 be strictly hyperbolic, and suppose lim

t!þ0

ϕð t Þ ¼ þ1: ln ð1=t Þ

Then all the assertions of the previous theorem hold true for the space D'. Proofs of these theorems are similar to the proofs of the corresponding assertions for operators containing a single convolution (see Theorems 2.1.1 and 2.1.3). Conditions of hyperbolicity given in Theorems 2.2.3 and 2.2.4 are close to necessary and sufficient ones. It is possible to specify these results in dependence on the character of growth of ϕ(t) as t ! + 0.

2.3 Multidimensional Case: Preliminary Lemma

2.3

61

Multidimensional Case: Preliminary Lemma

This section is devoted to geometrical constructions of which we shall have heed below when using the Paley–Wiener type theorem about analyticity in a tube domain (Theorem 1.3.2). We consider λ and σ ¼ (σ 1, . . ., σ n) coordinates in ℝ1  ℝn, n > 1. Consider a cone N ¼ fλ, σjtλ þ x:σ > 0 for ðt, xÞ 2 K g where K is the influence cone for the strictly hyperbolic operator  V0 ¼ V0

∂ ∂ ∂ , ..., , : ∂t ∂x1 ∂xn

As we have noted in Sect. 1.2 N ¼ ∘N \ fλ, σjλ > 0g

ð2:3:1Þ

Here ∘N denotes the core of the normal cone for the operator V0. Let us make a rotation in the λ, σ-space g : λ, σ ! λ0 , σ 0

ð2:3:2Þ

under the condition that the semi-axis λ0 > 0 belongs to the interior of the cone N. By virtue of (2.3.1), the last condition is equivalent to two following ones: (a) The semi-axis λ0 > 0 belongs to the interior of ∘N. (b) The angle between semi-axes λ > 0 and λ0 > 0 is acute. The set of all rotations satisfying the above conditions, (a) and (b), will be denoted by G. A compact subset of G, consisting of all rotations under which the minimal (acute) angle between the semi-axis λ0 > 0 and the surface of the cone N is greater than or equal to ε > 0, will be denoted by Gε. Let Ω be a sphere in ℝn given by the equation jσ j ¼ 1. Let us draw through an arbitrary vector w 2 Ω and the λ0-axis a two-dimensional plane π(w, g). Furthermore, let ρ0 be the axis which lies in π(w, g) and is orthogonal to the λ0-axis, and let w0 be the unit directing vector of the ρ0-axis (supposing the angle between w and w0 is acute). Clearly, when w runs over the sphere Ω, the vector w0 runs over a unit sphere Ω', which is located in the hyperplane λ0 ¼ 0 and whose centre coincides with the origin of coordinates. Let us make the following rotation in π(w, g) Aðw, gÞ : λ0 , ρ0 ! λ00 , ρ00

ð2:3:3Þ

62

2 General Hyperbolic Operators with Memory

Fig. 2.2 A multidimensional case 00

00

where the λ -axis is the projection of the λ-axis on π(w, g) and the ρ -axis is the intersection of π(w, g) with the hyperplane λ¼ 0 (we define the positive direction on 00 the λ -axis as the projection of the positive direction on the λ-axis and consider w the 00 directing vector of the ρ ‐axis). Namely, λ00 ðλ0 , ρ0 Þ ¼ λðλ0 , w0 ρ0 Þ

1 , cos α

ð2:3:4Þ

ρ00 ðλ0 , ρ0 Þ ¼ w σ ðλ0 , w0 ρ0 Þ

ð2:3:5Þ

where α ¼ α(w, g) is the acute angle between the plane π(w, g) and the λ-axis (see Fig. 2.2). It is easy to see that σ ½λ0 ð0, ρ00 Þ, w0 ρ0 ð0, ρ00 Þ ¼ wρ00 ,

ð2:3:6Þ

σ ½λ0 ðλ00 , 0Þ, w0 ρ0 ðλ00 , 0Þ ¼ w⊥ λ00 sin α

ð2:3:7Þ 00

where w⊥ is the unit directing vector of the projection of the λ -axis on the hyperplane λ ¼ 0. From (2.3.4, 2.3.6, and 2.3.7), it evidently follows that λ½ðλ0 ðλ00 , ρ00 Þ, w0 ρ0 ðλ00 , ρ00 Þ ¼ λ00 cos α,

ð2:3:8Þ

σ ½λ0 ðλ00 , ρ00 Þ, w0 ρ0 ðλ00 , ρ00 Þ ¼ w⊥ λ00 sin α þ wρ00 :

ð2:3:9Þ

Let us introduce into consideration the following quantities:

2.3 Multidimensional Case: Preliminary Lemma

63

  V 1 ðλ, wρÞ  k j ð wÞ ¼ ∂  λ ∂λ V 0 ðλ, wρÞλ¼c

ð2:3:10Þ j ðwÞρ,ρ6¼0

where ρ 2 ℝ1, cj(w) are coefficient of the decomposition V 0 ðλ, wρÞ ¼ V 0 ð1, 0Þ

m  Y  λ  c j ðwÞρ ; w 2 Ω

ð2:3:11Þ

j¼1

From strict hyperbolicity of the operator V0, it follows that for w 2 Ω, the coefficients kj(w) are finite for w such that corresponding cj(w) 6¼ 0. If for some w 2 Ω there exists a cj(w) ¼ 0, then for w under consideration we shall say the corresponding coefficient kj(w) to be not determined. Let Ω0 ¼ fwj V 0 ð0, wÞ ¼ 0g: Then, on the set Ω0, the number of determined coefficients kj(w) equals m – 1 (since, on account of strict hyperbolicity of V0, exactly one of the coefficients cj(w) vanishes at each point of Ω0). On the set Ω\Ω0, the number of determined coefficients kj obviously equals m. Furthermore, let us put V sw,g ðλ00 , ρ00 Þ ¼ V s fλ½λ0 ðλ00 , ρ00 Þ, w0 ρ0 ðλ00 , ρ00 Þ, σ ½λ0 ðλ00 , ρ00 Þ, w0 ρ0 ðλ00 , ρ00 Þg ð2:3:12Þ  V 1w,g ðλ00 , ρ00 Þ  k j ðw, gÞ ¼ 00 ∂ w,g 00 ð2:3:13Þ  λ 00 V 0 ðλ , ρ00 Þ 00 00 00 ∂λ

λ ¼c j ðw,gÞρ ,ρ 6¼0

where cj(w) are coefficient of the decomposition V 0w,g ðλ00 , ρ00 Þ ¼ V 0w,g ð1, 0Þ

m  Y

λ00  c j ðw, gÞρ00



ð2:3:14Þ

j¼1

Lemma 2.3.1 Let the operator V0 be strictly hyperbolic. Then (a) The polynomial V0[λ(λ0, w0ρ0), σ(λ0, w0ρ0)] is strictly hyperbolic with respect to λ0. 00 (b) The polynomial V 0w,g ðλ00 , ρ00 Þ is strictly hyperbolic with respect to λ . 00 00 (c) The normal surface for the operator V0 if bounded then λ , ρ ‐plane the  00in 00the w,g  00 00 00 intersection of the straight line λ ¼ 0 with the cone λ , ρ jV 0 ðλ , ρ Þ ¼ 0 consist of the origin of coordinates.   (d) The semi-axis λ0 > 0 belongs to the core of the cone λ00 , ρ00 jV 0w,g ðλ00 , ρ00 Þ ¼ 0 ; 00 besides that, the angle between the semi-axes λ0 > 0 and λ > 0 is acute. (e) All the coefficients cj(w, g) are different.

64

2 General Hyperbolic Operators with Memory

(f) Let us fix an arbitrary w 2 Ω and enumerate each of two sets of numbers cj(w); j ¼ 1,. . ., m and cj(w, g); j ¼ 1,. . ., m in order of increase. Then c j0 ðwÞ ¼ 0 if and only if c j0 ðw, gÞ ¼ 0: (g) Let k j0 > 0 for w such that c j0 ðwÞ 6¼ 0 ðhere w 2 Ω Þ: Then also k j0 ðw, gÞ > 0 for the same w. Proof (a) In accordance with our construction, the semi-axis λ0 > 0 belongs to the core of the cone {λ, σ| V0(λ, σ) ¼ 0}, whence the required assertion follows. 00 (b) Since the core of the cone {λ, σ| V0(λ, σ) ¼ 0} is convex, the semi-axis λ > 0 (which is the orthogonal projection of the semi-axis λ > 0 on the plane π(w, g)) is contained inside of the core of the cone {λ0, ρ0| V0[λ(λ0, w0ρ0), σ(λ0, w0ρ0)] ¼ 0}, whence strict hyperbolicity of the polynomial V 0w,g ðλ00 , ρ00 Þ follows. In particular, V 0w,g ð1, 0Þ 6¼ 0. (c) From (2.3.12, 2.3.8, 2.3.9), it follows that     V 0w,g 0, ρ} ¼ V 0 0, wρ}

ð2:3:15Þ

Whence, by virtue of the assumption of boundedness of the normal surface for the V0, the required result follows. (d) The that the semi-axis λ0 > 0 is contained in the core of the cone  00 fact  λ , ρ00 jV 0w,g ðλ00 , ρ00 Þ ¼ 0 follows from the point (b). The angle between the 00 semi-axes λ0 > 0 and λ > 0 is acute, since it is evidently no greater than the acute angle between the semi-axes λ > 0 and λ0 > 0 (see Fig. 2.2). (e) The assertion of this point follows from the point (b). (f) Let us connect the mapping g with the identity mapping I by a continuous way of mappings g(Z ) 2 G, Z 2 [0, 1], providing g(0) ¼ I, g(1) ¼ g. For each fixed w 2 Ω, all numbers cj(w, g(Z )) are different, continuously depend on Z, and satisfy the condition cj(w, g(0)) ¼ cj(w); j ¼ 1, . . ., m. Furthermore, analogously to (2.3.15), we have w,gðZ Þ 

V0

   0, ρ} ¼ V 0 0, wρ}

whence w,gðZ Þ

V0

ð1, 0Þ

m Y j¼1

c j ðw, gðZ ÞÞ ¼ V 0 ð1, 0Þ

m Y

c j ðwÞ:

j¼1

Therefore, for each Z 2 [0, 1] and each w 2 Ω, there exists j0 such that c j0 ðw, gðZ ÞÞ ¼ 0 if and only if c j0 ðwÞ ¼ 0. Now, the required assertion is proved. Consider the expression

2.3 Multidimensional Case: Preliminary Lemma

65

Fig. 2.3 Figure showing an acute angle between the semi-axes

    w,gðZ Þ 00 00  V ðλ , ρ Þ

w,gðZ Þ

k

w, gðZ ÞÞ ¼

j0

V1 λ00

∂ ∂λ00

ðλ00 , ρ00 Þ

0

ð2:3:16Þ λ00 ¼c

00 00 j0 ðw,gðZ ÞÞρ ,ρ 6¼0

for Z 2 [0, 1] under the condition c j0 ðwÞ 6¼ 0 (or, which is the same, c j0 ðw, gðZ ÞÞ 6 ¼ 0Þ: It is clear that k j0 ðw, gð0ÞÞ ¼ k j0 ðwÞ; k j0 ðw, gð1ÞÞ ¼ k j0 ðw, gÞ: The required assertion will be established, if we demonstrate that for Z 2 ½0, 1, c j0 ðwÞ 6¼ 0, expression (2.3.16) is a non-vanishing continuous function of 00 Z. (Without loss of generality, we consider ρ > 0 in (2.3.16).) Notice, first of all, that from positiveness of k j0 ðwÞ for c j0 6¼ 0 it follows that the polynomial V1(λ, σ) is distinct from zero on the cone Q

j0

¼

[

w2Ω, c



j0 ðwÞ6¼0

 λ, σjλ ¼ c j0 ðwÞρ, σ ¼ wρ, ρ > 0 :

ð2:3:17Þ

6 0 the half-line Furthermore, it is geometrically evident that for Z 2 ½0, 1, c j0 ðwÞ ¼ λ00 ¼ c jo ðw, gðZ ÞÞρ00 ,

ρ00 > 0

belongs to the intersection of the plane π(w, g(Z )) with the cone (2.3.17) (see Fig. 2.3). Hence the fraction (2.3.16) numerator, which is equal to

66

2 General Hyperbolic Operators with Memory

n V 1 λ½λ0 ðλ00 , ρ00 Þ, w0 ρ0 ðλ00 , ρ00 Þ, σ ½λ0 ðλ00 , ρ00 Þ, w0 ρ0 ðλ00 , ρ00 Þgjλ00 ¼c 00

j0 ðw,gðZ ÞÞρ

00 ,ρ00 6¼0

is dis-

tinct from zero for Z 2 [0, 1], ρ > 0 by virtue of strict hyperbolicity of the w,gðZ Þ 00 00 ðλ , ρ Þ: Finally, the continuity of (2.3.16), regarded as a function polynomial V 0 of Z 2 [0, 1], is obvious. The lemma is proved.

2.4

The Basic Theorem

In this section, we consider λ and σ ¼ (σ 1, . . ., σ n) coordinates in the space ℂ1  ℂn, n > 1. As to constructions of the previous section, we suppose them to be made in the real part of ℂ1  ℂn. Below we shall prove the main result of Chap. 2. Namely, we shall obtain conditions of hyperbolicity in S' for the operator W with the symbol W ðλ, σ Þ ¼

1 h X

is e ðλÞ V s ðλ, σ Þ ϕ

s¼0

providing V0 is a strictly hyperbolic operator with a bounded normal surface. From our assumptions it follows, in particular, that the numbers cj(w), j ¼ 1, . . ., m, defined in (2.3.11) are all different and distinct from zero and that the coefficients   V 1 ðλ, wρÞ  k j ðwÞ ¼ ∂  λ ∂λ V 0 ðλ, wρÞλ¼c

;

j ¼ 1, . . . , m

j ðwÞρ,ρ6¼0

are determined for each w 2 Ω, where Ω is the unit sphere |Reσ| ¼ 1 in the space Reσ. Theorem 2.4.1 Let V0 be a strictly hyperbolic operator with a bounded normal surface. Then the operator W is hyperbolic in S0 and preserves this property under small real arbitrary perturbations of the coefficients of operators Vs; s ¼ 0, 1, . . . if and only if k j ðwÞ > 0; j ¼ 1, . . . , m

ð2:4:1Þ



for each w 2 Ω. The condition (2.4.1) if satisfied, suppEðt, xÞ ⊆ ∘K and Eðt, xÞ  0 in a however small neighbourhood of an arbitrary point P 2 ∂∘K: Here E(t, x) is the operator W fundamental solution describing finite speed wave propagation, ∘K is the propagation cone for the operator V0. Note We have already mentioned that in the multidimensional case (n > 1), boundedness of the normal surface for V0 yields the coincidence of the propagation cone

2.4 The Basic Theorem

67

for V0 with the influence cone for V0. This circumstance enables us to describe supp E(t, x) more precisely than in case where K 6¼ ∘K: Proof of the Theorem For simplicity, we shall, as usual, suppose Vs(λ, σ)  0 for s  3; transition to the general case is obvious. Necessity Suppose the operator W has a fundamental solution E(t, x) describing finite speed wave propagation and such that eMt E(t, x) 2 S' for some M. Let us prove the validity of (2.4.1), for example, for w ¼ (1,0,. . .,0). Consider the function Z uðt, x1 Þ ¼

Eðt, x1 , x2  ξ2 , . . . , xn  ξn Þ dξ2 . . . dξn ℝn1

On the one hand, it is easy to see that supp uðt, x1 Þ ⊆ ft, x1 j t  const jx1 jg,

const > 0,

and that eMt u(t, x1) 2 S0. On the other hand, notice that the function u(t, x1) satisfies the equation 

 ∂ ∂ ∂ ∂ , 0, . . . , 0 uðt, x1 Þ þ ϕðt Þ  V 1 , 0, . . . , 0 uðt, x1 Þ , , V0 ∂t ∂x1 ∂t ∂x1  ð2:4:2Þ ∂ ∂ , , 0, . . . , 0 uðt, x1 Þ ¼ δðt Þδðx1 Þ þϕðt Þ  ϕðt Þ  V 2 ∂t ∂x1 Hence the operator on the left-hand side of (2.4.2) must satisfy the necessary conditions of Theorem 2.2.3. Therefore, taking into account the requirement of stability, we obtain the inequalities (2.4.1) for w ¼ (1, 0,..., 0). The case of arbitrary w obviously reduces to the case of w(1,0,. . .,0) by means of a rotation. Sufficiency We are going to show that the function 1 im W ðλ, σ Þ is a Fourier–Laplace transform of a distribution E(t, x) such that eMtE(t, x) 2 S0 (where M > 0 is large enough) and supp E ðt, xÞ ⊆ ∘K, whence it will evidently follow that E(t, x) is the desired fundamental solution for the operator W. Now, note the following circumstance. Real homogeneous linear mappings (2.3.2) and (2.3.3) can be naturally extended to the complex space. Under such an extension, the imaginary parts of variables will be transformed by the same law as the real ones. For the extended, in such a way, mappings (2.3.2, 2.3.3), we shall use the former notation. If we demonstrate that under conditions: (a) g E Gε (where ε > 0 is sufficiently small)

68

2 General Hyperbolic Operators with Memory

(b) Imλ0   B (where B ¼ B(ε) > 0 is sufficiently large);σ 0 2 ℝn the estimate    1 W λðλ0 , σ 0 Þ, σ ðλ0 , σ 0 ÞÞj  const

ð2:4:3Þ

holds true, then by Theorem 1.4.2, we shall obtain the existence of E(t, x) with the required properties. Let w 2 Ω, g 2 Gε. Consider the expression W w,g ðλ00 , ρ00 Þ ¼ V 0w,g ðλ00 , ρ00 Þþ h i2 e ðλ cos αÞ V w,g ðλ00 , ρ00 Þ þ ϕ 2 00

where α ¼ α(w, g) is the acute angle between the axis Reλ and the planeπ(w, g). We recall that by virtue of Lemma 2.3.1 (e),(f), all the numbers cj(w, g) are different and distinct from zero and all the coefficients kj(w, g) are determined. Hence from 00 Lemma 2.2.1, for continuity and compactness reasons, it follows that for  Im λ 00 00 large enough, the roots ρ ¼ ρj(λ , w, g) of the equation W w,g ðλ00 , ρ00 Þ ¼ 0 satisfy the following relations c j ðw, gÞ Imρ j ðλ00 , w, gÞ ¼ Imλ00 þ h i   e ðλ00 cos αÞ ; for j ¼ 1,. . ., m, where o(1) goes to zero k j ðw, gÞ þ oð1Þ Im λ00 ϕ 00

00

uniformly with respect to Reλ , w, g, as Imλ !  1 . Furthermore, from (2.4.1) and Lemma 2.3.1(g), we easily obtain, for continuity and compactness reasons, that k j ðw, gÞ  k ðεÞ > 0; j ¼ 1, . . . , m Therefore, by virtue of Lemma 1.5.4 c j ðw, gÞ Imρ j ðλ00 , w, gÞ  Imλ00 ,

for Imλ00  B1

where B1 ¼ B1(ε) is large enough. Therefore  w,g  00 } 1 W λ ,ρ    h i2 e ðλ00 cos αÞV w,g ð0, 1Þ þ ϕ e ðλ00 cos αÞ V w,g ð0, 1Þ j ¼j V 0w,g ð0, 1Þ þ ϕ 1 2   Y m   1   ρ00  ρ j λ} , w, g    const   j¼1 on the set

ð2:4:4Þ

2.4 The Basic Theorem

69

  T ðw,g,δÞ¼fλ} ,ρ} jImλ} min ð1þδÞc1 ðw,gÞImρ00 ,......, ð1þδÞcm ðw,gÞImρ} ,B1 g, ð2:4:5Þ where δ > 0 is whatever small. Furthermore, by virtue of Lemma 2.3.1(d), for fixed w 2 Ω, g 2 Gε, one can choose B > 0 large enough and δ > 0 small enough for the inclusion 

 λ0 , ρ0 j Imλ0  B, ρ0 2 ℝ1 ⊆ T ðw, g, δÞ 00

ð2:4:6Þ

00

to be valid. Here λ0, ρ0 and λ , ρ are related by the rotation A(w, g). Moreover, by virtue of continuous dependence of A(w, g) on parameters w and g, one can always consider B ¼ B(ε) > 0 and δ ¼ δ(ε) > 0 to be such that (2.4.6) holds true for all w 2 Ω, g 2 Gε. Now, from (2.4.4, 2.4.5, and 2.4.6), it follows that    w,g 00 0 0 00 0 0 1 λ ðλ , ρ Þ, ρ ðλ , ρ ÞÞj  const W

ð2:4:7Þ

for Imλ'   B, ρ' 2 ℝ1, w 2 Ω, g 2 Gε..However W w,g ðλ00 ðλ0 , ρ0 Þ, ρ00 ðλ0 , ρ0 ÞÞ  W ðλðλ0 , w0 ρ0 Þ, σ ðλ0 , w0 ρ0 ÞÞ, hence (2.4.7) is equivalent to the desired inequality (2.4.3). Therefore, the operator W has a fundamental solution E(t, x) such that supp Eðt, xÞ ⊆ ∘K and eMt E(t, x) 2 S0 for some M > 0 (one can assume M to be equal to B(ε0), where ε0 > 0 is a sufficiently small fixed number). Now, let P be an arbitrary point of ∂∘K: Let us demonstrate that E(t, x) does not identically vanish in a however small neighbourhood of the point P. Suppose the Contrary Let E(t, x)  0 on the open set U containing the point P. Let us draw, through P, a hyperplane tangent to the cone ∘K and choose coordinate axes x2, . . ., xn in the intersection of hyperplanes π p and t ¼ 0; the x1- axis we direct along the normal to the mentioned above intersection. (The origin of coordinates is, as before, located at the vertex of the cone ∘K). Then the projection of supp E(t, x) on the t, x1-plane will not contain some neighbourhood U1 of the point P1 (P1 is the projection of the point P on the t, x1‐plane). Let Z uðt, x1 Þ ¼

E ðt, x1 , x2  ξ2 , . . . , xn  ξn Þ dξ2 . . . dξn : ℝ

n1

It is obvious that the function u(t, x1) identically vanishes on the set U1 defined above.

70

2 General Hyperbolic Operators with Memory

Fig. 2.4 Figure showing the position of point P1

As was noted in the proof of necessity, the function u(t, x1) satisfies the equation 

 ∂ ∂ ∂ ∂ , , , 0, . . . , 0 uðt, x1 Þ þ ϕðt Þ  V 1 , 0, . . . , 0 uðt, x1 Þ þ ϕðt Þ V0 ∂t ∂x1 ∂t ∂x1  ∂ ∂  ϕð t Þ  V 2 , 0, ⋯, 0 uðt, x1 Þ , ∂t ∂x1 ¼ δðt Þδðx1 Þ Besides that, it is geometrically evident that the influence cone for the operator V0(∂/ ∂t, ∂/∂x1, 0, . . ., 0) coincides with the propagation cone ∘K 1 for this operator. Therefore by virtue of Theorem 2.2.3, supp uðt, x1 Þ ⊆ ∘K 1 and uðt, x1 Þ does not identically vanish in a however small neighbourhood of an arbitrary point on ∂∘K 1 : However, it is easy to see that the cone ∘K 1 is just the projection of the cone ∘K on the t, x1‐ plane. It is also clear that the point P1 belongs to ∂∘K 1 (see Fig. 2.4). Hence the function u(t, x1) cannot identically vanish in the neighbourhood of the point P1, which gives the desired contradiction.

2.5 Generalization of the Basic Theorem to the Case of Unbounded Normal Surface

71

Corollary Suppose the relaxation kernel R(t) can be represented in the form Rðt Þ ¼ ϕðt Þ þ a1 ϕðt Þ  ϕðt Þ þ ⋯ where jaij is increasing not so rapidly as a geometrical progression. Then the wave operator with memory " # n n 2 2 2 X X ∂ ∂ ∂ 2 c  R ðt Þ  ∂t 2 ∂x2k ∂x2k k¼1 k¼1 is hyperbolic in S0. In fact, for this operator   c 2 ρ2 1   ¼ > 0: k1,2 ðwÞ ¼ ∂  2 λ ∂λ λ  c2 ρ2 λ¼ cρ,ρ6¼0 2

2.5

Generalization of the Basic Theorem to the Case of Unbounded Normal Surface

In the multidimensional case, unboundedness of the normal surface for the operator V0 leads us to difficulties, analogous to the ones which occur for hyperbolic differential operators with characteristics of variable multiplicity [7]. In what follows we shall use the former notation. Theorem 2.5.1 Let V0 be a strictly hyperbolic operator, Then the operator W is hyperbolic in S0 only if for each w 2 Ω (a) {w| V0(0, w) ¼ 0} ⊆ {w| Vs(0, w) ¼ 0}; s ¼ 1, 2, . . ., (b) kj(w)  0 for all j such that cj(w) 6¼ 0. To prove this theorem, it suffices to suppose the contrary and make use of the Hadamard’s method of descent, which leads us to the desired contradiction with the results obtained above in the one-dimensional case (see Theorem 2.2.3). Theorem 2.5.2 Let V0 be a strictly hyperbolic operator and let for each w 2 Ω (a) {w|V0(0, w) ¼ 0} ⊆ {w|Vs(0, w) ¼ 0}; s ¼ 1, 2, . . ., i2  n h P ∂  V ð 0, w Þ 6 0, ¼ (b) 0  ∂wi i¼1

V 0 ð0,wÞ¼0

72

(c)

2 General Hyperbolic Operators with Memory n P i¼1

" ∂ ∂wi

 2  V 1 c j ðwÞ, wÞ   j¼1

m Q



6¼ 0, V 0 ð0,wÞ¼0



(d) kj(w) > 0 for all j such that cj(w) 6¼ 0. Then the operator W is hyperbolic in S0 , supp Eðt, xÞ ⊆ K and E ðt, xÞ  0 in a however small neighbourhood of arbitrary point P 2 ∂K \ ∂∘K: Here E(t, x) is the operator W fundamental solution describing finite speed wave propagation; K and ∘K are the influence cone and the propagation cone for the operator V0, respectively. The proof of this theorem can be carried out in the manner of the proof of sufficiency in Theorem 2.4.1. Here we shall only show the changes which are to be introduced into the proof of the inequality (2.4.3) as case of unbounded normal surface for the operator V0; for simplicity we suppose g to be the identity mapping (the general case can be studied in the same manner). Namely, we shall demonstrate that for Imλ   M, where M > 0 is large enough, σ 2 ℝn, the inequality 1  const jW ðλ, σ Þj

ð2:5:1Þ

holds true. In the neighbourhood of points w 2 Ω such that V0(0, w) 6¼ 0, we have in fact established (2.5.1) in the proof of Theorem 2.4.1. Now, let us establish the validity of (2.5.1) in some neighbourhood of the set Ω0 ¼ fwjV 0 ð0, wÞ ¼ 0g Let us put σ ¼ wρ and decompose the symbol of the operator V0 into a product of first-order multipliers V 0 ðλ, wρÞ ¼ V 0 ð1, 0Þ

m  Y

λ  c j ðwÞρ



j¼1

(We suppose the coefficients cj(w), j ¼ 1, .., m to be enumerated in order of increase.) Let us fix an arbitrary w0 2 Ω0 and suppose, to be specific, c1(w0) ¼ 0. Furthermore, let us expand Vs(λ, wρ) in powers of λ  c1(w)ρ, then the characteristic equation will take the following form

2.5 Generalization of the Basic Theorem to the Case of Unbounded Normal Surface

73

W ðλ, wρÞ  ½λ  c1 ðwÞρfam1 ðwÞ½λ  c1 ðwÞρm1 þ am2 ðwÞ½λ  c1 ðwÞρm2 ρ þ ⋯ þ a0 ðwÞρm1 g e ðλÞfbm ðwÞ½λ  c1 ðwÞρm þ bm1 ðwÞ½λ  c1 ðwÞρm1 ρ þϕ

ð2:5:2Þ

þ ⋯ þ b0 ðwÞρm g h i2 e ðλÞ fdm ðwÞ½λ  c1 ðwÞρm þ ϕ þ dm1 ðwÞ½λ  c1 ðwÞρm1 ρ þ ⋯ þ d 0 ðwÞρm g ¼ 0 For simplicity, we have supposed, as usual, Vs(λ, σ)  0 for s  3. Letting Z¼

λ  c1 ðwÞρ e ðλ Þ ρϕ

ð2:5:3Þ

e ðλÞ, and inserting (2.5.3) into (2.5.2), we obtain by cancellation by ρm ϕ

m1  m m1 e e ðλÞ Z m þ . . . þ b0 ðwÞ Z þ ⋯ þ a0 ðwÞ þ bm ϕ Z am1 ðwÞ ϕðλÞ h  m i e ðλÞ d m ðwÞ ϕ e ðλÞ Z m þ . . . þ d0 ðwÞ ¼ 0 þϕ 

ð2:5:4Þ By virtue of strict hyperbolicity of the operator V0, it is evident that a0(w) 6¼ 0 in some neighbourhood of the point w0. Besides that, it follows from the condition (a) of the theorem that b0(w0) ¼ 0,d0(w0) ¼ 0. Therefore, by the theorem about implicit functions, it follows that for however small ε > 0, there do exist M ¼ M (ε) > 0 and δ ¼ δ(ε) > 0, such that for Imλ   M, j w  w0 j  δ, Eq. (2.5.4) has a continuous solution Z ¼ Z1(λ, w) satisfying the estimate jZ 1 ðλ, wÞj  ε:

ð2:5:5Þ

Now, assembling in (2.5.4) terms of first and second orders in Z and taking into account (2.5.5), we obtain that for Imλ   M, j w  w0 j  δ,   h i2 e ðλÞb1 ðwÞ þ ϕ e ðλÞ d1 ðwÞ þ b0 ðwÞ þ ϕ e ðλ Þ d 0 ðw Þ j jZ 1 ðλ, wÞ a0 ðwÞ þ ϕ  const j Z 21 ðλ, wÞ j where const >0 does not depend on λ, w. Hence, as ε ! 0,

74

2 General Hyperbolic Operators with Memory

e ðλÞ d 0 ðwÞ b0 ðwÞ þ ϕ h i2 e ðλÞ b1 ðwÞ þ ϕ e ðλ Þ d 1 ðw Þ a0 ð w Þ þ ϕ 92 1 08 > > = < e ðλÞ d 0 ðwÞ b0 ðwÞ þ ϕ C B þ O@ A h i2 > ; : a0 ð w Þ þ ϕ e ð λ Þ b1 ð w Þ þ ϕ e ðλÞ d1 ðwÞ>

Z 1 ðλ, wÞ ¼ 

ð2:5:6Þ

uniformly with respect to λ, w for Imλ  M,

j w  w0 j δ:

One can directly calculate that for w such that c1(w) 6¼ 0 the equality b0 ðwÞ ¼ k 1 ðwÞ a0 ðwÞ c1 ðwÞ

ð2:5:7Þ

holds. Furthermore, from the equality m Y

V 0 ð0, wÞ ¼ ð1Þm V 0 ð1, 0Þ

c j ðwÞ

j¼1

and condition (b) of the theorem, it follows that

2  n X ∂c1 ðwÞ    ∂wi i¼1

6¼ 0:

ð2:5:8Þ

c1 ðwÞ¼0

Besides that, the equality V1(c1(w), w) ¼ b0(w) and condition (c) of the theorem yield

2  n X ∂b0 ðwÞ    ∂wi i¼1

6¼ 0,

ð2:5:9Þ

c1 ðwÞ¼0

since, by virtue of condition (a) of the theorem, b0 ðwÞjc1 ðwÞ¼0 ¼ 0:

ð2:5:10Þ

From (2.5.8, 2.5.9, and 2.5.10), it easily follows that the function k1(w) can be extended by continuity to the set of those w for which c1(w) ¼ 0. Moreover, it is evident that the inequality k1(w) > 0, which, by condition (d), is valid for w such that c1(w) 6¼ 0, will hold true under such an extension. From continuity of the (extended)

2.5 Generalization of the Basic Theorem to the Case of Unbounded Normal Surface

75

function k1(w) and compactness of the sphere Ω, it follows that a stronger inequality holds k1 ðwÞ  const > 0; w 2 Ω:

ð2:5:11Þ

Since the function k1(w) is now determined for all w 2 Ω, it follows from (2.5.6, 2.5.7) that for Imλ   M, j w  w0 j  δ the following ratio is also determined:   Z 1 ðλ, wÞ ¼ k1 ðwÞ þ oð1Þ þ O jw  w0 j : c1 ðwÞ

ð2:5:12Þ

Here the quantity o(1) tends to zero uniformly with respect to Reλ, w as Imλ !  1 ; the quantity O(| w  w0| ) is uniform with respect to λ. The root Z1(λ, w) of (2.5.4) evidently corresponds to the root ρ1(λ, w) of (2.5.2) such that c1 ðwÞρ1 ðλ, wÞ ¼

λc1 ðwÞ e ðλÞ c1 ðwÞ þ Z 1 ðλ, wÞ ϕ

λ e ðλ Þ 1  ½k 1 ðwÞ þ oð1Þ þ Oðjw  w0 jÞϕ    e ðλÞ: ¼ λ þ k1 ðwÞ þ oð1Þ þ O jw  w0 j λϕ

¼

By virtue of Lemma 1.5.5, the last relation yields i    h e ðλ Þ : c1 ðwÞ Imρ1 ðλ, wÞ ¼ Imλ þ k1 ðwÞ þ oð1Þ þ O jw  w0 j Im λϕ

ð2:5:13Þ

In what follows, we consider M > 0 sufficiently large and δ > 0 sufficiently small. From (2.5.11, 2.5.13) and Lemma 1.5.4, it follows that c1 ðwÞ Imρ1 ðλ, wÞ  Imλ for Imλ  M, j w  w0 j δ

ð2:5:14Þ

Furthermore, by virtue of strict hyperbolicity of the operator V0, cj(w) 6¼ 0 for jw  w0 j  δ; j ¼ 2, . . ., m, whence on account of Lemma 2.2.1, it follows that for large  Im λ and jw  w0 j  δ, the characteristic equation W(λ, σ) ¼ 0 has the roots ρ ¼ ρj(λ, w); j ¼ 2, . . ., m satisfying the relations h i   e ðλ Þ c j ðwÞImρ j ðλ, wÞ ¼ Imλ þ k j ðwÞ þ oð1Þ Im λϕ Here the quantity o(1) tends to zero uniformly with respect to Reλ, w as Imλ !  1 . Hence, in accordance with condition (d) and Lemma 1.5.4, we obtain the following inequalities analogous to (2.5.14)

76

2 General Hyperbolic Operators with Memory

  c j ðwÞ Imρ j ðλ, wÞ  Imλ for Im λ  M, w  w0   δ;

ð2:5:15Þ

for j ¼ 2,. . ., m Thus from (2.5.14, 2.5.15) and condition (a) of the theorem, we obtain that the function 1 W ðλ,wρÞ

m Q

c j ðwÞ

j¼1

m ¼  2  Q e ðλÞV 1 ð0,wÞþ ϕ e ðλÞ V 2 ð0,wÞ c j ðwÞρc j ðwÞρ j ðλ,wÞ V 0 ð0,wÞþϕ j¼1

ð1Þm V 0 ð0,wÞ

¼ h i2  Q  e ðλÞV 1 ð0,wÞþ ϕ e ðλÞ V 2 ð0,wÞ m c j ðwÞρc j ðwÞρ j ðλ,wÞ V 0 ð1,0Þ V 0 ð0,wÞþϕ j¼1 is bounded for Imλ   M, ρ 2 ℝ1, j w  w0 j  δ. Finally, covering Ω0 with neighbourhoods of the jw  w0 j  δ type and choosing a finite sub-covering from this covering, we arrive at the desired inequality (2.5.1). Note Chap. 2 is based on the results of [8, 9].

References 1. Vladimirov, V.S.: Distributions in Mathematical Physics, pp. 1–318. Nauka, Moscow (1979) 2. Marcushevitc, A.l.: Theory of Analytic Functions, vol. 1, pp. 1–486. Nauka, Moscow (1967) 3. Lions, J.-L. J. Anal. Math. 2, 369, 1952–1953. 4. Atiah, M.F., Bott, R., Garding, L.: Acta Math. 24, 109 (1970) 5. Lax, A.: Comm. Pure Appl. Math. 9, 135 (1956) 6. Münster, M. Rocky Mountain J. of Math. 3, 443 (1978). 7. Courant, R., Lax, A.: Comm. Pure Appl. Math. 8, 497 (1955) 8. Lokshin, A.A., Suvorova, J.V.: Mathematical Theory of Wave Propagation in Media with Memory, pp. 1–151. Moscow University Press, Moscow (1982) 9. Lokshin, A.A.: Trudy Sem. Petrovskogo, vol. 7, p. 148 (1982)

Chapter 3

The Wave Equation with Memory

In this chapter, we study wave front asymptotics of solutions of wave equations with memory. Sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, and 3.9 are devoted to the one-dimensional case. In Sects. 3.10 and 3.11, we deal with the case of two and three spatial variables, respectively.

3.1

Formulation of the Problem

Let us consider a linear hereditary elastic homogeneous rod located on the semi-axis x
0. Then, as it easily follows from (1.1.9), the deformation in such a rod satisfies the following wave equation with memory 2

½1 þ Λðt Þ

2

∂ ε ∂ ε  c2 2 ¼ 0 ∂t 2 ∂x

ð3:1:1Þ

Here c ¼ (A/ρ)1/2, ρ ¼ const > 0 is the rod density, and A ¼ const >0 is the instantaneous module of elasticity. Let us set the following problem for Eq. (3.1.1): εðt, xÞ ¼ 0, for x > 0, t  0; ( 1 for t > 0 εðt, 0Þ ¼ 0 for t < 0

ð3:1:2Þ

If we impose geometrical restrictions 1–5 from Sect. 1.1.8 on the kernel Λ(t), then (3.1.1) will evidently describe the finite speed wave propagation. However, in this chapter, we shall use another approach to the problem (3.1.1, 3.1.2). Namely, © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. A. Lokshin, Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity, https://doi.org/10.1007/978-981-15-8578-4_3

77

78

3 The Wave Equation with Memory

following [1], along the whole of Chap. 3, we suppose that the creep kernel Λ(t) can be represented as 1 Λðt Þ ¼ φðt Þ þ φðt Þ  φðt Þ 4

ð3:1:3Þ

where the function φ(t) is supposed satisfying the following conditions: φðt Þ ¼ 0 for t < 0; (a) φ(t) is non-negative, non-increasing and concave for t > 0. (b) φ(t) is n-smooth for t > 0 and satisfies the estimate   n d φðt Þ nδ   ,  dt n   const t

0 < δ < 1,

for small t > 0, where it is supposed that n(1  δ) > 1/2; (c) φ(t) ! + 1 as t ! + 0, φ(t) ! 0 as t ! + 1 ; (d) φ(t) is locally integrable on [0, 1]. By applying the Fourier–Laplace transform to (3.1.1, 3.1.2, and 3.1.3), one can easily obtain eεðλ, xÞ ¼

e

iλ

pffiffiffiffiffiffiffiffiffiffiffi 1þe Λ ð λÞ

ðx=cÞ

iλ

¼

eiλf1þ½eφðλÞ=2gðx=cÞ iλ

ð3:1:4Þ

where λ ¼ μ  ip, p > 0. Note that from conditions (a) and (b), it follows that Im ½λe φðλÞ  0, for p > 0

ð3:1:5Þ

(See the proof of Lemma 1.5.4.) Therefore, for p
const > 0 jeεðλ, xÞj 

const px=c e jλj

ð3:1:6Þ

whence it follows that εðt, xÞ ¼ 0, for t
0 and some small δ > 0,

ð3:1:7Þ

3.1 Formulation of the Problem

εðt, xÞ ¼ 0,

79

for t
0 is large enough: j eεðλ, xÞ j C ð1þjλjÞυ ep½x=ðcδÞ ;

ð3:1:8Þ

where C ¼ const > 0, υ ¼ const > 0. Note, however, that putting in (3.1.4) λ ¼  ip, we have     e ðipÞ x φ 1 eεðip, xÞ ¼ exp p 1 þ : 2 p c

ð3:1:9Þ

e ðipÞ ! 0 as p ! 1, hence (3.1.9) contradicts By virtue of Lemma 1.5.1 φ (3.1.8). Finally, note that by virtue of the estimate (3.1.6), the function eεðλ, xÞ is square integrable over dμ along each straight line p ¼ γ, γ > 0, whence by Plancherel’s theorem, it follows that eγtε(t, x) is square integrable over dt on [0, 1). Hence, in its turn, it follows that the function eδtε(t, x) is Lebesgue integrable over dt on [0, 1) for each δ > γ > 0 (and hence for each δ > 0). Therefore, for p > 0, one can consider the Fourier–Laplace transform eεðλ, xÞ in the classical sense. Thus we have arrived at the following result. Theorem 3.1.1 The solution ε(t, x) of (3.1.1, 3.1.2, and 3.1.3) vanishes for t < x/c, does not identically vanish in a however small neighbourhood of an arbitrary point on the wave front t ¼ x/c and is locally integrable over dt. Moreover, the classical Laplace transform Lt!p ε ¼ εðp, xÞ makes sense for p > 0. In terms of the Laplace transform formula (3.1.9) evidently assumes the form     φðpÞ x 1 εðp, xÞ ¼ exp p 1 þ 2 c p

ð3:1:90 Þ

Thus using the well-known properties of the Fourier and Laplace transforms, one can write the following formula for the solution of (3.1.1, 3.1.2, and 3.1.3): εðt, xÞ ¼ F 1 λ!tðx=cÞ

h i h i 1 x 1 x ¼ L1 ð3:1:10Þ exp iλe φð λ Þ exp φðpÞ p!t ð x=c Þ iλ 2c p 2c

Our main purpose is to give a description of the asymptotic behaviour of (3.1.10) in the vicinity of the wave front t ¼ x/c in dependence on the character of singularity of φ(t). Lemma 3.1.1 demonstrates the fact that the sought for asymptotic behaviour of ε(t, x) in the vicinity of t ¼ 0 depends only on the behaviour of φ(t) in the vicinity of t ¼ 0. Lemma 3.1.1 Let ψ(t) be a locally integrable function equal to zero for t < 0, smooth for t > 0 and satisfying the estimate

80

3 The Wave Equation with Memory

j ψ 0 ðt Þ j const eγt , γ > 0, for large t > 0: (a) Let ψ ðt Þ ¼ φðt Þ, for t < t 0 , t 0 > 0

ð3:1:11Þ

Then the function ε(t, x) determined by (3.1.10) will not change in the domain t < x/c + t0 if one substitutes φ for ψ in (3.1.10). (b) Let φ1(t) ¼ φ(t) + υ ln t+ where υ ¼ const is a real number:  ln t þ ¼

ln t

for t > 0

0

for t < 0

Then iλe φ11 ðλÞ F 1 ¼ 0, λ!t e

for t < 0:

ð3:1:12Þ

Furthermore, let ψ ðt Þ ¼ φ1 ðt Þ,

for t < t 0 , t 0 > 0:

ð3:1:13Þ

Then iλe ψ ðλÞ iλe φ11 ðλÞ F 1 ¼ F 1 , λ!t e λ!t e

for t < t 0 :

ð3:1:14Þ

Proof (a) For λ ¼ μ  ip, p > γ > 0, we have         e ðλ Þ x e ðλ Þ x φ ψ 1 1 exp iλ 1 þ  exp iλ 1 þ 2 c 2 iλ iλ c     n o e ðλ Þ x φ 1 x e ðλ Þ  ψ e ðλ Þ ¼ exp iλ 1 þ 1  exp iλ½φ 2 c iλ 2c Note that by virtue of (3.1.11)

ð3:1:15Þ

3.1 Formulation of the Problem

81

Z1 e ðλ Þ  ψ e ðλÞ ¼ iλ iλ½φ

iλt  e ½φðt Þ  ψ ðt Þ d iλ

t0

Z1 ¼

½φ0 ðt Þ  ψ 0 ðt Þ eiλt dt

t0

Z1 ¼

½ðφ0 ðt Þ  ψ 0 ðt ÞÞ eMt eiλtþMt dt

t0

It is easy to see that for M > γ, the function j [φ0(t)  ψ 0(t)] eMtj is integrable on the whole of t-axis. Therefore, it follows from the previous equality that for p   Im λ
M > γ, e ðλ Þ  ψ e ðλ Þ j  e jλ½φ

t 0 ðpM Þ

Z1

 0  ½φ ðt Þ  ψ 0 ðt Þ eMt  dt ¼ const et0 p

t0

Hence for p large enough and x bounded above  n o x   e ðλ Þ  ψ e ðλÞ  1  exp iλ½φ   const et0 p 2c

ð3:1:16Þ

Finally, it follows from (3.1.5) that  h i x   φð λ Þ  exp iλe   1; 2c

p > 0:

ð3:1:17Þ

Now, (3.1.15, 3.1.16, and 3.1.17) yield      h  i  e ðλÞ x  x eεðλ, xÞ  1 exp iλ 1 þ ψ  const exp  þ t 0 p  2 c  iλ c

ð3:1:18Þ

for p large enough and x bounded above. Hence by Theorem 1.3.1, one obtains the required equality as εðt, xÞ ¼

(b) As we know

F 1 λ!tðx=cÞ

    e ðλ Þ x ψ 1 , exp iλ 1 þ 2 iλ c

x for t < þ t 0: c

82

3 The Wave Equation with Memory

Im ½λe φðλÞ  0,

for p > 0

(See Lemma 1.5.4). Therefore      iλeφ1 ðλÞ   iλeφðλÞ iλυ F t!λ ln tþ  υ e e  ¼ e   const ð1 þ jλjÞ ;

p>0

whence, by Theorem 1.3.1, (3.1.12) follows. The proof of (3.1.14) is similar to the proof of the assertion (a) of the lemma. The lemma is proved.

3.2

Two Classical Tauberian Theorems

In this section, we give formulations of two classical results which will be employed later on. Definition A function g( p), 0 < p < 1 , is said to be completely monotone, if it is infinitely differentiable and ð1Þk

dk gð pÞ
0; dpk

k ¼ 0, 1, . . .

ð3:2:1Þ

Lemma 3.2.1 [2] Let f( p), 0 < p < 1 , be a real function with a completely monotone derivative. Then, ef( p) is a completely monotone function. Proof In accordance with the complete monotonicity of f0( p), we have ð1Þn1 f ðnÞ ðpÞ
0,

n ¼ 1, 2, . . .

ð3:2:2Þ

k ¼ 0, 1, . . .

ð3:2:3Þ

for p > 0. Let us check that ð1Þk

dk f ðpÞ e
0, dpk

for p > 0. For k ¼ 0, (3.2.3) is obvious. For k ¼ 1, we have 

d f ðpÞ e ¼ ef ðpÞ f 0 ðpÞ
0 dp

by virtue of (3.2.2) for n ¼ 1. For k ¼ 2, one obtains

3.2 Two Classical Tauberian Theorems

d2 f ðpÞ ð1Þ e ¼ ef ðpÞ dp2

83



2

d ½f ðpÞ dp

2

d2 þ 2 ½f ðpÞ dp

!
0

by virtue of (3.2.2) for n ¼ 2. For k ¼ 3 ð1Þ3

d3 f ðpÞ e dp3

3 d ½f ðpÞ dp   2 d d ½f ðpÞ ½f ðpÞ þ3 dp dp2 ¼ ef ðpÞ

d3 þ 3 ½f ðpÞ dp






0

by virtue of (3.2.2) for n ¼ 1, 2, 3. Proceeding with differentiation, one obtains the required result. Theorem 3.2.1 (S. Bernstein; See also [2, 3]) A function g( p),0 < p < 1, can be represented as the Laplace–Stieltjes integral Z1 gð pÞ ¼

ept dGðt Þ

ð3:2:4Þ

0

where G(t) is a non-decreasing function, which is supposed to be equal to zero for t < 0, if and only if g( p) is completely monotone. Definition (See [4]) A function l( p) is said to be slowly varying as p ! 1 if it is real valued, positive, and measurable on [p0, 1], p0 > 0 and if for each a > 1 lim

p!1

lðapÞ ¼1 lðpÞ

ð3:2:5Þ

Theorem 3.2.2 (J. Karamata; See also [2, 4]) Let G(t) be a monotone non-decreasing function equal to zero for t < 0 and such that the integral Z1 gð t Þ ¼ 0

is finite for all p > 0. Then (a) If g( p) ¼ pρl(1/p) as p ! + 0, then

ept dGðt Þ

84

3 The Wave Equation with Memory

G ðt Þ 

t ρ l ðt Þ , Γðρ þ 1Þ

as t ! þ1;

ð3:2:6Þ

(b) if g( p) ¼ pρl( p), as p ! 1 , then t ρ l 1t Gðt Þ  , as t ! þ0 Γ ð ρ þ 1Þ

ð3:2:7Þ

Here Γ is the Gamma function. l( p) slowly varies as p tends to infinity; ρ > 0.

3.3

The Continuity and Monotonicity Theorem

In this section, we prove a general result which is independent of the character of singularity of the kernel φ(t). Lemma 3.3.1 There exists a non-decreasing function U(t) such that eρφðpÞ ¼

Z1

ept dU ðt Þ, p > 0

ð3:3:1Þ

0

U(t) is supposed to be equal to zero for t < 0. Proof First of all, note that pφðpÞ is non-negative, since φ(t)
0. Furthermore, let us demonstrate that d ½pφðpÞ ¼  dp

Z1

tφ0 ðt Þ ept dt:

ð3:3:2Þ

0

Usually this formula is proved under the assumption of the function φ(t) being bounded. To prove (3.3.2) for φ(t) satisfying the mentioned conditions (a)–(e) from Sect. 3.1.1, let us consider the equality

3.3 The Continuity and Monotonicity Theorem

2 d 4 p dp

3

Z1 φð t Þ e

pt

d dt 5 ¼  dp

δ

Z1

85

φðt Þ dept

δ

t¼1 Z1  d d pt  ¼  ½φðt Þ e   þ φ0 ðt Þ ept dt dp dp t¼δ ¼ δφðδÞ epδ 

Z1

δ

tφ0 ðt Þ ept dt;

δ>0

δ

Notice that there must exist a sequence δn ! + 0 such that δnφ(δn) ! 0, since otherwise φ(t) would obviously have a non-integrable singularity at t ¼ 0. Passing to the limit in the last equality, one easily gets the desired result. Note that we have also shown that the function tφ0(t) is locally integrable on [0, 1). Now, let us demonstrate that for p > 0, the following sequence of inequalities ð1Þk1

dk ½pφðpÞ
0; k ¼ 1, 2, . . . , dpk

ð3:3:3Þ

holds. Really, (3.3.2) yields (3.3.3) for k ¼ 1, as in accordance with our assumptions, φ0(t)  0 for t > 0. Now, it is clear that dk dk1 ½pφðpÞ ¼  k1 k dp dp

Z1

tφ0 ðt Þ ept dt ¼ ⋯ ¼ ð1Þk

0

Z1

t k φ0 ðt Þ ept dt

0

whence (3.3.3) follows. Thus we have proved the function ½pφðpÞ0 being completely monotone. Hence pφðpÞ is also a completely monotone function (see Lemma 3.2.1). Whence, in e accordance with Theorem 3.2.1, the stated result follows. Lemma 3.3.2 (a) The function U(t) determined in (3.3.1) can be represented as U ðt Þ ¼ L1 p!t

epφðpÞ p

ð3:3:4Þ

(b) Moreover U(t) ! 0 as t ! + 0, and U(t) ! 1 as t ! + 1 . Proof The assertion (a) of the lemma immediately follows from (3.3.1) after integration by parts. The assertion (b) follows from (3.3.4) by virtue of wellknown limiting theorems for the Laplace transform.

86

3 The Wave Equation with Memory

Now, our purpose is to demonstrate the continuity of the function (3.3.4). To prove this fact, we have to establish two following preparatory lemmas which employ property (c) of φ(t). Lemma 3.3.3 Suppose a locally integrable function ψ(t) equals zero both for t < 0 and for t > const >0, is n times differentiable for t > 0, and for small t
0 satisfies the condition   n d ψ ðt Þ nδ   ;  dt n   const t

0 0 can be chosen however small. Note From the condition (3.3.5), it evidently follows that for small t > 0, the sequence of inequalities   j d ψ ðt Þ jδ   ;  dt j   const t

j ¼ 0, 1, . . . , n  1,

ð3:3:7Þ

holds. Proof of the Lemma First of all, let us establish (3.3.6) for j ¼ 0. Taking γ 1 > 0 however small and making use of the Parseval’s theorem, we have Z1 e ðλ Þ ¼ ψ

t δγ1 t δþγ1 ψ ðt Þ eiλt dt

0

0 1 1 2 1 3 Z Z 1 @  ¼ t δγ1 eiμtpt dt A μ4 t δþγ1 ψ ðt Þ eiμt dt 5 2π 0

ð3:3:8Þ

0



where μ denotes convolution with respect to μ. Let us evaluate the first integral on the right-hand side of (3.3.8). We have Z1

t δγ1 eiμtpt dt ¼ Γð1  δ  γ 1 Þ ðp þ iμÞδþγ1 1

0

where Γ is the Gamma function. Whence

3.3 The Continuity and Monotonicity Theorem

87

  1  Z    t δγ 1 eiμtpt dt   const jμjδþγ1 1    

ð3:3:9Þ

0

Furthermore, let us evaluate the second integral on the right-hand side of (3.3.8). Note that by virtue of (3.3.7) with j ¼ 0 t δþγ1 ψ ðt Þ ! 0,

as t ! þ0:

Therefore, integration by parts yields Z1 t

δþγ 1

ψ ðt Þ e

iμt

1 dt ¼ iμ

0

Z1

 d  δþγ 1 t ψ ðt Þ dt: dt

ð3:3:10Þ

0

On the one hand, the module of the left-hand side of (3.3.10) is obviously bounded above by a constant independent of μ. On the other hand, the module of the right-hand side of (3.3.10) is bounded above by const/ j μ j , where const is again independent of μ. Thus we arrive at the inequality   1  Z    t δþγ1 ψ ðt Þ eiμt dt   const :  1 þ jμj   

ð3:3:11Þ

0

Now, (3.3.8, 3.3.9, 3.3.11) yield Z1 e ðλÞj  const jψ 1 Z1

 const

jξjδþγ 1 1

dξ 1 þ jμ  ξj

jξjδþγ1 1 jμ  ξjγ 2 1 dξ

ð3:3:12Þ

1 δþγ 1 þγ 2 1

 constjμj

e ðλÞ is evidently a bounded function of μ for Here γ 2 > 0 is however small. But ψ p ¼ const >0. Hence, it follows from (2.3.12) that for p ¼ const >0. e ðλÞj  const ð1 þ jμjÞδþγ1 þγ2 1 jψ

ð3:3:13Þ

Redenoting γ 1 + γ 2 by γ, we arrive at the desired result. Now, let us establish (3.3.6) for j
1. We have

88

3 The Wave Equation with Memory

dj e ðλÞ ¼ ψ dλ j

Z1

ðit Þ j ψðt Þeiλt dt

ð3:3:14Þ

0

But by virtue of (3.3.7)  dk  j t ψ ðt Þ ! 0, k dt

as t ! þ0; k ¼ 1, 2, . . . , j  1

Hence, by integration by parts, we successively obtain from (3.3.14) dj e ðλÞ ¼ðiÞ j ψ dλ j

Z1 t t ψ ðt Þ d

eiλt iλ

0

¼

ðiÞ iλ

j

Z1

 d j t ψ ðt Þ eiλt dt dt

ð3:3:15Þ

0

¼ ⋮ ðiÞ j ¼ ð1Þ ðiλÞ j

Z1

j

 dj  j t ψ ðt Þ eiλt dt dt j

0

As before, let us choose γ 1 > 0, however small. Then for p ¼ const >0, we can rewrite the integral on the right-hand side of (3.3.15) as follows Z1 0

 dj  j t ψ ðt Þ eiλt dt ¼ j dt

Z1

 dj  j t ψ ðt Þ eiλt dt j dt 0 0 1 1 0 1 1 Z Z j   1 @ d  ¼ t δγ1 eiμtpt dt A μ@ t δþγ 1 j t j ψ ðt Þ eiμt dt A 2π dt t δγ1 t δþγ1

0

0

ð3:3:16Þ Note that by virtue (3.3.7) t δþγ1

 dj  j t ψ ðt Þ ! 0, dt j

as t ! þ0:

3.3 The Continuity and Monotonicity Theorem

89

Hence one can use integration by parts in the second integral on the right-hand side of (3.3.16). Now, evaluation of (3.3.16), which is just like in the case of j ¼ 0 and formula (3.3.15), give the required result. Lemma 3.3.4 For p > 0

e

pφðpÞ

Z1 ¼

uðt Þ ept dt

ð3:3:17Þ

0

where u(t) is a locally integrable function. Proof By virtue of Lemma 3.1.1, one can evidently consider the function φ(t) as having a compact support. Furthermore, by virtue of the same Lemma 3.1.1 iλe φðλÞ F 1 ¼ 0, for t < 0: λ!t e

ð3:3:18Þ

Now, supposing p > 0, consider the expression h F t!λ t

n

F 1 λ!t

e

iλe φðλÞ

i

n d eiλeφðλÞ ¼ i dλ

ð3:3:19Þ

As we know, by virtue of Lemma 1.5.4, which obviously holds true for the function φ(t) Im ½λe φðλÞ  0,

for p > 0:

Furthermore, by virtue of the previous lemma   j  d φ  e ðλÞ  const ð1 þ jμjÞj1þδþγ ; j ¼ 0, 1, . . . , n  dλ j  for p
const > 0, where γ > 0 is however small. Then one easily obtains   n   d nð1δγ Þ iλe φ ð λÞ   i   const ð1 þ jμjÞ  dλ e

ð3:3:20Þ

for p
const > 0. Since by property (c) of the function φ(t) (see Sect. 3.1) n (1  δ  γ) > 1/2 for γ > 0 small enough, it follows from (3.3.20) that the function (3.3.19) turns out to be square integrable over dμ on ð1, 1Þ for p ¼ const > 0:

90

3 The Wave Equation with Memory

iλe φ ð λÞ Hence the only possible singularity of the function F 1 may be located at λ!t e t ¼ 0. As is well-known from the theory of distributions, the mentioned singularity must have the form s X

ck δðkÞ ðt Þ

k¼0

where δ(t) is the Delta function. Thus in accordance with (3.3.18), one can write

e

iλe φ ð λÞ

Z1 ¼

uðt Þ eiλt dt þ

s X

ck ðiλÞk

ð3:3:21Þ

k¼0

0

where u(t) is a locally integrable function such that eγt u(t)is Lebesgue integrable on [0, 1) for however small γ > 0. Hence by putting in (3.3.21), λ ¼ ip, p > 0, we obtain

e

pφðpÞ

Z1 ¼

uðt Þ ept dt þ

s X

c k pk :

ð3:3:22Þ

k¼0

0

Now, from the fact that φ(t) ! 1 as t ! + 0, from the property (d) of the function φ(t), it follows that pφðpÞ ! 1,

as p ! 1:

Therefore, letting in (3.3.22) p ! 1 , one easily obtains ck ¼ 0; k ¼ 0, 1, . . . , s: Hence follows the required result. Lemma 3.3.5 The function U(t) determined by (3.3.1) (or, which is the same, by (3.3.4)) is continuous for all t. Proof Comparing formulas (3.3.4) and (3.3.17), one can easily see that Zt U ðt Þ ¼

uðτÞ dτ 0

whence by virtue of Lemma 3.3.4, follows the assertion of the lemma.

ð3:3:23Þ

3.3 The Continuity and Monotonicity Theorem

91

Corollary In the representation (3.3.17) uð t Þ
0

ð3:3:24Þ

for almost all t
0. Lemma 3.3.6 Let 0  a  b Then L1 p!t

epφðpÞa epφðpÞb
L1 : p!t p p

ð3:3:25Þ

Proof At first, let a ¼ 0. Then (3.3.25) easily follows from Lemmas 3.3.1 and 3.3.2. Now, let a > 0. Then we have L1 p!t



 epφðpÞa epφðpÞb  p p   epφðpÞðbaÞ 1 pφðpÞa 1  ¼ Lp!t e p p   h i  1 epφðpÞðbaÞ 1 pe φðpÞa 1 ¼ Lp!t e  Lp!t  p p

ð3:3:26Þ

By virtue of Corollary to Lemma 3.3.4 pφðpÞa L1
0: p!t e

Furthermore, it easily follows from Lemmas 3.3.1 and 3.3.2 that for t > 0 L1 p!t



 1 epφðpÞðbaÞ epφðpÞðbaÞ 
0:  1  L1 p!t p p p

Hence the right-hand side of (3.3.26) presents a convolution of two non-negative functions, which gives the result required. Theorem 3.3.1 For x > 0, the solution of the problem (3.1.1, 3.1.2, and 3.1.3) εðt, xÞ ¼ L1 p!tðx=cÞ

h i 1 x , exp pφðpÞ p 2c

Possesses the following properties: (a) ε(t, x) is a non-decreasing in t function. (b) ε(t, x) is a non-increasing in x function.

92

3 The Wave Equation with Memory

lim

t!ðx=cÞþ0

εðt, xÞ ¼ 0,

lim εðt, xÞ ¼ 1;

ð3:3:27Þ ð3:3:28Þ

t!1

(c) ε(t, x) is continuous in t, x. Proof The assertions (a) and (c) of the theorem directly follow from Lemmas 3.3.1 and 3.3.2. As regards the assertion (b), let us note the following:

By virtue of Lemma 3.3.6, for each t 0 > 0, ε xc þ t 0 , x proves to be a non-increasing in x function. Let x2 > x1 > 0, then ε

  x1 x þ t 0 , x1
ε 2 þ t 0 , x2 : c c

ð3:3:29Þ

On the other hand, by virtue of the assertion (a) of the theorem ε

  x1 x þ t 0 , x1  ε 2 þ t 0 , x1 : c c

ð3:3:30Þ

Hence it turns out from (3.3.29, 3.3.30) that ε

  x2 x þ t 0 , x1
ε 2 þ t 0 , x2 : c c

Redenoting (x2/c) + t0 by t gives the result required. Finally, the assertion (d) of the theorem easily follows from Lemma 3.3.5 and assertions (a) and (b).

3.4

The Approximation Theorem

Theorem 3.4.1 Let functions φj(t), j ¼ 1, 2, . . . be smooth, non-increasing, and concave for t > 0, integrable [0, 1) and let φj(t) ¼ 0 for t < 0. Furthermore, suppose Z1 0

Then for each fixed x

  φðt Þ  φ j ðt Þ dt ! 0,

as j ! 1:

ð3:4:1Þ

3.4 The Approximation Theorem

εðt, xÞ  F 1 λ!tðx=cÞ

n

93

h io 1 x ! 0, exp iλe φ j ðλÞ iλ 2c

as j ! 1

ð3:4:2Þ

uniformly with respect to t. Proof Let us denote ε j ðt, xÞ  F 1 λ!tðx=cÞ

n

h io 1 x exp iλe φ j ðλÞ iλ 2c

First of all, let us demonstrate that for however small δ > 0 Z1

 e2δt εðt, xÞ  ε j ðt, xÞj2 dt ! 0 as j ! 1:

ð3:4:3Þ

 

  1 λ x eεðλ, xÞ  eε j ðλ, xÞ ¼ exp iλ 1 þ φ e iλ 2 c  n   x o e ðλ Þ  φ e j ðλÞ 1  exp iλ φ 2c

ð3:4:4Þ

1

We have

By virtue of (3.4.1) e ðλ Þ  φ e j ðλÞ ! 0 as j ! 1 φ

ð3:4:5Þ

uniformly in the half-plane p
δ > 0. Hence for p ¼ δ, the difference (3.4.4) goes to zero uniformly on each segment M  μ  M, as j ! 1. Furthermore, as we know Im ½λe φðλÞ  0,

for p > 0

(see the proof of Lemma 1.5.4), whence jeεðλ, xÞj 

1 , for p > 0 jλj

Similarly, one can prove that

Im λe φ j ðλÞ  0 for p > 0 whence

ð3:4:6Þ

94

3 The Wave Equation with Memory

j eε j ðλ, xÞ j

1 , jλj

for p > 0

ð3:4:7Þ

Now, from (3.4.4, 3.4.5, 3.4.6, and 3.4.7), one easily gets that for p ¼ δ > 0, Z1

 eεðλ, xÞ  eε j ðλ, xÞj2 dμ ! 0,

as j ! 1

1

whence by Plancherel’s theorem, (3.4.3) follows. Furthermore, as we know from Theorem 3.3.1, for x > 0, ε(t, x) is a continuous, non-decreasing in t, bounded above function equal to zero for t < x/c. In the same manner, one can prove that for x > 0 all εj(t, x)are non-decreasing in t bounded above functions equal to zero for t < x/c. Since the values φj(+0) may be finite, the functions εj(t, x) may suffer jumps across the wave front t ¼ x/c. Now, from above-mentioned, follows the required result for x > 0. Finally, in case of x ¼ 0, the assertion of the theorem is evident. Note To evaluate the difference ε(t, x)  εj(t, x), one can make use of the wellknown Esseen’s inequality (see [5]). Note One can easily see that for φ(t) tending to a finite limit as t ! + 0 Theorem 3.4.1 is valid only under the additional assumption of φ j ðþ0Þ ! φðþ0Þ,

3.5

as j ! 1:

Memory Function with a Singularity, Which Is Weaker than the Logarithmic One

In Sects. 3.5 and 3.6, we have studied the wave front asymptotics of ε(t, x) by purely real methods. As usual, the function φ(t) is supposed satisfying conditions (a)–(e) in Sect 3.1. Lemma 3.5.1 (Abelian) Let φ0 ð t Þ ¼ o

 1 , t

as t ! þ0:

ð3:5:1Þ

3.5 Memory Function with a Singularity, Which Is Weaker than the Logarithmic One

95

Then pφðpÞ  φ

 1 ! 0, p

as p ! þ1

ð3:5:2Þ

Note From the condition (3.5.1), it evidently follows that  1 φðt Þ ¼ o ln , as t ! þ0: t Proof of the Lemma We have



 Z1 1 1 ¼ p φðt Þept dt  φ pφðpÞ  φ p p 0

 Z1  t t 1 ¼ e dt  φ φ p p 0



 Z1   Z1   t 1 t 1 t ¼ φ e dt þ φ et dt φ φ p p p p 0

1

ð3:5:3Þ Let

 Z1   t 1 I1 ¼ φ et dt, φ p p 0

 Z1   t 1 I2 ¼ φ φ et dt: p p 1

Let us evaluate the integral I1. For 0 < t  1, we have



 t 1 φ
0 φ p p since φ(t) is non-increasing for t > 0. Therefore

96

3 The Wave Equation with Memory

 Z1   t 1 φ φ dt p p

0  I1 

0

Z1=p ¼p

 1 φðτÞ dτ  φ p

0 τ¼1 pτφðτÞjτ¼0p

¼

Z1=p p

 1 τφ ðτÞdτ  φ p 0

0

Z1=p ¼ p

τφ0 ðτÞ dτ

0

From (3.5.1), it is clear that Z1=p p

τφ0 ðτÞ dτ ! 0,

as p ! þ1

0

Hence I1 also goes to zero as p ! + 0. Now, let us evaluate the integral I2. For t
1, we have





 Zt=p t 1 t 1 0
φ φ ¼ φ0 ðτÞ dτ
φ0 p p p p

ð3:5:4Þ

1=p

as φ(t) is non-increasing and concave for t > 0. From (3.5.4), it follows that

 Z1 1 0 1 0
I2
φ tet dt p p 1

whence, by virtue of (3.5.1), it is clear that I2 ! 0 as p ! + 1 . Thus we have demonstrated that the difference (3.5.3) goes to zero as p ! + 1 . The lemma is proved. In Sect. 3.6, we shall make use of the following variant of the previous lemma. Lemma 3.5.10 Let φ1(t), t 2 (0, +1), be a function whose Laplace–transform makes sense for p > 0 and suppose

3.5 Memory Function with a Singularity, Which Is Weaker than the Logarithmic One



 t 1  φ1 ! 0, φ1 p p

as p ! þ1

97

ð3:5:5Þ

uniformly with respect to t on each segment of the sort 0 < a  t  b; (a) For large p > 0  

    φ1 t  φ 1 1   ψ ð t Þ  p p 

ð3:5:6Þ

where ψ(t)
0 is some function whose Laplace transform makes sense for p > 0. Then pφ1 ðpÞ  φ1

 1 ! 0, as p ! þ1: p

ð3:5:7Þ

Lemma 3.5.2 (Tauberian) Let φ(t) satisfy the condition of Lemma 3.5.1. Then the function epφðpÞ slowly varies as p ! 1 and L1 p!t

epφðpÞ  eφðtÞ , p

as t ! þ0:

ð3:5:8Þ

Proof For t > 0, we have 1 epφðpÞ

h  i  h i p p p p ¼ exp  φ þ pφðpÞ exp  φ t t t t   

  t 1 ¼ exp  φ φ þ oð 1Þ p p

ð3:5:9Þ

by virtue of Lemma 3.5.1. Let us demonstrate that



 t 1 φ φ ! 0, p p In fact, in case of 0 < t  1, we have

as p ! þ1:

ð3:5:10Þ

98

3 The Wave Equation with Memory



 Z1=p t 1 φ ¼ 0φ φ0 ðτÞ dτ p p t=p

 t 1t  φ p p 0

 φ0



 t 1 t t 1 ¼  φ0 ! 0, p p p p t

as p ! þ1

by virtue of (3.5.1), which proves (3.5.10). In case of t > 1, (3.5.10) follows, for example, from (3.5.4). So, it follows from (3.5.10) that the ratio on the left-hand side of (3.5.9) tends to 1, as p ! þ1, that is epφðpÞ slowly varies as p ! + 1. Furthermore, as we know from the Note to Lemmas 3.3.1 and 3.3.2

e

pφðpÞ

Z1 ¼

ept dU ðt Þ

ð3:5:11Þ

0

where U(t) is a monotone non-decreasing function, whence by Karamata’s theorem, see formula (3.2.5) h  i 1 1 , as t ! þ0:: U ðt Þ  exp  φ t t But by virtue of Lemma 3.5.1  1 1  φðt Þ ! 0, φ t t

as t ! þ0:

Therefore U ðt Þ  eφðtÞ ,

as t ! þ0:

But, as we know, (3.5.11) can be rewritten as U ðt Þ ¼ L1 p!t whence follows the required result.

epφðpÞ p

ð3:5:12aÞ

3.5 Memory Function with a Singularity, Which Is Weaker than the Logarithmic One

99

Theorem 3.5.1 Let φ0 ð t Þ ¼ o

 1 , as t ! þ0 t

Then h  i x x , εðt, xÞ  exp φ t  c 2c

x as t ! þ 0: c

ð3:5:13Þ

Proof Formula (3.5.13) immediately follows from Lemma 3.5.2. Now let us pass to the evaluation of ε(t, x) by using the method of Tauberian inequalities. Lemma 3.5.3 Let (3.5.1) be valid, and suppose tφ0(t)increases for 0 < t < t0, where t0 > 0. Then L1 p!t

epφðpÞ
eφðtÞ , p

for 0  t < t 0 :

ð3:5:14Þ

Proof As we know from Sect. 3.3 pφðpÞ uðt Þ  L1
0, p!t e

for t
0

ð3:5:15Þ

See Eq. (3.3.24). Furthermore, by means of the Laplace transform, one can easily verify that for t
0 the function u(t) satisfies the following integral Volterra type equation Zt tuðt Þ ¼ 

τφ0 ðτÞ uðt  τÞ dτ:

ð3:5:16Þ

0

Note that by virtue of property (b) of Sect. 3.1, φ0(t)  0. Therefore, taking into account (3.3.15) and the increase and non-negativeness of tφ0(t), one easily obtains from (3.5.16) that 0

Zt

tuðt Þ  tφ ðt Þ

uðτÞ dτ, 0

or, which is the same

for 0 < t < t 0

100

3 The Wave Equation with Memory

d ½φðt Þ þ ln U ðt Þ  0, dt

for 0 < t < t 0

ð3:5:17Þ

where Zt U ðt Þ ¼

uðτÞdτ ¼ L1 p!t

epφðpÞ p

0

Hence, by integration over dt along the interval (γ, t), where 0 < γ < t < t0, one easily obtains Zt γ

d ½φðt Þ þ ln U ðt Þdt ¼ φðt Þ þ ln U ðt Þ  φðγ Þ  ln U ðγ Þ  0 dt

whence U ðt Þ  eφðtÞ U ðγ Þ eφðγÞ :

ð3:5:18Þ

However, by virtue of Lemma 3.5.2 U ðγ Þ eφðγÞ ! 1,

as γ ! þ0:

Therefore, letting γ ! + 0 in (3.5.18), we obtain the desired estimate U ðt Þ  eφðtÞ ,

0  t < t 0:

The Lemma is proved. Thus we arrive at the following result. Theorem 3.5.2 Let (3.5.1) be valid and suppose tφ0(t) increases for 0 < t < t0, where t0 > 0. Then h  i x x , 0  εðt, xÞ  exp φ t  c 2c

3.6

for

x x  t < þ t0 : c c

ð3:5:19Þ

Memory Function with the Logarithmic Singularity

Now, let us pass to the case where the creep kernel has a singularity (as t ! + 0) of the logarithmic type. As we shall see below, the asymptotic behaviour of the solution of the problem (3.1.1, 3.1.2, and 3.1.3)

3.6 Memory Function with the Logarithmic Singularity

101

epφðpÞ2c , c p

ð3:6:1Þ

for 0 < t < t 0

ð3:6:2Þ

x

εðt, xÞ ¼ L1 p!tx has a somewhat unexpected character. Theorem 3.6.1 Let φðt Þ ¼ k ln

1 , t

where t0 > 0, k > 0. Then  ekCx=ð2cÞ x kx=ð2cÞ

t , εðt, xÞ ¼ kx c Γ 2c þ 1

for

x x  t  þ t0 c c

ð3:6:3Þ

where C ¼ 0.57... is the Euler’s constant, Γ the Gamma function.. Proof By virtue of Lemma 3.1.1, when calculating ε(t, x) for t  (x/c) + t0, one can consider φðt Þ ¼ k ln

1 t

ð3:6:4Þ

for all t > 0. Then φð p Þ ¼ k

C þ ln p : p

ð3:6:5Þ

After inserting (3.6.5) into the right-hand side of (3.6.1), we have for t < (x/c) + t0, εðt, xÞ ¼

L1 p!tðx=cÞ

 1 k C þ ln p x exp p p 2 p c

whence follows the required result. Note It follows from (3.6.3) that ε(t, x) becomes more smooth in the vicinity of the wave front t ¼ x/c with the growth of x. Now we shall extend the result of Theorem 3.6.1 to a class of memory functions, which are equivalent to k ln (1/t) but do not necessarily coincide with k ln (1/t) for small t > 0.

102

3 The Wave Equation with Memory

Lemma 3.6.1 (Abelian) Let for t > 0 φ



 t 1 1 φ ! k ln , p p t

as p ! þ1

ð3:6:6Þ

where k > 0, and let for large p > 0  

  t 1  φ  p  φ p   f ðt Þ

ð3:6:7Þ

where f(t)
0 is a function whose Laplace transform is determined for p > 0. Then pφðpÞ  φ

 1 ! k C, p

as p ! þ1

ð3:6:8Þ

Here C ¼ 0.57. . . is the Euler’s constant. Proof Let φ1 ðt Þ ¼ φðt Þ  k ln

1 t

ð3:6:9Þ

Let us demonstrate that φ1(t) satisfies the conditions of Lemma 3.5.10 . In fact, by virtue of (3.6.6) φ1



  

  t 1 t 1 p  φ1 ¼ φ φ  k ln  ln p p p p p t



 t 1 1 ¼φ φ  k ln ! 0, p ! 1, p p t

uniformly with respect to t on each segment of the sort 0 < a  t  b,since the function φ(t/p)  φ(1/p) is monotone in t and the function k ln (1/t) is continuous for t > 0. Furthermore  

  

       φ1 t  φ1 1   φ t  φ 1  þ k ln p  ln p     p p p p t    1  f ðt Þ þ k ln : t From the last two formulas, it follows that the conditions of Lemma 3.5.10 are satisfied. Therefore

3.6 Memory Function with the Logarithmic Singularity

 1 ! 0, pφ1 ðpÞ  φ1 p

103

as p ! þ1:

However by virtue of (3.6.9) pφ1 ðpÞ  φ1

      1 C þ ln p 1 ¼ pφðpÞ  pk  φ  k ln p p p p

 1 ¼ pφðpÞ  φ  kC, p

which proves the lemma. The condition (3.6.6) imposed above on the function φ(t) may seem somewhat artificial. To dispel this impression, we adduce the following modification of a proposition taken from [4]. Proposition (See [4]) Let φ(t) be defined for t > 0 and suppose that for each t > 0, there exists  

 t 1 lim φ φ ¼ f ðt Þ p!þ1 p p

ð3:6:10Þ

where f(t) is a continuous function. Then f(t) ¼ const. ln t. Proof Consider the expression  

  

 t1 1 t2 1  φ : φ φ φ p p p p

ð3:6:11Þ

On the one hand, from (3.6.10), it is clear that (3.6.11) tends to f(t1)  f(t2) as p ! + 1 . On the other hand, by opening the brackets in (3.6.11) and letting q ¼ p/t2, one obtains φ









 t =t t1 t 1 t !f 1 , φ 2 ¼φ 1 2 φ q p p q t2

as q ! þ1

whence

 t f 1 ¼ f ðt 1 Þ  f ðt 2 Þ, t2 which gives the required result. Lemma 3.6.2 (Tauberian): Suppose φ(t) satisfies the conditions of Lemma 3.6.1. Then the function

104

3 The Wave Equation with Memory

lðpÞ  pk epφðpÞ slowly varies as p ! + 1 and L1 p!t

l ð pÞ epφðpÞ ekC  L1  eφðtÞ , p!t kþ1 p Γ ð k þ 1Þ p

t ! þ0:

ð3:6:12Þ

Here C ¼ 0.57... is the Euler’s constant, the constant k > 0 is defined in (3.6.6). Proof We have kC k pφðpÞ

e l ð pÞ  e p e kC



  C þ ln p ¼ exp p φðpÞ  k ¼ epφ1 ðpÞ p

where φ1 ðt Þ ¼ φðt Þ  k ln

1 : t

As was noted in the proof of Lemma 3.6.1, the function φ1(t) satisfies the conditions of Lemma 3.5.10 . Therefore by virtue of Lemma 3.5.10   p p l ekC l t ¼ t l ð pÞ ekC lðpÞ h  i 1 p p ¼ pφ ðpÞ exp φ1 t e 1  t

  t 1 ¼ exp  φ1  φ1 þ oð 1Þ , p p as p ! + 1 . However φ1



 t 1  φ1 ! 0, p p

as p ! þ1

from condition (a) of Lemma 3.5.10 . Hence, the ratio l( p/t)/l( p) tends to 1, as p ! + 1 . That is, l( p) slowly varies as p ! + 1. Furthermore by virtue of Lemma 3.3.1 lðpÞ  epφðpÞ ¼ pk

Z1 0

ept dU ðt Þ

3.6 Memory Function with the Logarithmic Singularity

105

where U(t) is a non-decreasing function (which is supposed to be equal to zero for t < 0). Hence, by Karamata’s theorem U ðt Þ 

  k h  i tk 1 tk 1 1 1 ¼ , t ! þ0, exp  φ l t t Γðk þ 1Þ t Γðk þ 1Þ t

whence by virtue of Lemma 3.6.1 U ðt Þ 

ekCφðtÞ , Γðk þ 1Þ

t ! þ0:

The lemma is proved. Lemma 3.6.2 obviously leads us to the following result. Theorem 3.6.2 Suppose φ(t) satisfies the conditions of Lemma 3.6.1. Then h  i ekCx=ð2cÞ x x

exp φ t  , εðt, xÞ  kx c 2c Γ 2c þ 1

x as t ! þ 0 c

ð3:6:13Þ

where C ¼ 0.57. . . is the Euler’s constant. Now, let us pass to the evaluation of ε(t, x) by means of Tauberian inequalities in the manner of Lemma 3.5.3 and Theorem 3.5.2. Lemma 3.6.3 Let φðt Þ ¼ k ln

1 for 0 < t  γ t

ð3:6:14Þ

where k > 0, γ > 0, and suppose a  tφ0 ðt Þ  b,

for γ < t < t 0

ð3:6:15Þ

where 0 < a  k  b. Then ta

ekC epφðpÞ ekC ,  tb γ ka  U ðt Þ ¼ L1 p!t p Γ ð k þ 1Þ Γðk þ 1Þγ bk

for 0  t

< t0

ð3:6:16Þ

where C ¼ 0.57... is the Euler’s constant, Γ the Gamma function. Proof As we know, from Eq. (3.5.16), the function pφðpÞ uðt Þ ¼ L1 p!t e

satisfies the Volterra equation

106

3 The Wave Equation with Memory

Zt tuðt Þ ¼ 

τφ0 ðτÞuðt  τÞdτ

ð3:6:17Þ

0

Furthermore, since 0 < a  k  b, it follows from (3.6.14, 3.6.15) that 0 < a  tφ0 ðt Þ  b,

for 0 < t < t 0 :

Hence (3.6.17) yields Zt

Zt uðτÞdτ  tuðt Þ  b

a 0

uðτÞdτ,

for 0  t < t 0

ð3:6:18Þ

0

Since Zt

uðτÞdτ ¼ L1 p!t

epφðpÞ  U ðt Þ, p

0

Eq. (3.6.18) can obviously be rewritten as ad ln t  d ln U ðt Þ  bd ln t whence integrating over dt on (γ, t) (where t < t0), one easily obtains ln

U ðt Þ ta tb  ln  ln b : a γ U ðγ Þ γ

ð3:6:19Þ

Now, note that Eq. (3.6.14) yields U ðγ Þ ¼ ekC

γk Γðk þ 1Þ

ð3:6:20Þ

where C is the Euler’s constant. Formulas (3.6.19, 3.6.20) yield the result required. Now, substituting in (3.6.16) t for (t – x)/c and k, a, b for kx/(2c), ax/(2c), bx/(2c), one easily gets the following result.

3.7 Memory Function with a Singularity Stronger than the Logarithmic One

107

Theorem 3.6.3 Under the assumptions of Lemma 3.6.3

ðkaÞx

ax2c e kCx 2c γ 2c

 kx þ1 Γ 2c  εðt, xÞ  x bx=ð2cÞ ekCx=ð2cÞ

  t , c kx þ 1 γ ðbkÞx=ð2cÞ Γ 2c t  xc

ð3:6:21Þ for

x x  t < þ t0 c c

where C ¼ 0.57... is the Euler’s constant, Γ the Gamma function.

3.7

Memory Function with a Singularity Stronger than the Logarithmic One

In this section, we consider the case where the memory function φ(t) satisfies the condition lim

t!þ0

φð t Þ ¼ þ1 ln ð1=t Þ

ð3:7:1Þ

As we shall see below, in this case, the method of computation of the wave front asymptotics of ε(t, x), based on the theory of regularly varying functions, leads to comparatively weak results. Fortunately, one can employ, the approach based on Tauberian inequalities by use of which one can evaluate the solution ε(t, x). We shall begin with the following assertion of general character. Theorem 3.7.1 Suppose Eq. (3.7.1) holds. Then for x > 0, the solution ε(t, x) of the problem (3.1.1, 3.1.2, and 3.1.3) is infinitely differentiable and, in particular, is infinitely differentiable in the vicinity of the wave front t ¼ x/c, x > 0. Proof As we know from Eq. (3.1.4)     e ðλ Þ x φ 1 eεðλ, xÞ ¼ exp iλ 1 þ : 2 iλ c Furthermore, by virtue of Lemma 1.5.4, for p > 0

108

3 The Wave Equation with Memory



π Im ½λe φðλÞ   φ 2jμj





 pπ exp  , 2jμj

μ 6¼ 0,

ð3:7:2Þ

0, μ ¼ 0 Now, let us fix p ¼ p0 > 0 large enough. Then for λ ¼ μ  ip0, one obtains      1 e ðλÞ x  φ  jeεðλ, xÞj ¼ exp iλ 1 þ 2 c  iλ

 

 p π x π  const exp  φ exp  0 , 2c 2jμj 2jμj

 const,

μ 6¼ 0, μ¼0

whence by virtue of Eq. (3.7.1), it follows that for x 6¼ 0 and for k > 0, however large, there exists ak ¼ ak(x, p0), such that jeεðλ, xÞj  ak ð1 þ jμjÞk ,

λ ¼ μ  ip0 :

ð3:7:3Þ

As is well-known [6], it follows from Eq. (3.7.3) that ε(t, x) is infinitely differentiable in t (for x > 0). Let us pass to the demonstration of ε(t, x) being infinitely differentiable in x for x > 0. By virtue of Eq. (3.7.2), we have for λ ¼ μ  ip0,      k        k e ðλÞ   e ðλÞ x  φ φ 1  iλ ∂ eεðλ, xÞ  1þ exp iλ 1 þ ¼  2  2 c   ∂xk  jiλj  c 

k

   p0 π e ðλÞ  φ jxj 1 λ π  2jμj   1þ exp  e , μ 6¼ 0, φ 2  2c 2jμj jλj c    k e ðλÞ  φ 1  , μ¼0 λ 1 þ  2  jλj 

ð3:7:4Þ

where k ¼ 0, 1, 2, .... Hence by virtue of Eq. (3.7.1), it is clear that for x > δ (where k δ > 0 is however small) and for each k ¼ 0, 1, 2, . . ., the function ∂ eεðλ, xÞ=∂xk is Lebesgue integrable over dμ and the following estimate holds  Z1  k  ∂  dμ  bk ¼ bk ðδ, p0 Þ; λ ¼ μ  ip0 :  e ε ð λ, x Þ  ∂xk

1

Hence for each k

ð3:7:5Þ

3.7 Memory Function with a Singularity Stronger than the Logarithmic One

F 1 μ!t

k

109

k

∂ ∂ eεðλ, xÞ ¼ ep0 t k εðt, xÞ ∂xk ∂x

is a uniformly continuous in t equicontinuously depending on x function, provided x is bounded away from zero. But from Eq. (3.1.4), it evidently follows that the function ∂kε(t, x)/∂xk considered as a distribution in t, continuously depends on the parameter x for x > 0. Hence by virtue of the above-mentioned follows the continuity (in the usual sense) of the function ∂kε(t, x)/∂xk, for x > 0. Since k can be chosen however large, the infinite differentiability of ε(t, x), x > 0, is proved. Corollary Under the condition (3.7.1), the function iλe φðλÞ pe φðpÞ uðt Þ ¼ F 1 ¼ L1 λ!t e p!t e

is infinitely differentiable on the whole of t-axis. Note Let us adduce the following asymptotic result, which is based on a Tauberian theorem by Feller. Suppose φð t Þ ¼

 t α 1 , l Γð1  αÞ t

0 0. Then 0  L1 p!t

epφðpÞ  eφðtÞ , p

for t
0:

ð3:7:11Þ

Example 0

L1 p!t

X 1 exp  c j pα j p j

"

!

 exp 

X j

# t α j

, for t
0 cj Γ 1  αj

ð3:7:12Þ

Here cj > 0, 0 < αj < 1. Proof of the Lemma As we know, the function pφðpÞ uðt Þ ¼ L1 p!t e

satisfies the equation Zt tuðt Þ ¼ 

τφ0 ðτÞuðt  τÞ dτ

ð3:7:13Þ

0

(see Sect.3.5). Furthermore, u(t)
0, for t > 0 (see Corollary to Lemma 3.3.5) and φ0(t)  0, for t > 0 by virtue of property (b) from Sect. 3.1.1. Hence, the integrand in (3.7.13) does not change its sign. So, by virtue of tφ0(t) being non-increasing for t > 0, one obviously gets from (3.7.13)

3.7 Memory Function with a Singularity Stronger than the Logarithmic One

0

111

Zt

t uðt Þ
tφ ðt Þ

uðτÞ dτ, t > 0, 0

whence d ½φðt Þ þ ln U ðt Þ
0 dt

ð3:7:14Þ

where Zt U ðt Þ ¼

uðτÞ dτ ¼ L1 p!t

epφðpÞ : p

0

Thus, it follows from Eq. (3.7.14) that Z1 d½φðτÞ þ ln U ðτÞ
0, t > 0:

ð3:7:15Þ

t

Note now that by virtue of Lemma 3.3.2 φð1Þ þ ln U ð1Þ  0 Hence (3.7.15) yields φðt Þ þ ln U ðt Þ  0, That is U ðt Þ  eφðtÞ , which proves the right-hand inequality in Eq. (3.7.11). The left-hand inequality was established in Sect. 3.3. Now, let us pass to the evaluation of u(t). Lemma 3.7.2 Suppose (3.7.1) holds, the function tφ0(t) is non-increasing for t > 0, and the function φ(t)  ln (1/t) is non-increasing on (0, t0) and does not possess this property on any larger interval. Then pφðpÞ 0  uðt Þ ¼ L1  p!t e

eCφðtÞ , t

for 0  t < t 0

ð3:7:16Þ

and u(t) is non-decreasing for 0  t < t0. Here C ¼ 0.57. . . is the Euler’s constant.

112

3 The Wave Equation with Memory

Proof Since Lt!p ln

1 C þ ln p ¼ , t p

we have epφðpÞ ¼ eC

n h  io 1 1 : exp pLt!p φðt Þ  ln p t

ð3:7:17Þ

Let 1 , for 0 < t < t 0, t 1 ¼φðt 0 Þ  ln , for t
t 0 : t0

ψ ðt Þ ¼φðt Þ  ln

ð3:7:18Þ

Then by virtue of Lemma 3.1.1 L1 p!t

n h  io 1 1 epψ ðpÞ ¼ L1 exp pLt!p φðt Þ  ln , for 0  t < t 0 : ð3:7:19Þ p!t p t p

Furthermore, in the manner of Lemma 3.3.1 and Theorem 3.7.1, one can easily establish that L1 p!t

epψ ðpÞ p

is a non-decreasing, non-negative, infinitely differentiable function. Moreover, since tφ0(t) is non-increasing for t > 0, it follows from (3.7.18) that tψ 0(t) is also non-increasing for t > 0. Hence in the manner of Lemma 3.7.1, one obtains 0  L1 p!t

epψ ðpÞ  eψ ðtÞ , p

for t
0

ð3:7:20Þ

whence by virtue of (3.7.17, 3.7.18, and 3.7.19), it follows that for 0  t < t0, pφðpÞ  eC eψ ðtÞ ¼ eC L1 p!t e

eφðtÞ , t

which gives the right-hand inequality in Eq. (3.7.16). The left-hand inequality was established in Sect. 3.3. Thus we arrive at the following result.

3.8 The Power Memory Function

113

Theorem 3.7.2 (a) Suppose Eq. (3.7.1) holds and the function tφ0(t) is non-increasing for t > 0. Then h  i x x x , for t
: 0  εðt, xÞ  exp φ t  c 2c c

ð3:7:21Þ

(b) Moreover, suppose also that for each x > 0 φð t Þ

x 1  ln 2c t

is a non-increasing in t function on the interval (0, t0(x)) and does not possess this property on any larger interval. Then 0  ε0t ðt, xÞ 

x x eC x x eφðtcÞ2c , for  t < þ t 0 ðxÞ c c t  ðx=cÞ

ð3:7:22Þ

Here C ¼ 0.57... is the Euler’s constant.

3.8

The Power Memory Function

Let φð t Þ ¼

k t α , for t > 0 Γ ð1  αÞ

ð3:8:1Þ

where k > 0, 0 < α < 1, α

p : uα ðt Þ ¼ L1 p!t e

ð3:8:2Þ

By virtue of Corollary to Theorem 3.7.1, uα(t) is infinitely differentiable on the whole of t-axis. Furthermore, from Corollary to Lemma 3.3.5, it follows that uα(t)
0 (for the first time this fact was established in [7]). Let us adduce the following result concerning the behaviour of uα(t). Lemma 3.8.1[7] uα ð t Þ ¼

1 sin ðπnαÞ Γðnα þ 1Þ 1X ð1Þn1 , n! π n¼1 t nαþ1

Γ is the Gamma function.

for t > 0

ð3:8:3Þ

114

3 The Wave Equation with Memory

The expansion (3.8.3) converges non-uniformly for small t > 0. So, of interest is the following result giving the asymptotic representation for uα(t), as t ! + 0 [8] h i α α1=2ð1αÞ uα ðt Þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 12ð1αÞ exp ð1  αÞαα=ð1αÞ t α=ð1αÞ : 2π ð1  αÞ

ð3:8:4Þ

Note that in case of α ¼ 1/2 (3.8.4) passes into a well-known precise equality u1=2 ðt Þ ¼

1 2π 1=2 t 3=2

 1 exp  : 4t

ð3:8:5Þ

The following simple theorem immediately follows from formula εðt, xÞ ¼ L1 p!tðx=cÞ

ekp

α

x=ð2cÞ

p

:

Theorem 3.8.1 [1] Under the (3.8.1) tx=c

Z

ðkx=2cÞ1=α

εðt, xÞ ¼

uα ðξÞ dξ

ð3:8:6Þ

0

Note: It is known (see [9]) that uα(t) has a unique maximum on the semi-axis t
0. Let us show that this maximum is achieved in some point tm of the segment [t0, t1], where 

α t0 ¼ Γð1  αÞ

1=α



,

α t1 ¼ Γ ð2  α Þ

1=α ð3:8:7Þ

At first, let us prove that the maximum of uα(t) is located to the right of t0. In fact, one can easily see that t α 1  ln t Γð1  αÞ is monotone decreasing on the interval (0, t0), where the value of t0 is given by (3.8.7). Hence, by Lemma 3.7.2, it follows that tm > t0. Now, let us prove that tm < t1. We have

3.9 Application of the Nonlinear Laplace Transform to Calculating the Wave Front. . .

α tuα ðt Þ ¼ Γð1  αÞ

Zt

ðt  τÞα uα ðτÞ dτ:

115

ð3:8:8Þ

0

This equality is a special case of (3.5.16). It can also be checked by direct application of the Laplace transform. Since uα(t)
0, it follows from Eq. (3.8.8) and monotone increase of uα(t) on (0, tm) that α u ðt Þ t m uα ð t m Þ  Γð1  αÞ α m

Ztm

ðt m  τÞα dτ

0 1α

¼

ðt Þ α u ðt Þ m Γð1  αÞ α m 1  α

whence 

1=α α tm  ¼ t1 : Γ ð2  αÞ

3.9

3.9.1

Application of the Nonlinear Laplace Transform to Calculating the Wave Front Velocity in an Inhomogeneous Hereditary Rod Formulation of the Problem

Consider an inhomogeneous rod located on the semi-axis x
0 and having the density ρ ¼ ρ(x). We suppose that in the rod under consideration stress σ and deformation ε are related by σ ¼ AðxÞ ½ε  Rðt Þ  ε

ð3:9:1Þ

1 ½σ þ Λðt Þ  σ  AðxÞ

ð3:9:1’0 Þ

or which is the same, ε¼

Here A(x) is the instantaneous module of elasticity, R(t) the relaxation kernel, and Λ(t) the creep kernel. The relation between R(t) and Λ(t) is given by the formula (see Sect. 1.1.1)

116

3 The Wave Equation with Memory

1 ¼ 1 þ Λðt Þ 1  Rðt Þ As everywhere in this chapter 1 Λðt Þ ¼ φðt Þ þ φðt Þ  φðt Þ 4

ð3:9:2Þ

where φ(t) satisfies conditions (a)–(e) from Sect. 3.1. This section is devoted to the problem of calculating the longitudinal wave front velocity in the rod under consideration. Let us write out equations of the rod motion: 1 ∂σ ∂v ∂v ∂ε ¼ ; ¼ ρðxÞ ∂x ∂t ∂x ∂t

ð3:9:3Þ

where v is the velocity of the rod element. Eliminating v from the system (3.9.3) and substituting σ for its expression (3.9.1), we arrive at the following wave equation with memory and variable coefficients: 2

∂ ε ∂ 1 ∂  fAðxÞ ½ε  Rðt Þ  εg ¼ 0 ∂t 2 ∂x ρðxÞ ∂x or which is the same, 2

½1 þ Λðt Þ

∂ ε ∂ 1 ∂ ½AðxÞε ¼ 0  ∂t 2 ∂x ρðxÞ ∂x

ð3:9:4Þ

Let us set the following problem for Eq. (3.9.4): εðt, xÞ ¼ 0 for x > 0, t  0 εðt, 0Þ ¼ ε0 ðt Þ

ð3:9:5Þ

where ε0(t) ¼ 0, for t < 0 and ε0(+0) > 0. In what follows, it will be convenient for us to take ε0(t) as the function 1 1 1 h i ε0 ðt Þ ¼ L1 p!t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Lp!t p 1 þ φð2pÞ P 1 þ ΛðpÞ

ð3:9:6Þ

It is easy to see that when calculating the wave front velocity, we can restrict ourselves to the special case (3.9.6) without loss of generality. Note For completeness of exposition, let us recall the classical method of calculating the wave front velocity in the problem under consideration in case of regular

3.9 Application of the Nonlinear Laplace Transform to Calculating the Wave Front. . .

117

function of memory. In this case, the line of wave front, t ¼ t(x), issues out of the origin of coordinates. In front of the wave front ε ¼ σ ¼ 0, while across the line t ¼ t (x), ε and σ suffer finite jumps (see [10]). Let us introduce the following notation for the jump of f(t) across the wave front t ¼ t(x): ⟦ f⟧  f ðtðxÞ þ 0, xÞ  f ðtðxÞ  0, xÞ: As is well known, the equations of motion (3.9.3) must be supplemented with the following conditions on the wave front 1 ⟦σ⟧ ¼ c⟦υ⟧; ⟦υ⟧ ¼ c⟦ε⟧ ρð x Þ

ð3:9:7Þ

where c is the front velocity. Besides, by virtue of continuity of the convolution R (t)  ε, it follows from Eq. (3.9.1) that ⟦σ⟧ ¼ AðxÞ⟦ε⟧:

ð3:9:8Þ

Eliminating ⟦σ⟧ and ⟦υ⟧ from (12.6, 12.7), we have AðxÞ ⟦ε⟧ ¼ c2 ⟦ε⟧: ρð x Þ

ð3:9:9Þ

Now, by cancellation by the nonzero jump ⟦ε⟧, we obtain the wave front velocity, sffiffiffiffiffiffiffiffiffiffiffi AðxÞ c¼ : ρð x Þ

ð3:9:10Þ

(We have chosen the sign “plus” in front of the square root because, in the problem under consideration, the wave evidently propagates to the right.) Now, note that in case of singular memory ⟦σ⟧ ¼ ⟦ε⟧ ¼ 0 (see Sect. 3.3). Therefore, the relation (3.9.9) takes on the form AðxÞ

0 ¼ c2 0 ρð x Þ from which one cannot determine the wave front velocity. Surely, by analogy with the homogeneous case (see Sect. 3.1), it is natural to expect that for an inhomogeneous rod with singular memory the wave front velocity will be equal to the instantaneous elastic one, that is, (3.9.10). But how should one prove this fact in absence of an explicit formula for εðp, xÞ?

118

3.9.2

3 The Wave Equation with Memory

Solution of the Problem

Let us apply the Laplace transform Lt ! p to the problem (3.9.4, 3.9.5, and 3.9.6). On account of Eq. (3.9.2), we have  2 φð p Þ d 1 d p2 1 þ εðp, xÞ  ½AðxÞ εðp, xÞ ¼ 0 2 dx ρðxÞ dx

ð3:9:11Þ

and 1 i: εðp, 0Þ ¼ h p 1 þ φð2pÞ

ð3:9:12Þ

Since Eq. (3.9.11), generally speaking, cannot be integrated in quadratures, at the first glance, it is impossible to make use of the techniques developed above. Our main idea is to introduce a nonlinear change of the Laplace variable by formula   φð p Þ : q¼p 1þ 2

ð3:9:13Þ

Then (3.9.11, 3.9.12) takes the form q2 wðq, xÞ 

d 1 d ½AðxÞw ¼ 0, dx ρðxÞ dx wðq, 0Þ ¼

ð3:9:14Þ

1 q

where it is denoted wðq, xÞ  εðpðqÞ, xÞ:

ð3:9:15Þ

However, it is easy to see that Eq. (3.9.14) coincides with the Laplace transform Lt ! q of the following problem for a purely differential equation with variable coefficients

3.9 Application of the Nonlinear Laplace Transform to Calculating the Wave Front. . .

119

2

∂ w ∂ 1 ∂ ½AðxÞw ¼ 0;  ∂t 2 ∂x ρðxÞ ∂x ∂w ¼ 0, for x > 0, t ¼ 0; w¼ ∂t ( 1 for t > 0 wðt, 0Þ ¼ , 0 for t < 0

ð3:9:16Þ

We suppose A(x) and ρ(x) to be such that Eq. (3.9.16) has a solution w(t, x) smooth behind the wave front Zx  t ¼ t ð xÞ ¼

AðxÞ ρð x Þ

1=2 dx

ð3:9:17Þ

0

and satisfying the estimate w0 t ¼ o eMt , M > 0, for large t. As is known, on the wave front the solution w(t, x) suffers the jump  ⟦w⟧ ¼

ρðxÞ ρð0Þ

1=4 

A ð 0Þ AðxÞ

3=4

> 0:

However, from the above-mentioned equation, it follows that jwðq, xÞj 

const tðxÞ Re q e , j qj

for Re q > M > 0

ð3:9:18Þ

where const >0. Besides, for real q, the following estimate holds wðq, xÞ 

eqtðxÞ ⟦w⟧; q ! þ1 q

ð3:9:19Þ

(see [11]). Now, let us return to the variable p by formula (3.9.13). Then (3.9.18) yields  

    φð p Þ const  i  exp t ðxÞ Re p 1 þ jεðp, xÞj  wðqðpÞ, xÞj   h φðpÞ  2  p 1 þ 2  ð3:9:20Þ for

120

3 The Wave Equation with Memory

   φð p Þ >M Re p 1 þ 2 But by virtue of Lemma 1.5.4 Re ½pφðpÞ
0,

for Re p > 0:

ð3:9:21Þ

Clearly (3.9.20, 3.9.21) yield const  tðxÞ i e jεðp, xÞj   h φðpÞ  p 1 þ 2 

Re p

, for Re p > M:

ð3:9:22Þ

Hence by Theorem 1.3.1, it follows that ε(t, x) ¼ 0 for t < t(x). (This fact was established in [12].) Finally, let us demonstrate that t ¼ t(x) is the precise equation of the wave front for ε(t, x). Suppose the contrary. Then for some x > 0, there must exist δ1 > 0 such that ε(t, x) ¼ 0 for t  t(x) + δ1. Then by Theorem 1.3.1, we must have the following estimate jεðp, xÞj  C 1 ð1 þ jpjÞυ1 e½tðxÞþδ1  Re p ,

for Re p > M 1

where C1, ν1, M1 are some positive constants. But, the last estimate contradicts Eq. (3.9.19). The contradiction obtained demonstrates that in inhomogeneous rods with singular memory wave fronts propagate precisely at instantaneous elastic velocity. Note that we have obtained the solution of our problem by reducing it to a classical one by applying the nonlinear Laplace transform εðt, xÞ ¼ L1 p!t Lt!qðpÞ wðt, xÞ:

3.10

ð3:9:23Þ

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Line Source

As is well known, the classical two-dimensional Cagniard–de Hoop method (see [13, 14]) enables us to analytically solve problems about wave propagation generated by a line source in linear elastic layered media (see [13]). In this section, we give a generalization of the mentioned method to the hereditary case. For simplicity, we demonstrate the idea of our generalization in solving the simplest problem about shear wave propagation from a line source in an isotropic homogeneous hereditary

3.10

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Line. . .

121

space. It should be noted that our exposition is not mathematically rigorous and presents a refined variant of [15]. So, let us consider an isotropic homogeneous hereditary elastic xyz-space of density ρ ¼ const > 0. As we know, equations of motion of the medium under consideration have the form (see 1.1.10) ρ

2!  ∂ u ! ½ ð Þ  ½ ð Þ  ¼ f λ 1  q t þ 2μ 1  h t g∇ ∇ u ∂t 2  !

!

ð3:10:1Þ

μ½1  hðt Þ∇ ∇ u þ f





! ! Here u ¼ ux , uy , uz is the vector of displacement, f ¼ f x , f y , f z the body force;λ > 0, and μ > 0 are the instantaneous constants of Lamé, and q(t) and h(t) are the corresponding relaxation kernels. Let !

f ¼ ð0, A δðxÞ δðzÞ δðt Þ, 0Þ

ð3:10:2Þ

where A ¼ const 6¼ 0, and suppose !

u ¼ 0,

for t < 0:

ð3:10:3Þ

Clearly, in such a setting ux and uz will remain identically equal to zero. For uy (3.10.1) yields 2

2

2

∂ uy ∂ uy ∂ uy  β20 ½1  hðt Þ þ 2 ∂t 2 ∂x2 ∂z

! ¼

A δ ð t Þ δ ð xÞ δ ð z Þ ρ

ð3:10:4Þ

where β0 ¼

1=2 μ ρ

Let Λ(t) be the creep kernel corresponding to the relaxation kernel h(t) 1 ¼ 1 þ Λðt Þ 1  hðt Þ Then Eq. (3.10.4) can evidently be rewritten in the form

ð3:10:5Þ

122

3 The Wave Equation with Memory 2

2

2

∂ uy ∂ uy ∂ uy ½1 þ Λðt Þ  β20 þ 2 ∂t 2 ∂x2 ∂z

! ¼

A ½1 þ Λðt Þ δðt Þ δðxÞ δðzÞ ρ

ð3:10:6Þ

Just like Eq. (3.1.3), we suppose Λ ð t Þ ¼ φð t Þ þ

1 φð t Þ  φð t Þ 4

ð3:10:7Þ

where φ(t) satisfies conditions (a)–(e) from Sect. 3.1. In the manner of Theorem 2.4.1, one can demonstrate that under the condition (3.10.7), the operator in (3.10.6) is hyperbolic and describes wave propagation at speed β0. Our purpose is to derive, in the vicinity of the wave front, an asymptotic formula for the solution uy of (3.10.3, 3.10.6, 3.10.7) without applying any integral transformation with respect to z. We leave it for the reader to show that our approach really proves to be applicable for solving wave problems in layered (with respect to z) hereditary media. Let us apply the Laplace transform Lt ! p and the Fourier transform Fx ! ξ to (3.10.6, 3.10.7). After solving the resulting ordinary differential equation (with respect to z), one easily obtains on account of (3.10.3): F x!ξ Lt!p uy ¼

A enjzj 2ρβ2 ðpÞn

ð3:10:8Þ

where it is denoted as  1=2 β0 p2 n  ξ2 þ 2 , β ð pÞ  : β ð pÞ 1 þ φð2pÞ

ð3:10:9Þ

Note Here n
0 for ξ and p real. Below, we extend the definition of n to the case of complex ξ by using the restriction Re n > 0. On applying the inverse Fourier transform F 1 ξ!x to (3.10.8), we have A uy ðp, x, zÞ ¼ 4πρβ2 ðpÞ

Z1 1

eiξxnjzj dξ: n

ð3:10:10Þ

Furthermore, just like in the classical Cagniard–de Hoop method, we introduce the ray parameter, s, by formula ξ ¼ isp: From (3.10.9, 3.10.10, and 3.10.11), it follows that

ð3:10:11Þ

3.10

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Line. . .

Ai uy ðp, x, zÞ ¼  4πρβ2 ðpÞ

Zi1 i1

123

epðsxþηjzjÞ ds η

where η  [β2( p)  s2]1/2, Re η > 0. As in [13], from the last formula, one easily obtains Zi1

A uy ðp, x, zÞ ¼ Im 2πρβ2 ðpÞ

epðsxþηjzjÞ ds: η

ð3:10:12Þ

0

Now, we fix some p > 0, and some real x and z, and introduce the time-like variable τ ¼ τ(s) by formula  1=2 τ ¼ sx þ β2 ðpÞ  s2 jzj:

ð3:10:13Þ

In doing this, we suppose τ to run over the interval 0 < τ < 1 . Let H  (x2 + z2)1/2. Then solving Eq. (3.10.13) for s gives, in particular, the root h 2 i1=2 xτ  jzj βH2 ðpÞ  τ2 H s¼ , , for 0 < τ  β ð pÞ H2 h i1=2 2 xτ þ ijzj τ2  βH2 ðpÞ H ¼ : , for τ > 2 β ð pÞ H

ð3:10:14Þ

The equalities (3.10.14) determine the generalized Cagniard’s contour cp : s ¼ s (τ). Furthermore, just like in [13], (3.10.12, 3.10.13, 3.10.14) yield uy ðp, x, zÞ ¼

A Im 2πρβ2 ðpÞ

Z

epðsxþηjzjÞ ds: η

Cp

The part of Cp between τ ¼ 0 and τ ¼ H/β( p) gives no contribution into the last expression for uy ðp, x, zÞ, since the integrand is real along it. Besides, we have ds ¼ dτ h

iη 2

τ2  βH2 ðpÞ

i1=2

Along Cp, for τ > H/β( p). Therefore, on account of (3.10.14), the previous expression for uy can be rewritten as

124

3 The Wave Equation with Memory

A uy ðp, x, zÞ ¼ 2πρβ2 ðpÞ

Z1 H=βðpÞ

epτ dτ h i1=2 : 2 τ2  βH2 ðpÞ

ð3:10:15Þ

Now, making a change τ!τ

β0 β ð pÞ

in (3.10.15) and taking into account (3.10.9), one easily obtains n h i o Z1 exp p 1 þ φðpÞ τ 2 A up ðp, x, zÞ ¼ dτ:  2 1=2 2πρβ2 ðpÞ τ2  Hβ2 H=β0

ð3:10:15’Þ

0

Clearly, the integral on the right presents the nonlinear Laplace transform. Our idea is to qualitatively describe the behaviour of uy(t, x, z) near the wave front t ¼ H/β0 by means of asymptotics of uy ðp, x, zÞ for large p > 0. It is clear that for large p > 0, only a small part of the integration path H H < τ < þ δ, β0 β0 gives the main contribution into the integral (3.10, 3.150 ). Along the mentioned part of the integration path, one can consider H2 τ  2 β0 2

!1=2

¼

τ

H β0

1=2

1=2 1=2

1=2 H 2H H τþ  τ : β0 β0 β0

Hence for p ! + 1 , A

uy ðp, x, zÞ 

2πρβ ðpÞ 2

 1=2 2H β0

n h i o Z1 exp p 1 þ φðpÞ τ 2 dτ:  1=2 H τ  H=β0 β 0

Letting τ1 ¼ τ  H/β0 in the last integral and integrating, one easily obtains for p! + 1,

3.10

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Line. . .

uy ðp, x, zÞ 

AΓ 2πρβ

3=2

n h i o exp p 1 þ φð2pÞ βH

1 2

ð3:10:16Þ

0

ð2H Þ

1=2

125

p1=2

Now, to be specific, suppose φð t Þ ¼

kt α , Γð1  αÞ

for t > 0

ð3:10:17Þ

where k > 0, 0 < α < 1. Then φðpÞ ¼ k pα1 and (3.10.16) assumes the form

uy ðp, x, zÞ 

AΓ ð1=2Þ 3=2

2πρβ0 ð2H Þ1=2



 exp  p kH 2β pH 0 exp  , β0 p1=2

ð3:10:18Þ

for p ! + 1 . The first multiplier on the right-hand side of (3.10.18) is independent of p. The second multiplier corresponds to the time shift t ! t  H/β0. Therefore, it suffices to find the asymptotics of the inverse Laplace transform of the third multiplier (as t ! + 0). Making use of technique of the method of the steepest descent [8], we obtain

L1 p!t

p   1 kH Hkα 2ð1αÞ exp  1 2β0 2β0  t 2ð1αÞ 1=2 p1=2 ½2π ð1  αÞ ( 1 ) α α Hk 1α 1α exp  α ð1  αÞ t 1α , 2β0

ð3:10:19Þ

for t ! + 0. Therefore, neglecting the difference between the two sides of (3.10.18), we can expect that α

" 1

exp const2 H 1α

uy ðt, x, zÞ  const1 2ð1αÞ H # α

1α H t , β0



H t β0

for t ! (H/β0) + 0, where 1

const1 ¼ and

A ðkαÞ2ð1αÞ 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , πρð2β0 Þ2þ2ð1αÞ 2ð1  αÞ

1 2ð1α Þ

ð3:10:20Þ

126

3 The Wave Equation with Memory

const2 ¼

k 2β0

1 1α

α

α1α ð1  αÞ:

Note By applying L1 p!t (3.10.15’), one easily obtains the following exact formula A uy ðt, x, zÞ ¼ ½1 þ Λðt Þ 2πμ

Z1 H=β0

A ½1 þ Λðt Þ  2πμ

Zt H=β0

pφðpÞ ðτ=2Þ L1 p!tτ e dτ  2 1=2 τ2  Hβ2

ð3:10:21Þ

0

L1 p!tτ

e

pφðpÞ ðτ=2Þ

 2 1=2 τ2  Hβ2

dτ:

0

Since pφðpÞðτ=2Þ L1 ¼ 0, p!tτ e

for t < τ,

the integral on the right-hand side of Eq. (3.10.21) vanishes for t < H/β0. Therefore, Eq. (3.10.21) can be rewritten as 2

 Zt A H 6 uy ðt, x, zÞ ¼ ½1 þ Λðt Þ4Θ t  2πμ β0 H=β0

pφðpÞð2τ Þ L1 p!tτ e  2 1=2 τ2  Hβ2 0

3 7 dτ5: ð3:10:22Þ

In case where Eq. (3.8.1) takes place, Eq. (3.10.22) obviously yields 8 >  Zt <

A H uy ðt, x, zÞ ¼ ½1 þ Λðt Þ Θ t  2πμ β0 > :

H=β0

 9 tτ > = uα ðkτ=2Þ1=α : dτ  1=2 > 2 ; τ2  Hβ2

ð3:10:23Þ

0

Here pα uα ðt Þ  L1 (see Sect.3.8). p!t e

3.11

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Point Source

This section presents a continuation of Sect. 3.10. Just like in [13], we consider the SH-wave propagation generated by a point source buried in a hereditary elastic space.

3.11

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Point. . . 127

Let x, y, z be Cartesian coordinates in the hereditary space under consideration, r,  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ, z the corresponding cylindrical coordinates r ¼ x2 þ y2 : Let the point source of the rotation center type be located at the origin of coordinates. Then in the Cartesian coordinates, the body force can be represented as !

f ¼ ∇ ð0, 0, X Þ

ð3:11:1Þ

where X is an axisymmetric potential (that is, X ¼ X(t, r, z) in the cylindrical coordinates). In the Cartesian coordinates, the corresponding field of displacements can be written as [13] !

u ¼ ∇ ð0, 0, χ Þ

ð3:11:2Þ

where χ also is an axisymmetric potential. From (3.11.2), one easily obtains the ! cylindrical coordinates of u ur ¼ 0, uψ ¼ 

∂χ , uz ¼ 0: ∂r

ð3:11:3Þ

Using (3.11.1, 3.11.2) into Eq. (3.10.1), we arrive at the following wave equation with memory for the potential χ 2

∂ χ X  β20 ð1 þ hðt ÞÞΔχ ¼ ρ ∂t 2

ð3:11:4Þ

Here Δ denotes the Laplace operator with respect to x, y, z, and β0 ¼ (μ/ρ)1/2. As in Sect.3.10, we suppose 1 ¼ 1 þ Λðt Þ 1  hðt Þ where the creep kernel Λ(t) can be represented as 1 Λðt Þ ¼ φðt Þ þ φðt Þ  φðt Þ 4 with φ(t) satisfying conditions (a)–(e) from Sect.3.1. To be specific, just like in [13], we suppose X ¼ AΘðt ÞδðxÞδðyÞδðzÞ  AΘðt Þ where A ¼ const 6¼0

δ ðr Þ δ ðzÞ 2πr

ð3:11:5Þ

128

3 The Wave Equation with Memory

 Θðt Þ ¼

for t > 0 for t < 0

1 0

Moreover, we suppose !

u ¼ 0 for t < 0:

ð3:11:5’Þ

As in Sect 3.10, our purpose is to construct the near front asymptotics for uψ without applying any integral transformation with respect to z. Just like in the classical approach [13], we apply the multiple transform Z1 e

iξx

Z1 dx

1

e

iηy

1

Z1 dy

ept dt

0

to (3.11.4, 3.11.5, 3.11.6). Then we arrive at the ordinary equation d2 A F χ¼ 2 δðzÞ þ n2 F x,y!ξ,η χ dz2 x,y!ξ,η ρβ ðpÞp

ð3:11:6Þ

where n2  ξ2 þ η2 þ

β0 p2 : , β ð pÞ ¼ 2 β ð pÞ 1 þ φð2pÞ

Furthermore, under the restriction Re n
0, the solution of (3.11.6) describing finite speed wave propagation obviously has the form F x,y!ξ,η χ ¼

A enjzj 2ρβ2 ðpÞpn

whence A χ ðp, x, y, zÞ ¼ 2 2 8π ρβ ðpÞp

Z1

Zþ1 dξ 1

1

exp ðiξx þ iηy  njzjÞ dη: n

ð3:11:7Þ

Now, let us try to asymptotically calculate the right-hand side of Eq. (3.11.7), as p ! + 1 . We make use of the following de Hoop’s change of variables (see (3.15, 3.16):

3.11

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Point. . . 129

ξ ¼ ðw cos ψ  q sin ψ Þp,

η ¼ ðw sin ψ þ q cos ψ Þ p:

ð3:11:8Þ

Here ψ is the azimuth in the cylindrical coordinates introduced above in the xyzspace; q and w are new variables introduced instead of ξ and η. Since x ¼ r cos ψ, y ¼ r sin ψ, dξ dη ¼ p2dw dq (3.11.7, 3.11.8) yield A χ ðp, x, y, zÞ ¼ 2 2 2π ρβ ðpÞ

Z1

Zi1 dq Im

0

exp ðipwr  pNjzj Þ dq N

ð3:11:9Þ

0

where  1=2 N ¼ β2 ðpÞ þ q2 þ w2 ,

Re N > 0:

Furthermore, just like in [13], we set s ¼ iw. Then Eq. (3.11.9) assumes the form A χ ðp, x, y, zÞ ¼ 2 2 2π ρβ ðpÞ

Z1

Zi1 dq Im

0

exp ½pðsr þ Njzj Þ ds N

ð3:11:10Þ

0

where now  1=2 N ¼ β2 ðpÞ þ q2  s2 ,

Re N > 0:

Now, we recall that in Sect 3.10, it was shown that Zi1 Im 0

exp ½pðsx þ N 1 j z j ds ¼ N1

Z1 H=βðpÞ

epτ dτ h i1=2 2 τ2  βH2 ðpÞ

where  1=2

1=2 , Re N 1 > 0; H ¼ x2 þ z2 N 1 ¼ β2 ðpÞ  s2 Replacing here x by r ¼ (x2 + y2)1/2 and β2( p) by β2( p) + q2, one can easily obtain

130

3 The Wave Equation with Memory

Zi1 Im

Z1

exp ½pðsr þ Njzj Þ ds ¼ N

0

R



pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 β ðpÞþq2

epτ dτ  1=2 τ2  R2 β2 ðpÞ þ q2

where R ¼ (x2 + y2 + z2)1/2. Therefore, Eq. (3.11.10) can be rewritten as A χ ðp, x, y, zÞ ¼ 2 2 2π ρβ ðpÞ 

Z1

Z1 dq

0

 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 β

R

ðpÞþq2

epτ dτ

τ2  R β2 ðpÞ þ q2

1=2

ð3:11:11Þ

By interchanging the order of integration in Eq. (3.11.11), one obtains h A χ ðp, x, y, zÞ ¼ 2 2 2π ρβ ðpÞ 

τ2

Z1 dτ e

pτ

R=βðpÞ



dq 2

 R β ð pÞ þ q2 2

τ2  1 R2 β2 ðpÞ

i1=2

Z 0

1=2

ð3:11:12Þ

Clearly, the external integral in Eq. (3.11.12), which has the type of Z

 ! ! F p, τ, x epτ dτ, x ¼ ðx, y, zÞ,

Gðp, x Þ !

can be treated as the nonlinear Laplace transform, if one makes the following change of the integration variable  ! G 1, x τ !  ! τ: G p, x Such an approach is rather natural when one is considering the case of a general layered hereditary medium. However, in the particular case of a homogeneous hereditary space, the appropriate integral can be calculated exactly. In fact, one can easily see that the inner integral in Eq. (3.11.12) equals π/2R [13]. Now, by inserting this value of the inner integral into Eq. (3.11.12), one obtains

3.11

Asymptotic Generalization of the Cagniard–de Hoop Method: The Case of a Point. . . 131

A χ ðp, x, y, zÞ ¼ 4πRρβ2 ðpÞ

Z1

epτ dτ ¼

R=βðpÞ

A epR=βðpÞ 4πRρβ2 ðpÞp

or which is the same,

χ ðp, x, y, zÞ ¼

h i2 A 1 þ φð2pÞ 4πRμp

  φð p Þ R exp p 1 þ 2 β0 



ð3:11:13Þ

Therefore, by virtue of the relation R ¼ (r2 + z2)1/2, (3.11.3, 3.11.13) yield 1 0 h i2 φðpÞ     A 1 þ 2 φðpÞ R C ∂B exp p 1 þ uψ ¼  @ A 2 β0 4πRμp ∂x Ar epR=β0 epφðpÞ R=2β0 , 4πR2 μβ0



ð3:11:14Þ

as p ! þ1,

since φðpÞ ! 0, as p ! þ1:. Now, as in Sect.3.10, we consider the special case where φð t Þ ¼

kt α , Γð1  αÞ

for t > 0:

ð3:11:15Þ

Here k > 0, 0 < α < 1, Γ is the Gamma function. Then φðpÞ ¼ kp1α , whence (3.11.14) takes on the form uψ ðp, x, y, zÞ 

α Ar pR p kR e β0 e 2β0 , as p ! þ1: 2 4πR μβ0

Hence uψ ðt, x, y, zÞ 

α Ar L1 ep kR=ð2βo Þ , 4πR2 μβo p!tR=βo

as

t

R ! þ0: β0

ð3:11:16Þ

Therefore, (3.11.16) yields

1 α 2ð1αÞ 1 R uψ ðt, x, y, zÞ  const1 rR2þ2ð1αÞ t  β0 "

 α # 1 R 1α 1α exp const2 R t β0 as t ! R/β0 + 0, where

ð3:11:17Þ

132

3 The Wave Equation with Memory 1

const1 ¼

A α2ð1αÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4πμβ0 2π ð1  αÞ



k 2β0



1 2ð1αÞ

and α

const2 ¼ ð1  αÞ α1α



k 2β0

1 1α

:

Note The contents of Chap. 3 is based on the results of [12, 15–25].

References 1. Lokshin, A. A. and Rok, V.E. Dokl. Akad. Nauk. SSSR 239, 1305 (1978). 2. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971) 3. Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946) 4. Seneta, E. (1976). Regularly Varying Functions. pp. 1–112. Lecture Notes in Mathematics, Vol. 508 Berlin: Springer. 5. Esseen, G.G.: Acta Math. 77, 1 (1945) 6. Hormander, L.: Linear Partial Differential Operators. Springer, Berlin (1963) 7. Follard, H.: Bull. Amer. Math. Soc. 10, 908 (1946) 8. Ibragimov, I.A., Linnik, J.V.: Independent and Stationarily Linked Random Variables, pp. 1–524. Nauka, Moscow (1965) 9. Lukacs, E.: Characteristic Functions. Griffin, London (1970) 10. Rabotnov, J.N.: Elements of Hereditary Mechanics of Solids, pp. 1–383. Nauka, Moscow (1977) 11. Fedoriuk, M.V.: Asymptotics: Integrals and Series. Nauka, Moscow (1987) 12. Lokshin, A.A., Sagomonyan, E.A.: Vestnik Mosc. Univ. ser. Mat. Mekh. 2, 87 (1989) 13. Aki, K., Richards, P.: Quantitative Seismology, Vol. 1. W. H. Freeman and Company, San Francisco (1980) 14. de Hoop, A.T.: Appl.Science Research. B8, 349 (1960) 15. Lokshin, A. A., Lopatnikov, S. L. and Rok, V. E. Izv. Akad. Nauk. SSSR MTT 5, 188 (1990). 16. Lokshin, A.A.: Prikl, Mat. Mekh. 1, 162 (1994) 17. Lokshin, A.A., Suvorova, J.V.: Mathematical Theory of Wave Propagation in Media with Memory, pp. 1–151. Moscow University Press, Moscow (1982) 18. Lokshin, A. A. Dokl. Akad. Nauk. SSSR 240, 43 (1978). 19. Lokshin, A. A. Dokl. Akad. Nauk. SSSR 247, 812 (1979). 20. Lokshin, A. A. Vestnik Mosc. Univ. Ser, Mat. Mekh. 2, 93 (1979). 21. Lokshin, A.A.: Usp. Mat. Nauk. 1, 231 (1979) 22. Lokshin, A.A.: Vestnic Mose.Univ. Ser. Mat. Mech. 3, 70 (1979) 23. Lokshin, A.A.: Vestnik Mosc.Univ. Ser. Mat. Meckh. 1, 42 (1981) 24. Lokshin, A. A., Lopatnikov, S. L. and Sagomonyan, E. A. Izv. Akad. Nauk. SSSR. Fiz. Zemli 3, 80 (1991). 25. Vinigradova, O. S., Lokshin, A. A. and Rok, V. E. Izv. Akad. Nauk. SSSR MTT. 1, 152 (1989).

Appendix: The Near Source Behaviour of Fundamental Solutions for Wave Operators with Memory

In what follows we deal with a specific sort of singularities of fundamental solutions for wave operators with memory. These singularities do not travel along characteristics and remain located at the origin of the space coordinates.

A.1. The One-Dimensional Case Let E1(t, x) be the finite speed fundamental solution for the one-dimensional wave operator with memory. That is 2

2

∂ E1 ∂ E1
c2 ½1
hðt Þ ¼ δðt Þ δðxÞ ∂t 2 ∂x2

ðA:1:1Þ

where h(t) is the relaxation kernel and c the instantaneous elastic velocity. Let Λ(t) be the creep kernel corresponding to h(t). Then (A.1.1) is obviously equivalent to 2

½1 þ Λðt Þ

2

∂ E1 ∂ E1
c2 ¼ ½1 þ Λðt Þ δðt Þ δðxÞ: ∂t 2 ∂x2

ðA:1:10 Þ

As in Chap. 3, we suppose Λðt Þ ¼ φðt Þ þ

1 φð t Þ  φð t Þ 4

ðA:1:2Þ

where φ(t) satisfies conditions (a)–(e) from Sect. 3.1. In the manner of Sect. 3.1, one easily obtains the following formula for E1,

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2020 A. A. Lokshin, Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity, https://doi.org/10.1007/978-981-15-8578-4

133

Appendix: The Near Source Behaviour of Fundamental Solutions. . .

134

E 1 ðt, xÞ ¼ F
1 λ!t

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  e ðλÞ 1þΛ e ðλÞ jxj , exp
iλ 1 þ Λ 2iλc c

ðA:1:3Þ

or, which is the same by virtue of (A.1.2) E 1 ðt, xÞ ¼ F
1 λ!t

    1 þ eφð2λÞ e ðλÞ jxj φ exp
iλ 1 þ 2 2iλc c

ðA:1:30 Þ

where λ = μ
ip, p > 0.It can be shown that in the vicinity of x = 0 E1(t, x) is continuous in x for t > 0 (see the proof of Theorem 3.3.1). For simplicity, during the whole of the Appendix we suppose φ(t) to have a singularity(as t ! + 0), which is stronger than the logarithmic one. Then, from (A.1.30 ) and the results of Sect. 3.7, it follows that E1(t, x) is infinitely differentiable in t, x, for x 6¼ 0. Therefore, (A.1.3) yields

lim

x!0

8qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 1þΛ e ðλÞ

∂ ∂ E 1 ðt, xÞ ¼ lim F
1 λ!t x!0 ∂x ∂x : ¼  F
1 λ!t

2iλc

9 pffiffiffiffiffiffiffiffiffiffiffi = e jx j e
iλ 1þΛðλÞ c ;

e ðλÞ 1þΛ 2c2

whence it follows that lim

x!0

Λ ðt Þ ∂ E1 ðt, xÞ ¼  2 , 2c ∂x

for t > 0:

ðA:1:4Þ

Thus, we have found a specific singularity of E1(t, x) which is located at x = 0.

A.2. The Three-Dimensional Case Let E3(t, x, y, z) be a finite speed fundamental solution for a three-dimensional wave operator with memory. That is 2

∂ E3
c2 ½1
hðt ÞΔE3 ¼ δðt Þ δðxÞ δðyÞ δðzÞ, ∂t 2 or

ðA:2:1Þ

Appendix: The Near Source Behaviour of Fundamental Solutions. . . 2

½1 þ Λðt Þ

∂ E3
c2 Δ E3 ¼ ½1 þ Λðt Þ δðt Þ δðxÞ δðyÞ δðzÞ, ∂t 2

135

ðA:2:10 Þ

Here Δ is the Laplace operator with respect to x, y, 0z; h(t) and Λ(t) are the relaxation and creep kernel, respectively; c is the instantaneous elastic velocity for the appropriate type of waves. By applying the Fourier–Laplace transform to (A.2.10 ) and transition to spherical coordinates, one can easily obtain E3 ðt, x, y, zÞ ¼ F
1 λ!t

e ðλÞ
iλpffiffiffiffiffiffiffiffiffiffiffi 1þΛ 1þe ΛðλÞð R=cÞ e 4πRc2

ðA:2:2Þ

where R = (x2 + y2 + z2)1/2. Since E3 proves to be a spherically symmetric function, in what follows, we denote it as E3(t, R). From (A.1.4), (A.2.2), it follows that E 3 ðt, RÞ ¼


1 ∂ E ðt, RÞ 2πR ∂R 1

ðA:2:3Þ

Hence (A.1.4) yields E3 ðt, RÞ 

Λ ðt Þ , 4πRc2

as R ! þ0, t > 0,

ðA:2:4Þ

provided Λ(t) 6¼ 0, for t > 0.

A.3. The Two-Dimensional Case Now, let E2(t, x, y) be a finite speed fundamental solution for a two-dimensional wave operator with memory. That is 2

∂ E2
c2 ½1
hðt Þ Δ E 2 ¼ δðt Þ δðxÞ δðyÞ, ∂t 2

ðA:3:1Þ

or 2

½1 þ Λðt Þ

∂ E2
c2 Δ E2 ¼ ½1 þ Λðt Þ δðt Þ δðxÞ δðyÞ: ∂t 2

ðA:3:10 Þ

Here Δ is the Laplace operator with respect to x, y. We are going to construct the fundamental solution E2(t, x, y) by the Hadamard’s method of descent. That is

Appendix: The Near Source Behaviour of Fundamental Solutions. . .

136

Zþ1 E 2 ðt, x, yÞ ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E3 t, x2 þ y2 þ ðz
ξÞ2 dξ


1

Z1 ¼

Z1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 E 3 t, x þ y þ z dz ¼ E 3 ðt, RÞ dz


1


1

Z ¼

E 3 ðt, RÞ dz Rct

pffiffiffiffiffiffiffiffiffiffiffiffi ðZ ct Þ2
r2 ¼2

E 3 ðt, RÞ dz

ðA:3:2Þ

0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where R ¼ x2 þ y2 þ z2 , r ¼ x2 þ y2 :Let us introduce the following change of variables in (A.3.2) z¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2
r 2 ,

r ¼ const:

Then, on account of the relation RdR dz ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , R2
r 2 Eqs. (A.3.2) and (A.2.3) yield Zct E2 ðt, x, yÞ ¼ 2 r

RE3 ðt, RÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dR ¼
2 π 2 R
r

Zct r

1 ∂ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 1 ðt, RÞ dR: 2 2 ∂R R
r

ðA:3:3Þ

In accordance with Eq. (A.3.3), we denote E2 as E2(t, r). Then, from Eq. (A.3.3) and (A.1.4), it easily follows that E 2 ðt, r Þ

Λ~ðt Þ ln r, 2πc2

r ! þ0, t > 0,

ðA:3:4Þ

provided Λ(t) 6¼ 0, for t > 0. Note From Eqs. (A.1.4), (A.2.4), and (A.3.4), it is clear that wave operators with memory possess certain properties of the Laplace operators.