World Congress of Nonlinear Analysts '92: Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, August 19-26, 1992 [Reprint 2011 ed.] 9783110883237, 9783110132151

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World Congress of Nonlinear Analysts '92

World Congress of Nonlinear Analysts '92 Proceedings of the First World Congress of Nonlinear Analysts Tampa, Florida, August 19-26,1992

Editor

V. Lakshmikantham

Volume I

W DE

Walter de Gruyter · Berlin · New York 1996

Editor

V. Lakshmikantham Program of Applied Mathematics Florida Institute of Technology 150 West University Boulevard Melbourne, FL 32901-6988 USA 1991 Mathematics Subject Classification: OOBxx, 34-XX, 35-XX, 46-XX, 47-XX, 49-XX, 60-XX, 65-XX, 73-XX, 90-XX, 92-XX, 93-XX Keywords: Nonlinear analysis, nonlinear problems in diverse disciplines © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication Data World Congress of Nonlinear Analysts (1st · 1992. Tampa, Florida); proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida. August 19-26, 1992 / editor : V. Lakshmikantham. p. cm. Includes bibliographical references and indexes. ISBN 3-11-013215-X (alk. paper) 1. Nonlinear theories-Congresses. 2. Mathematical analysis - Congresses. I. Lakshmikantham, V. II. Title QA427.W67 1996 003'.75-dc20 95-44204 CIP

Die Deutsche Bibliothek - Cataloging-in-Publication Data World Congress of Nonlinear Analysts : World Congress of Nonlinear Analysts '92 : proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, August 19 - 26, 1992 / ed. V. Lakshmikantham. - Berlin ; New York: de Gruyter. ISBN 3-11-013215-X NE: Lakshmikantham, Vangipuram [Hrsg.] Vol. 1 (1996)

© Copyright 1995 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Printing: Ratzlow-Druck, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface

The problems of modern society axe both complex and interdisciplinary. Despite the apparent diversity of problems, however, often tools developed in one context are adaptable to an entirely different situation. Thus, in advancing the frontiers of knowledge on a variety of fronts, the mathematical scientists, in particular, can bring harmony amidst this diversity by exploring the ever present world of nonlinear phenomena. As we approach the end of the twentieth century, the evolution of our social and technological systems and subsystems is becoming more complex and interdependent. Moreover, we are on the threshold of a high-tech revolution as radical as the Industrial Revolution. Although this high-tech revolution that began after the end of the second world war and continues to accelerate today, has helped solve many technological and social problems, it has created new problems that are different in nature and global in character. For example, now there are many serious problems of worldwide consequence, such as the energy crisis, acid rain, environmental pollution, global warming, the spread of unknown diseases and the ever widening gap between the developed and developing countries. The interest in mathematical analysis of nonlinear phenomena has lead to increased interdependency and cooperation among scientists across traditional disciplinary barriers, and this collaboration has proved to be rewarding and stimulating. Also, industrial recognition of using nonlinear mathematical models for solving problems of technology is increasing because of the fast development of large scale computers. It is, therefore, clear that the attempt to understand the nonlinear world will dominate, in future years, large parts of mathematics in most areas of science, technology and various other disciplines. Future researchers will be able to utilize ultra high speed computers coupled with massive electronic data libraries to address complex global problems with resources that are more powerful and effective than those available today. Thus there is a definite need to have a forum, on a global basis, to forge unity in diversity and for bringing more cooperation and collaboration into the world community of nonlinear analysts. It is with this spirit, an "International Federation of Nonlinear Analysts" (IFNA) is established and the "First World Congress of Nonlinear Analysts" was held during August 19-26, 1992 in Tampa, Florida, USA under the auspices of IFNA. The present four volumes consist of the Proceedings of the "First World Congress of Nonlinear Analysts". The main themes of the Congress are: (i) to promote, encourage and influence more cooperation, understanding and collaboration in the world community of nonlinear analysts from various disciplines;

vi

V. Lakshmikantham

(ii) to forge unity in diversity by bringing together a variety of disciplines that attempt to solve the nonlinear phenomena and to understand the nonlinear world; (iii) to influence unification of ideas in order to shrink the boundaries of major disciplines; (iv) to create a sense of wholeness and participation in a common enterprise; (v) to assist interaction on a broader and deeper scale for the benefit of all disciplines; (vi) to highlight recent advances, to stimulate discussions and to share ideas for future trends in research from various disciplines from mathematics to medicine.

By any measure, the First World Congress was a great success. Despite unfavorable economic conditions and the threat of hurricane Andrew, over one thousand scientists from eighty countries participated. There were about forty principle speakers, over eighty organized sessions by experts from various disciplines, and several short communications. All in all, there were over seven hundred presentations. A remarkable aspect of the World Congress is its broad scope of topics covered. The topics include nonlinear operators, fixed point theory, bifurcation, critical point theory, variational methods, inertial manifolds, dynamical systems, evolution equations, various quantitative aspects of PDE and ODE, nonconvex analysis, set-valued analysis, stochastic analysis, numerical and computational aspects, control theory and optimization, nonlinear waves, fluid mechanics, combustion theory, computer vision, structural mechanics, manufacturing systems, environmental problems, neural networks, chaos, artificial intelligence, superconductivity, microelectronic devices, population dynamics, immune systems, biosystems, compartmental systems, bursting rhythms in biosciences, biochemical systems, ecology, enzyme systems, epidemiology and biomedicine. The first volume of the proceedings consists of various aspects of partial differential equations including problems in physics, fluid mechanics, combustion theory, structural mechanics, hysteresis, and atmospheric and marine sciences. Volume two incorporates a variety of problems in ordinary differential equations (deterministic and stochastic) as well as difference equations and dynamic systems. In addition, this volume includes neural networks, problems in superconductivity and microelectronic devices, computer vision and numerical and computational methods. Volume three is devoted to the theoretical aspects of nonlinear analysis such as nonlinear operators, nonconvex analysis, control theory and optimization, fixed point theory and evolution equations. Moreover, this volume also contains models in economic theory, manufacturing systems, engineering applications, chaos, bifurcation and artificial intelligence in nonlinear electronics. Finally, volume four deals with several aspects of biomathematics and ecology including biochemical systems, environmental problems, bursting rhythms and biomedicine.

Preface

vii

The proceedings also contains papers presented at "Lyapunov's and Poincare's Centenary sessions" as well as "Round Table meetings" in which experts from industry and academic institutions participated. The World Congress was supported by Embry-Riddle Aeronautical University, Florida Institute of Technology, Oak Ridge National Laboratory, Tampa Mayor's Office, UNESCO, University of Texas at San Antonio, U.S. Army Research Office and Walter de Gruyter Publishers. The organizers are immensely thankful for the institutions for their help. I wish to express my sincere thanks to all the organizers of the sessions of the congress, the referees, the proofreaders, and to Professors N.S. Papageorgiou and L. Krishnamurthy for their continued help for the proceedings and to Ms. Donn Harnish, without whose dedicated work, it would have been impossible to complete the four volumes of the proceedings. Finally, it is with great pleasure, I extend my thanks to Walter de Gruyter Publishers for undertaking this huge project and offering necessary cooperation. I would like to offer special thanks to my faculty, staff and students who did contribute their time, labor, and above all enthusiastic support. V. Lakshmikantham

Table of Contents

Volume I

Investigation of Oldrojt's viscoelastic model when viscous coefficient tends to zero P. E. Sobolevskii Nonlinear PDE problems in electrophotography A. Friedman

1

13

A special class of maximum principles with applications to nonlinear boundary value problems L. E. Payne

25

Nonlinear mathematics in nonacademic settings J. Bownds, D. Ross, J. Chandra, and P. Davis

41

Motion of a graph by nonsmooth weighted curvature T. Fukui and Y. Giga

47

Some results on heat flow in electric conductors S. N. Antontsev and M. Chipot

57

Global classical solutions to fully nonlinear wave equations T.-t. Li (Li Da-qian)

67

Solitons dipole chains E. Infeld,and G. domains Rowlands,in J. Μ. T. Thompson, and H. Zorski Nonlinear instabilities of steady and oscillatory wave convection in a rotating system

73

D. N. Riahi

79

Bifurcations and chaos in predator-prey models with delay S. Roy Choudhury On a new variant degree theory of mapping and its application M. A. Efendiev Solvability of semilinear operator equations and periodic solutions of differential equations P. S. Milojevic

87 111

125

χ

Table of Contents

Solutions of semilinear equations in cones and wedges J. R. L. Webb

137

On a nonlinear elliptic problem with subcritical and critical Sobolev exponent M. Zuluaga

149

Existence and bifurcation results for some semilinear elliptic equations on RN W. Rother

161

Some remarks about the existence of positive solutions for elliptic systems A. Canada and J. L. Gdmez

167

Sub- and supersolutions of nonlinear elliptic and parabolic problems K. Singh

175

Positive solutions and boundary value problems of singular and nonsingular type D. Ο'Regan

185

On the asymptotic behavior of laminar flow through a porous pipe C. Lu

201

The monotone method for third order boundary value problems A. Cabada

211

Duality and variational principles for nonlinear hyperbolic equations A. Nowakowski and A. Rogowski

223

Decay and global existence for some nonlinear dissipative wave equations M. Nakao

233

On the philosophy of the spectral variable R. Carroll

241

Wave propagation in uniaxial nonconducting nonlinear thermoelastic solids M. C. Singh and D. V. D. Tran

253

Approximation of dissipative hereditary systems R. C. MacCamy

269

Stefan problems in several space variables with dynamic boundary conditions T. Aiki

281

Existence and multiplicity of periodic solutions for semilinear parabolic equations N. Hirano

291

Boundary integral solution of a nonlinear heat conduction problem G. C. Hsiao and J. Saranen

299

On the existence of traveling waves in reaction-diffusion systems K. R. Schneider

311

Table of Contents

xi

The strong maximum principle for cooperative periodic-parabolic systems and the existence of principal eigenvalues I. Anton and J. Lopez-Gomez

323

Blowing-up of solutions of semilinear parabolic equations T. Itoh

335

An existence and uniqueness theorem for solutions of nonlocal parabolic partial differential equations D. Jackson

341

Open problems in nonlinear ordinary boundary value problems arising from the study of large-amplitude periodic oscillations in suspension bridges A. C. Lazer and P. J. McKenna

349

One species extinction in an autonomous competition model S. Ahmad and A. C. Lazer

359

Periodic stability and reachability of nonlinear parabolic equations under boundary periodic perturbations K. Naito

369

A criterion for asymptotic stability based on topological degree R. Ortega

383

Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type T. Ding and F. Zanolin

395

Periodic boundary value problems for impulsive integro-ordinary differential equations J. Nieto

407

Some multiplicity results for subharmonic solutions to second order nonautonomous systems M. Girardi and M. Matzeu

417

New results in quenching C. Y. Chan

427

On the life span of solutions to nonlinear wave equations M. A. Rammaha

435

Conversion I. Stakgold and penetration fronts in combustion The transition from decay to blow-up in some reaction-diffusion-convection equations M. Fila and P. Sacks

445

455

xii

Table of Contents

Numerical experiments on the Ginzburg-Landau equations Μ. K. Kwong

465

Counterexamples to the existence of inertial manifolds J. Mallet-Paret, G. Sell, and, Z. Shao

477

Inverse problems versus an algebraic spectral method for nonlinear evolution equations P. C. Sabatier

487

A finite difference scheme for an inverse heat equation J. R. Cannon, Y. Lin, and S. Xu

497

Nonlinear ill-posed problems A. N. Tikhonov, A. S. Leonov, and A. G. Yagola

505

Maximum entropy regularization of nonlinear ill-posed problems H. W. Engl and G. Landl Variational derivatives inZ.function E. P. Hamilton and Μ. Nashed Radial and nonradial solutions of aspaces nonlinear scalar field equation T. Bartsch

513 527 537

A geometric method for the periodic problem R. Srzednicki

549

Large perturbations of Toda lattices B. Ruf

561

Distribution of mass principle and its applications to nonlinear elliptic equations W. Chen

571

Attractors of bipolar and non-Newtonian viscous fluids F. Bloom

583

Nonlinear dynamics in the low Reynolds number cylinder wake D. J. Olinger

597

The present status of the theory of the flow of a compressible viscous fluid through a pipe or parallel-wall channel J. M. Russell

611

Laminar and turbulent film flows R. J. Gribben

623

Table of Contents

xiii

Sensitivity equations for the k — e turbulence model for ill-posed inverse problems in computational fluid dynamics L. Krishnamurthy

631

Large-time behavior of reaction-diffusion systems modeling combustion with complex chemistry J. D. Avrin

651

Stability analysis of the traveling wave solutions in burning solid propellants C. D. Pagani and M. Verri

661

Stability of travelling waves in combustion problems: a fully nonlinear approach C.-M. Brauner, A. Lunardi, and CI. Schmidt-Laine

667

On the numerical analysis of buckling of general thin shells M. Bernadou and B. Lalanne

681

Random fields in nonlinear structural problems E. Bielewicz, J. Gorski, and H. Walukiewicz

691

Geometrically nonlinear analysis of inelastic damaged shells I. Kreja, R. Schmidt, 0. Teyeb, and D. Weichert

701

Moderate of shells under circulatory loading W. Altmanrotations and A. F. Palmerio Further results on postbuckling behavior of shear deformable plates and shallow shells accounting for initial geometric imperfections Μ. - Y. Chang and L. Librescu

711

723

Failure prediction in composite laminates according to the layerwise plate theory Y. S. N. Reddy and J. N. Reddy

739

A theory for nonlinear static and transient analysis of composite laminates R. Schmidt

753

Numerical methods for ODEs with hysteresis L. Tavernini

765

P.D.E.s with hysteresis operators A. Visintin

775

Some limit results for relays Τ. I. Seidman The Mroz model: A hysteresis operator for rate-independent plasticity M. Brokate, Κ. Dressler, and P. Krejci

787 797

xiv

Table of Contents

A differential model of fatigue in elastoplastic materials P. Krejci

807

Orientable and nonorientable minimal surfaces J. Jost

819

Continuation S. S. Antman methods in nonlinear elasticity

827

Bifurcation for variational inequalities and applications to elasticity F. Schuricht

839

Infinite dimensional integrals in classical, stochastic and quantum nonlinear problems S. Albeverio

849

Asymptotic formulas in capillarity E. Miersemann

861

An envelope theorem for weak solutions of quasilinear elliptic boundary value problems S. Carl

869

Group theoretical methods for finite-difference modeling V. A. Dorodnitsyn

879

Two problems for the p-Laplacian P. A.parameter Binding and Υ. X. Huang Positive solutions of semilinear Neumann problems and Sobolev type inequalities Adimurthi and S. L. Yadava

891 901

Α free boundary eigenvalue problem R. Meyer-Spasche

911

Measure valued solution of non-Newtonian compressible heat conductive fluid S. Matusu-Necasovd

925

Asymptotics for a class of weakly nonlinear wave equations with applications to some problems W. T. van Horssen

945

Path optimization in flat and wrinkled labyrinths using arrays of active nonlinear circuits: traveling wave and its failure V. Perez-Munuzuri, V. Perez- Villar, J. L. F. Porteiro, and L. O. Chua

953

Table of Contents An Oregonator based model for the influence of a radial electric field on the dynamics of spiral waves J. L. F. Porteiro, V. Perez-Munuzuri, M. Gomez-Gesteira, A. Perez-Munuzuri, and, V. Perez-Villar

xv

965

Theorem about a weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities L. Byszewski

977

Products of distributions in mathematical physics A. Palanques-Mestre

989

A qualitative analysis of the potential and current driven semiconductor equations F. Alabau

1001

Author Index

1011

xvi

Table of Contents

Table of Contents of Volume II Differential-geometric computational methods for nonlinear problems W. Rheinboldt

1017

Set-valued analysis, viability theory and partial differential inclusions J. P. Aubin and H. Prankowska

1039

Global stability, injectivity, and the Jacobian conjecture G. H. Meisters and C. Olech

1059

Theory of multipolar fluids J. Necas

1073

Theory vs. relativity E. Messere

1083

Difference equations and inequalities: a survey R. P. Agarwal

1091

Invariant fiber bundles for nonautomous difference systems B. Aulbach

1109

Sign properties of Green's functions for difference equations P. W. Eloe and T. Peil

1121

Global stability of difference equations S. Elaydi Singular boundary value problems for higher order difference equations J. Henderson Asymptotic behavior and oscillation of general nonlinear difference equations E. Thandapani and B. S. Lalli

1131 1139 1151

Permanence and global attractivity in nonlinear difference equations V. L. Kode and G. Ladas

1161

Comparison theorems for a difference equation with memory A. Peterson and J. Schneider

1173

Dichotomies for differential systems with impulsive effect M. Pinto

1181

Soliton-like of the discrete wave equation F. Iavernaro solutions and D. Trigiante

1193

Examples of Lyapunov functionals for non-differentiated equations T. A. Burton

1203

Table of Contents

xvii

A Lyapunov-Razumikhin method for functional differential equations T. Furumochi

1215

Attractivity in nonautonomous equations by Lyapunov's method T. Yoshizawa

1223

New directions in the method of vector Lyapunov functions via cone valued Lyapunov functions A. S. Vatsala

1233

A conjecture in Lyapunov method for functional differential equations J. Kato

1239

Stability criteria for impulsive integro-differential equations M. Rama Mohana Rao and S. K. Srivastava

1247

Current state of the method of vector Lyapunov functions S. Leela

1255

New approach to the dynamic programming for optimal control problems with state constraints Μ. M. Khrustalev

1269

Methods of stability theory for singularly perturbed problems with applications to dynamics L. K. Kuzmina

1279

On stability of steady motions of rigid body with liquid-filled cavity V. V. Rumyantsev

1287

Some new trends in Lyapunov's direct method C. Corduneanu

1295

Uniform asymptotic stability in functional differential equations with finite delay B. Zhang

1303

Stability analysis of an HIV transmission model X. Liu

1313

Stability of general ecological systems and neural networks systems L. X. Xin

1325

On the asymptotic stability in differential systems by Lyapunov direct method L. Hatvani

1341

Stability and attraction in impulsive semidynamical systems S. K. Kaul

1349

xviii

Table of Contents

A universal integral invariant of non-autonomous dynamical systems V. Vujicic

1357

Convergence in neutral equations with infinite delay arising from active compartmental systems J. Wu

1361

An invaxiance principle of Lyapunov-Razumikhin type and compartmental systems T. Krisztin 1371 On the existence of almost periodic solutions of nonlinear compartmental systems with time delays H. Maeda, T. Takahashi, and S. Kodama

1381

On numerical solutions for a class of nonlinear delay equations with timeand state-dependent delays I. Györi, F. Härtung, and J. Turi

1391

Asymptotic periodicity in impulsive differential equations of retarded type with applications to compartmental models I. Györi

1403

Variation of parameters formula and Lipschitz stability of nonlinear matrix differential equations D. Fausett and S. Köksal

1415

Stability of a class of nonlinear continuous neural networks L. X. Xin

1427

Roughness properties of (h, ^-dichotomies R. Naulin and M. Pinto On the existence and qualitative properties of solutions for a nonlinear Neumann problem with critical exponent Z. Q. Wang

1437

1445

Behavior of nonoscillatory solutions of forced second order neutral delay equations J. R. Graef and P. W. Spikes

1457

Regularity and asymptotics of a similarity solution of an equation arising in boundary layer theory V. Marie and Z. Radasin

1469

On solutions of a neutral equation with an oscillatory coefficient J. R. Graef and P. W. Spikes

1475

Asymptotic behavior of nonlinear neutral differential equations Q. Kong

1483

Table of Contents On oscillation of nonlinear first order differential equations with its application H. Onose

xix

1497

Topological and variational approaches for nonlinear oscillation: an extension of a Bhatia result M. Cecci and M. Marini, and G. Villau

1505

Oscillation and nonoscillation of a class of neutral equations B. G. Zhang and K. Gopalsamy

1515

Positive nonlinear operators associated with isoperimetric eigenvalue problems in algebra P. J. Bushell

1523

Positive nonlinear systems: Some results and applications U. Krause

1529

Random products of contractions J. M. Dye, T. Kuczumow, and S. Reich

1541

Functional-differential equations with non-Volterra operator M. Drakhlin and E. Litsyn

1549

Searching for periodic orbits in a four dimensional symplectic map A. Olvera and C. Vargas

1557

Nonlinear Gravito-electrodynamics - An Einstein's dream A. Yu Sun (Y. Xin)

1565

A geometric approach to optimal control H. Schüttler A R. generalized I. Becker implicit function theorem for equations with symmetry Bifurcation analysis of nonuniform flow patterns in axial-flow gas compressors R. A. Adomaitis and Ε. H. Abed

1579 1591

Analysis of energy balance climate models G. Hetzer and P. G. Schmidt

1597 1609

A numerical method for solving nonlinear singular problems and application to bifurcation problems Z. Yang and R. Ye

1619

On the analysis of dynamic bifurcations and stability of solutions K. Huseyin and S. Wang

1627

XX

Table of Contents

Limit theorems for dependent random variables A. Bozorgnia, R. F. Patterson, and, R. L. Taylor

1639

The stochastic zero-sum differential games with measurement uncertainties Κ. M. Ramachandran

1651

On certain nonlinear difference and differential equations arising from optimal choice problems Z. Govindarajulu

1663

Near-optimal control of a class of stochastic systems under integral constraints Α. Ν. V. Rao and C. P. Tsokos

1677

Stability analysis of stochastic bilinear systems P. L. Wing and Α. Ν. V. Rao

1689

Existence of solutions of nonlinear stochastic differential inclusions of Banach space N. U. Ahmed 1699 Bimodal nonparametric density estimation H. Qiao and C. P. Tsokos

1713

Conditional percentiles and their estimates for the additive model Y. Wang

1727

Sequential allocation for a nonlinear estimation problem K. Rekab

1739

Ergodic theorems for modulated stochastic processes J. H. Dshalalow

1745

Some results in stochastic reaction-diffusion equations Μ. H. new Chang Stochastic evolution equations with non-coercive monotone operators F. Flandoli

1757 1765

Optimal control of arrival and service rates for two M / M / l queues in series M. S. Moustafa

1779

Repairable system with differential failure modes S. A. A. Emara and K. M. El-Said

1785

Differential inclusions and implicit equations on closed subsets of M" M. Frigon, L. Gorniewica, and T. Kaczynski

1797

Recent trends in the theory of nonlinear functional differential equations Ν. V. Azbelev

1807

Table of Contents

xxi

Spherically symmetric solutions of semilinear wave equations Y. Ebihara

1815

Exchange of stability and bifurcation of tori in four dimensions S. R. Bernfeld

1819

Multivalued differential equations with time-dependent constraints D. Bothe

1829

Weighted inequalities with an arbitrary convex function S. A. A. Emara

1841

Lower semicontinuity of multivalued mappings and equilibrium points K.-K. Tan andX. Z. Yuan

1849

Periodicity in delay difference systems S. Zhang

1861

A convexification algorithm for the modeling of nonlinear dynamic processes D. B. Silin and N. G. Trinko

1871

On the pulse phenomenon for impulsive differential equations Α. V. Anokhin and A. I. Chindyapin

1881

Fourier spectral methods for a class of generalized ATdV-Burgers equations with variable coefficients Z. Xiao and F. Huang

1887

Structure of elastoplastic shock wave and numerical computation A. Noussair

1899

Automatic contours detection in image processing F. Dibos, V. Caselles, T. Coll, and F. Catte

1911

Second generation J. Froment and S. image Mallat coding and wavelet transform Segmentation by minimizing functionals and the merging methods G. Koepfler

1923 1933

Equivalence of various characterizations of the oscillatory/nonoscillatory classification for singular endpoints of fourth order self-adjoint differential equations C. T. Fulton, S. Pruess, and L. Wu

1943

Taylor recurrence in analytic O.D.E.s F. S. Munger

1955

Asymptotic behavior of difference equations R. Medina and M. Pinto

1969

xxii

Table of Contents

Oscillation of solutions of arbitrary order neutral functional differential equations with forcing terms A. Zafer and R. S. Dahiya

1977

Method of Lyapunov-Lagrange functions for optimization of dynamic systems with information constraints S. V. Savastuk

1989

Approximation methods for eigenvalue problems of Fredholm integral equations Ε. B. Lin and X. Shen

2001

Mode interaction at a triple zero point of 0(2)-symmetric nonlinear systems with two parameters W. Wu

2011

Author Index

2023

Table of Contents

xxiii

Table of Contents of Volume III Nonlinear elliptic systems arising from control theory and differential games A. Bensoussan and J. Frehse

2029

Aspects of singular perturbation theory in physiology J. Cronin

2045

Quasi-variational systems J. Serrin

2055

New spectral I.representations I. Antoniou, Prigogine, andofS.mixing Tasakidynamical systems

2069

Differential inclusions without convexity: A survey of directionally continuous selections A. Bressan

2081

Some problems of existence,uniqueness and regularity in the calculus of variations P. Marcellini

2089

On the existence problem in the Calculus of Variations A. Cellina

2097

A class of upper semicontinuous differential inclusions without convexity G. Colombo

2107

Constructive aspects of fixed point theory for nonexpansive mappings T. Büber and W. A. Kirk

2115

On the growth properties of the homology of free loop spaces on configuration spaces E. Fadell and S. Husseini

2127

Borsuk-Ulam theorems, the Lusternik-Schnirelmann category and index theories T. Bartsch

2135

Monotone methods with applications to nonlinear analysis S. Heikkilä

2147

Spherical algorithms and evaluation of the fixed point set 9 E. A. Galperin

2159

Fixed points of acyclic maps on topological vector spaces S. Park

2171

Convergence Y. S. Lee andofS.accretive Reich operators

2179

xxiv

Table of Contents

Existence of solutions of nonlinear equations with perturbations of monotone operators in reflexive Banach spaces Z. Guan

2187

Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces A. G. Kartsatos

2197

Braess's paradox and power-law nonlinearities in networks II B. Calvert and G. Keady

2223

On the invertibility of J ± exp(—tA), t > 0, with A maximal monotone V. Barbu and Ν. H. Pavel

2231

A global existence theorem on nonlinear evolution equations Y. You

2239

Existence of solutions of nonlinear functional differential equations in Lp spaces K. Sik Ha and K. Shin

2249

The global attractor of the fifth order KdV equation J. Cao

2261

On the blow-up of solutions to some semilinear parabolic systems arising in chemical reaction modelling S. L. Hollis and J. J. Morgan

2273

Asymptotic behavior and periodic forcing in time for the solutions of a nonlinear system representing a phase change with dissipation A. Damlamian and N. Kenmochi

2281

Implicit functions defined by generalized equations A. L. Dontchev and W. W. Hager

2289

Non-smooth A. D. Ioffe subdifferentials: their calculus and applications Generalized differentiation and stability of solution maps to parametric variational systems

2299

B. Mordukhovich

2311

The linear nonconvex generalized gradient J.S. Treiman

2325

Second order necessary conditions for nonsmooth optimum problems with constraints Zs. Ρdies and V. M. Zeidan

2337

Table of Contents Wellposedness of second-order nonlinear abstract differential equations with multivalued boundary conditions: Applications to von Karmän systems I. Lasiecka

xxv

2347

Some stability results for vector optimization problems in partially ordered topological vector spaces Ε. M. Bednarczuk

2371

Some nonlinear results for abstract differential equations S. Zaidman

2385

Some applications and extensions of P-coercive variational inequalities D. D. Ang and L. K. Vy

2391

On semilinear wave equations at resonance J. Berkovits and V. Mustonen

2399

Approximate solvability of nonlinear equations and related results in control theory M. C. Joshi 2409 The Newton-Kantorovich method for nonlinear integral operators J. Appell

2419

A uniqueness theorem for a nonlinear integral equations of gravimetry D. D. Ang, R. Gorenflo, and L. K. Vy

2423

Applications of Pareto optimality of a mapping to mathematical economics E. Tarafdar

2431

On ^-minimal, P-monotone trajectories of nonautonomous differential inclusion in R ^ S. Hu and N. S. Papageorgiou

2441

On the solution of some nonlinear boundary value problem G. Bogndr

2449

Mathematical controllability theory of the growth of wealth of nations Ε. N. Chukwu

2459

Bilateral interactions and aggregate fluctuations S. N. Durlauf

2471

Short-run decisions on price, output, and inventories in a dynamic model of the firm R. J. Rossana

2481

Dynamical systems defined by infinite horizon optimal control problems M. Sieveking

2493

xxvi

Table of Contents

Geometry and nonlinear differential equations W. F. Shadwick

2503

Pareto solution of a cone variational inequality and Pareto optimality of a mapping E. Tarafdar

2511

Output tracking of nonanalytic trajectories for nonlinear systems L. R. Hunt, G. Meyer, and R. Su

2521

Linearization E. O. Roxin and approximation of control systems The shape design problem for systems with distributed parameters S. P. Ohezin

2531 2541

Optimality conditions for controls of infinite dimensional systems X. Li

2549

Two norm approach in stability analysis of optimization and optimal control problems K. Malanowski

2557

On nonlinear minimum norm control problems W. Krabs

2569

Generalizations of the Lusternik Theorem and applications to nonsmooth and abnormal problems in optimization and optimal control U. Ledzewicz and S. Walczak

2583

Optimal shape design of plates and shells with obstacles M. Rao and J. Sokolowski

2595

Some results on the viscosity solutions of fully nonlinear equations containing non-local terms K. Ishii and N. Yamada

2605

Optimal control and relaxation of systems governed by Volterra integral equations S. Aizicovici and N. S. Papageorgiou

2617

Stability of a class of nonlinear singularly perturbed systems that contain marginally stable boundary layer systems M. Corless and D. Da

2627

A universal stabilizer for a class of nonlinear systems E. P. Ryan

2633

Computational methods for boundary optimal control of incompressible flows L. S. Hou

2643

Table of Contents

xxvii

Computational hybrid-control techniques in hydrodynamics T. P. Svobodny

2655

Optimal control problems of viscous flow Y.-R. Ou

2667

A direct method for parameter estimation in distributed systems D. W. Brewer and R. K. Powers

2679

Uniformly approximation of the LQR problem Z. Liu and stable S. Zheng On a mathematical model of phase transitions in materials with memory R. D. Spies

2687 2697

Overview of optimal trajectories for aeroassisted orbital transfer A. Miele

2705

Long term planning of hydro-thermal power systems Κ. H. Wong and L. S. Jennings

2723

On the optimal drug administration of a discrete cancer chemotherapy model C. S. Lee and V. Rehbock

2735

The neural network approach to several nonlinear control problems C. J. Goh

2749

A unified computational approach to optimal control problems K. L. Teo

2763

Some trajectory-following algorithms for min-max optimization T. L. Vincent, B. S. Goh, and K. L. Teo

2775

On the usage and applications of parameter optimization software Op&\ in constrained optimal control J. Dolezal and J. Fidler

2779

Construction of numerical solutions of permanent heavy waves of two liquids N. Moussa

2791

Asymptotic theorems about compression and expansion of cones in Banach spaces for multivalued maps differentiable at the origin and infinity D. Violette and G. Fournier

2803

Bayesian statistical approach to design of reliability models V. P. Savchuk

2815

Determine optimal burn-in time for highly integrated circuit systems W.-T. K. Chien and W. Kuo

2827

xxviii

Table of Contents

A nonhomogeneous Poisson process for software reliability N. Suresh and Α. Ν. V. Rao

2837

Multiple category screening procedures A. Mouhab and R. A. Johnson

2851

Is your model chaotic? A consistence simulation-based test I. Domowitz and M. A. El-Gamal

2861

Economic models of survival under uncertainty S. Roy

2873

Aggregation in mathematical models of manufacturing systems A. A. Pervozvanskii

2885

Nonlinear interaction dynamics of multiple mobile automata P. K. C. Wang Some unsolved problems in optimal design theory: Interaction of mathematical modelling of large structural, mechanical or chemical systems with their design V. Komkov stability and bifurcation analysis of gated radar range trackers Stochastic R. E. Gover, Ε. H. Abed, and A. J. Goldberg

2897

2909 2931

The twist-and-flip map R. Brown and L. Chua

2943

Bifurcations and chaos from Chua's circuit: A state-of-the-art review M. J. Ogorzalek

2957

Nonlinear dynamics in cellular neural networks J. A. Nossek and F. Zou

2969

CNN: A new paradigm of nonlinear dynamics in space L. O. Chua and T. Roska

2979

Geometric invariants of chaotic attractors encountered in Chua's circuit M. J. Ogorzalek and Z. Galias

2991

Nonstandard methods in metric fixed point theory M. A. Khamsi

3003

The minimax principle of mechanics and its application to the optimal control problem for robot manipulators Ye. S. Pyatnitskiy

3013

On a fixed point theorem of Jungck Η. K. Pathak

3019

Table of Contents

xx ix

A nonlinear ergodic theorem for almost-orbits of nonlinear contraction semigroups in Banach spaces K.-K. Tan and H.-K. Xu

3025

Author Index

3037

Table of Contents

XXX

Table of Contents of Volume IV Α simple ODE with more than 20 strange attractors L. O. Chua

3043

Applications of generalized functions to propagating and deforming interfaces R. P. Kanwal

3055

Conservation laws for integro-differential equations V. S. Vladimirov

3069

Dynamics of neuro-muscular control with delayed displacement-dependent feedback J. G. Milton and T. Ohira 3085 Oscillations and chaos in nonlinear delay differential equations with applications to physiology U. an der Heiden

3095

Multiple-delayed differential equations as models for biological control systems S. A. Campbell and J. Belair

3109

Lattices with spatially extended coupling and the statistical properties of neural populations M. C. Mackey and J. G. Milton

3119

Dependence on initial conditions in nonlocal PDE's and hereditary dynamical systems R. Crabb, J. Losson, and M.C. Mackey

3125

Biased preference models for partnership formation J. M. Hyman and J. Li

3137

AIDS in the Heterosexual Population D. E. Kirschner

3149

Completion of mixing matrices for non-closed social networks S.-F. H. Schmitz and C. Castillo-Chavez Modelling vector-host disease transmission and food web dynamics through the mixing/pair formation approach J. X. Velasco-Hernandez and C. Castillo-Chavez

3163

3175

Simple models for interactions of the HIV virus and the immune system: Can we point out to one specific threshold for the onset of AIDS? L. E. Harnevo

3187

Variability infectivity in communicable disease models F. Brauer

3201

Table of Contents

xxxi

Stochastic and deterministic models in epidemiology C. Castillo-Chavez, S. Fridman, and X. Luo

3211

A pair formation approach to modeling inheritance of social traits S. Lubkin and C. Castillo-Chavez

3227

Information processing and practical implementation of "Biochemical neuron" M. Okamoto, Y. Maki, and T. Sekiguchi

3235

Nonlinear control and self-organization Η. V. Westerhoff, P. R. Jensen, D. Kahn, Β. Ν. Kholodenko, and P. Richard

3245

Analysis of two hypothetical mechanisms of photosynthetic oscillations by means of mathematical skeleton models C. Giersch and W. Rovers

3255

The theory of the control of metabolism J.-P. Mazat, C. Reder, and T. Letellier

3267

Mathematical modelling of enzymic systems; simulation, control analysis and optimization R. Heinrich

3277

Selection of enzymatic mechanisms which account for simplicity in the evolution of metabolic pathways E. Melendez-Hevia, T. G. Waddell, and F. Montero

3291

Mathematical description of regulation in metabolic systems: The regulatory potential J. H. S. Hofmeyr and A. Cornish-Bowden

3305

Self-activation and self-inactivation kinetics of enzymes: theoretical aspects and a mathematical model for the particulate Prostaglandine synthase from ovine vesicular glands H. G. Holzhütter, C. Scheme, and T. Scheme

3311

Power-law formalism: A canonical nonlineax approach to modeling and analysis M. A. Savageau

3323

S-system analysis of the TCA cycle in Dictyostelium

discoideum

F. Shiraishi

3335

A monomial-based version of Newton's method S. A. Burns

3347

Assessment of the regulatory properties of a metabolic pathway: Parameter estimation from steady-state measurements A. Sorribas and M. Cascante

3357

xxxii

Table of Contents

How many variables? Some comments on the dimensionality of nonlinear systems E. 0. Voit

3369

Simple dynamical systems for bursting and spindling J. L. Hindmarsh and R. M. Rose

3381

A two-component discrete dynamical model of the bursting rhythm caused by veratramine in the pacemaker of the heart C. D. Thron

3389

On the clonal inheritance model of cell proliferation D. N. Stivers and M. Kimmel

3401

Emergence of stable DNA repeats from random sequences under unequal sister chromatid exchange K. Baggerly and M. Kimmel

3409

A model for retrieving the dynamics of gene amplification from episomes to intrachomosomal amplification L. E. Harnevo and Z. Agur

3419

New mathematical methods for evaluating toxicity of anti-cancer drugs and AZT L. Cojocaru and Z. Agur

3433

A particle method for population shock waves with application to synchronization of bacterial culture growth C. Chiu and F. C. Hoppensteadt

3443

Simulation of drug-inhibited cell proliferation with a model of clonal inheritance of cell lifetimes Y. Gusev and D. E. Axelrod

3455

Resonances in periodic chemotherapy scheduling G. F. Webb

3463

A review of the effect of radiation transfer on the simulation of atmospheric circulations of different scales M. J. Leach and S. Raman

3475

Effects of thermal forcing in atmospheric shear flows with critical level Y.-L. Lin

3491

Baroclinic-barotropic instabilities and energy transfers in the Gulf Stream L. C. Sun and L. J. Pietrafesa

3515

Turbulence structure in the atmospheric boundary layer—data and models K. S. Rao

3533

Table of Contents A review of coupled ocean-atmosphere models N. Reddy and S. Raman

xxxiii

3545

Review of recent advances in the development of semi-Lagrangian numerical models in meteorology F. Η. M. Semazzi

3559

Atmospheric dispersion in low wind conditions A. K. Yadav, M. Sharan, and M. P. Singh

3567

A review of non-hydrostatic numerical models for the atmosphere L. Xu, S. Raman, and R. V. Madala

3595

The stagnant flow upstream of hills or mountains K. Kitabayashi

3611

Micro-meteorological studies in Gulati air quality management P. Goyal, M. P. Singh, and A. Nonlinear analysis of stability, oscillations and control of immunological processes V. M. Matrosov, N. A. Karatueva, and Ν. V. Banina

3619 3633

Mathematical modeling of cellular interaction dynamics in multicellular tumor spheroids Z. Bajzer, M. Marusic, and S. Vik-Pavlovic

3645

Polynomial networks for optimal nonlinear filtering J. T.-H. Lo, M. J. Sanders, Jr., and S. K. Ng

3655

Analysis of the computational characteristics of adaptive resonance theory neural nets L. V. Fausett

3669

Differentiating and pruning multilayer feedforward neural networks J. T.-H. Lo and L. Yu

3681

Stability analysis of neural networks using vector Lyapunov functions S. Köksal and D. W. Fausett

3695

Neural network approach to optimal filtering J. T.-H. Lo

3705

Signal recognition using a dynamic time warping neural network F. A. Unal and N. Tepedelenlioglu

3717

Development and analysis of a mathematical model for a plant-herbivore system L. J. S. Allen, Μ. K. Hannigan, and M. J. Strauss

3723

xxxiv

Table of Contents

The effects of asymmetric competition on competitive outcome Κ. M. Crowe

3733

Persistence in Kolmogorov population models with time delay Y. Cao, J. Fan, and T. C. Gard

3741

Global variables and population dynamics prey-predator models with individual activity sequences P. Auger

3753

The computer-aided study of dynamics in the system primary tumor, cytotoxic lymphocytes, metastas V. A. Kuznetsov and Ε. I. Alekseeva

3763

Folding at the genesis of chaos B. Deng

3765

Ginzburg-Landau type models for superconductivity Q. Du, Μ. Gunzburger, and J. Peterson

3779

Time-dependent Ginzburg-Landau models for superconductivity Q. Du Asymptotic analysis of the Ginzburg-Landau model of superconductivity: Reduction to a free boundary model

3789

S. J. Chapman

3803

Dynamics E. Weinan of vortices in superconductors

3811

Computation of singular solutions in the micromagnetics of thin films T. Brule and Μ. E. Henderson

3823

Energy transport systems for semiconductors: Analysis and simulation J. W. Jerome and C.-W. Shu

3835

Nonlineaxities in drift-diffusion semiconductor modeling T. Kerkhoven

3847

Some classes of iterative methods for finding zeros of polynomials and analytic functions M. S. Petkovic

3859

Fine and coarse grained parallel implementations of polynomial root finding algorithms E. Fleury and P. Fraigniaud

3871

Divide and conquer techniques for the polynomial root-finding problem D. Bini

3885

Table of Contents

xxxv

A unified approach to some classes of simultaneous inclusion methods for polynomial zeros L. Y. Atanassova

and J. P. Herzberger

3897

WKB procedure for Schrödinger type difference equations R. E. Mickens and I. Ramadhani

3907

Toward an efficient and stable determination of spectra of a general matrix and a more efficient solution of the Lyapunov equation G. Howell

3913

Interfacial wave theory of dendritic growth and a comparison with experiments J.-J. Xu

3929

Thomas-Fermi model calculations of one and two dimensional lattices A. Rabinovitch, J. Komlos, and M. Friedman

3941

Author Index

3949

Titles of Organized Sessions Qualitative Theory of Difference Equations R.P. Agarwal Periodic Solutions of Boundary Value Problems S. Ahmad and A. Lazer Nonlinear Parabolic Problems H. Amann Differential Delay Equations h Applications in Biology and Medicine U. an der Heiden and S.I. Andersson Critical Point Theory & Hamiltonian Systems D. Bend and A. Ambrosetti Special Session Lyapunov's 100th Anniversary V.M. Matrosov, L. Hatvani, J. Kato and T. Burton Math Epidemiology C. Castillo-Chavez Nonconvex Analysis A. Cellina Blow-up, Dead Cores & Quenching C. Y. Chan and M.K. Kwong Multiple Solutions of Nonlinear Differential Equations K.C. Chang Bifurcation, Chaos L· Artificial Intelligence in Nonlinear Electronics L. Chua and R. Marshall Control Theory in Economics E.N. Chukwu Special Session on Poincare's Centenary C. Corduneanu Nonlinear Waves L. Debnath Neural Networks in Biomedicine N. DeClaris Asymptotic Behavior of Nonlinear Systems Attractors and Confmors: Application to Biology J. Demongeot

xxxvii

Fixed Point Theory & Related Topics E. Fadell and A. Kirk Global Bifurcations & Chaos K. Gopalsamy Nonlinear Oscillations J.R. Graef Stability Problems in Nonlinear Compartmental Systems with Pipes and Delays I. Gyori Structural Models in Ecology Theory & Applications T. G. Hallam and J. Cushing Models in Ecology: Spatial Heterogenity T.G. Hallam and J. Cushing Math Modeling of Enzyme Systems R. Heinrich Computational Methods for Parameter Estimation and Control T. Herdman Nonlinear Controlled Systems H. Hermes Nonlinear Problems in Superconductivity M.K. Kwong and C.Y. Chan Accretive L· Monotone Operator Theory A.G. Kartsatos Evolution Equations N. Kenmochi and S. Oharu Nonlinear Aspects of the Analysis & Computations for Microelectronic Devices (In memory of F. Odeh) T. Kerkhoven Hereditary Systems and Their Applications V. Kolmanovskii Positive Discrete Dynamical Systems U. Krause and R. Nussbaum Nonlinear Fluid Mechanics L. Krishnamurthy Deterministic Control of Uncertain Systems G. Leitmann and M. Corless

xxxviii

Neural Networks: Algorithm and Applications D. Lainiotis Nonlinear PDE's and Nonlinear Waves T.T. Li Nonlinear Volterra Equations S.O. Londen Math Models in Combustion Theory A. Lunardi Nonlinear Dynamic Models in Economic Theory M. Majumdar Nonlinear Boundary Value Problems J. Mawhin and D. G. DeFigueiredo Nonlinear Dynamics in Manufacturing Systems S. Meerkov Nonsmooth Analysis and Optimization B. Mordukhovich Mathematics & Computer Vision J.M. Morele and S. Solomini Nonlinear Ill-posed/Inverse Problems M.Z. Νashed Nonlinear Operator and Set Valued Analysis M.Z. Νashed The Dynamics of Oscillatory & Excitable Bio Systems H.G. Othmer Semilinear Equations W. V. Petryshyn Nonlinear Stochastic Systems A.N. V. Rao and C.P. Tsokos Recent Developments of Reliability Analysis & Modeling A.N. V. Rao and C.P. Tsokos Canonical Nonlinear Modeling with Applications to Biochemical Systems M.A. Savageau Nonlinear Structural Mechanics R. Schmidt

xxxix

Atmospheric & Marine Sciences S. Raman and M.P. Singh Inertial Manifolds R. Temarn Numerical Nonlinear Control L· Applications K.L. Teo and, T.L. Vincent Bursting Rhythms in Bio Systems C.D. Thron Roots of Polynomials h Systems of Polynomials D. Trigiante Problems in Hysteresis L. Tavernini and A. Visintin Dynamics of Populations of Cells and Genes: Gene Amplifications and Related Topics M. Kimmel and G. Webb Nonlinear Problems in Physics E. Zeidler

List of Participants

Abed, E.H. (USA) Adomaitis, R.A. (USA) Adomian, G. (USA) Agarwal, R.P. (Singapore) Agur, Z. (Israel) Ahlebrandt, C. (USA) Ahmad, I.A. (USA) Ahmad, S. (USA) Ahmed, N.U. (Canada) Aiki, T. (Japan) Aizicovici, S. (USA) Alabau, F. (France) Albeverio, S. (Germany) Alekseeva, E. (Russia) Allen, L.J.S. (USA) Alt, W. (Germany) Altman, W. (Brazil) Amann, H. (Switzerland) Ambuehl, B. (Switzerland) An der Heiden, U. (Germany) Anderson, J.R. (USA) Andersson, S.I. (Sweden) Ang, D.D. (Vietnam) Anger, G. (Germany) Anokhin, A.V. (Russia) Antman, S.S. (USA) Antoniou, I. (Belgium) Appell, J. (Germany) Astumian, R.D. (USA) Atanassova, L. (Bulgaria) Aubin, J.P. (France) Auger, P. (France) Aulbach, B. (Germany) Avrin, J.D. (USA) Azbelev, N.V. (Russia) Babai, D. (USA) Badiale, M. (Italy) Baggerly, K. (USA) Bajzer, Z. (USA) Balis, U.J. (USA) Ballion, J.Β (France) Banks, H.T (USA) Bardou, A. (France) Bartsch, Τ. (Germany)

Battelli, F. (Italy) Baules, E. (Panama) Becker, R.I. (South Africa) Bednarczuk, E.M. (Poland) Belair, J. (Canada) Benci, V. (Italy) Bensoussan, A. (France) Bernadou, M. (France) Bernfeld, S.R. (USA) Bertram, R. (USA) Beschler, Ε. (USA) Bielewicz, Ε. (Poland) Binding, P.A. (Canada) Bini, D. (Italy) Bishop, A. (USA) Bloom, F. (USA) Boggs, P.T. (USA) Bognär, G. (Hungary) Boguslavsky, I. (Russia) Böhm, Μ. (Germany) Böhme, R. (Gemany) Bona, J. (USA) Borwein, J.M. (Canada) Bothe, D. (Germany) Bownds, J. (USA) Brauer, F. (USA) Brenner, S. (USA) Bressan, A. (Italy) Brewer, D.W. (USA) Brezis, H. (France/USA) Brokate, M. (Germany) Browder, F. (USA) Brown, R. (USA) Bruck, R.E. (USA) Burns, J. (USA) Burns, S.A. (USA) Burton, T.A (USA) Bushell, P.J. (England) Byszewski, L. (Poland) Cabada, A. (Spain) Calvert, B.D. (New Zealand) Campbell, S.A. (USA) Cantrell, S. (USA) Cannon, J.R. (USA)

List of Participants Cao, J. (USA) Carroll, R.W. (USA) Carl, S. (Germany) Cascante, M. (Spain) Castillo-Chavez, C. (USA) Cellina, A. (Italy) Cerami, G. (Italy) Chan, C.Y. (USA) Chandna, O.P. (Canada) Chandra, J. (USA) Chang K.C. (China) Chang M.H. (USA) Chapman, S.J. (USA) Chatelin, F. (France) Chen, Y. (USA) Chen, W. (USA) Chen, C. (USA) Cheng, S.S. (Taiwan) Cheng, Y. (Sweden) Chernousko, F.L. (Russia) Chipot, M. (France) Chiu, C. (USA) Chhabra, A. (USA) Choudhury, S.R. (USA) Chua, L.O. (USA) Chukwu, E.N. (USA) Clement, P. (Netherlands) Colombo, G. (Italy) Colonius, F. (Germany) Corduneanu, C. (USA) Corless, M. (USA) Cosner, C. (USA) Costa, D. (Brazil) Coti Zelati, V. (Italy) Cover, A. (USA) Cremins, C. (USA) Cronin-Scanlon, J. (USA) Crowe, K.M. (USA) Cushing, J. (USA) Damlamian, A. (France) Dancer, E.N. (Australia) Daprato, G. (Italy) Davis, P. (USA) DeBlasi, (Italy) Debnath, L. (USA) Debussche, A. (France) DeClaris N. (USA) Deguire, P. (Canada) Demetrius, L. (USA) Demongeot, J. (France) Dence, D. (USA) Deng, B. (USA)

Deng, K. (USA) Desch, W. (Austria) Dibos, F. (France) Ding, T. (China) Ding, Z. (China) Dolezal, J. (Czechoslovakia) Dong, G. (China) Dontchev, A.L. (USA) Dorodnitsyn, V.A. (Russia) Drici, Z. (Algeria) Dshalalow, J.H. (USA) Du, Q. (USA) Durlauf, S.N. (USA) Dye, J.M. (USA) Ebihara, Y. (Japan) Efendiev, M.A. (Germany) Elaydi, S.N. (USA) El-Gamal, M.A. (USA) Eloe, P.W. (USA) Emara, S.A.A. (Egypt) Emelyanov, S.V. (Russia) Engl, H.W. (Austria) Erbe, L.H. (Canada) Fabiano, R. (USA) Farkas, M. (Hungary) Fausett, D.W. (USA) Fausett, L.V. (USA) Felmer, P. (Chile) Fenske, C. (Germany) Fila, M. (USA) Fink, A.M. (USA) Finn, R. (USA) Fitzpatrick, B.G. (USA) Flandoli, F. (Italy) Fomenko, I. (Ukraine) Fonda, A. (Belgium) Forest, G. (USA) Fournier, G. (Canada) Fraigniaud, P. (France) Freedman, H.I. (USA) Friedman, A. (USA) Friedmein, Μ. (Israel) Friedman, Mark (USA) Frigon, M. (Canada) Froment, J. (France) Fujimoto, T. (Japan) Fulton, C.T. (USA) Furi, M. (Italy) Furumochi, T. (Japan) Furusho, Y. (Japan) Fushchyck, W. (Ukraine) Galperin, E.A. (Canada)

xlii Gämez, J.L. (Spain) Gard, T.C. (USA) Gardner, R. (USA) Gautier, S. (France) Giersch, C. (Germany) Gershwin, S. (USA) Giga, Y. (Japan) Girardi, M. (Germany) Glass, L. (Canada) Glimm, J. (USA) Goh, C.J. (Australia) Gomez, H.H. (Colombia) Gopalsamy, K. (Australia) Gorenflo, R. (Germany) Gossez, J.P. (Belgium) Goubet, O. (France) Gover, R.E. (USA) Goyal, P. (India) Graef, J.R. (USA) Grantham, W.J. (USA) Gribben, R.J. (Brunei) Guan, Z. (USA) Gunzburger, M.D. (USA) Gurney, D.R. (USA) Gyllenberg, M. (Sweden) Györi, I. (Hungary) Ha, K.S. (Korea) Haddock, J. (USA) Hallam, T. (USA) Hamead, A. (Kuwait) Hamilton, E.P. (USA) Han, Z.C. (USA) Hankerson, D. (USA) Harnevo, L.E. (Israel) Hashish, H. (Egypt) Hastings, S. (USA) Hatvani, L. (Hungary) Heikkilä, S.V. (Finland) Heinrich, R. (Germany) Helm, C.A. (Germany) Henderson, G. (USA) Henderson, J. (USA) Henderson, M.E. (USA) Herdman, T. (USA) Hering, R. (USA) Hernandez, G.E. (USA) Hernandez, J. (Spain) Hervagault, J.F. (France) Herzberger, J.P. (Germany) Hess, P. (Switzerland) Hess, D.P. (USA) Hetzer, G. (USA)

List of Participants Hindmarsh, J.L. (Wales) Hinterman, T. (Switzerland) Hiramo, N. (Japan) Hofmer, J.H. (South Africa) Hollis, S.L. (USA) Holzhütter, H.G. (Germany) Hou, L.S. (Canada) Howell, G. (USA) Hsieh, H.Y. (Hong Kong) Hu, S. (USA) Huang, W. (USA) Huang, Q. (China) Huang, Y.X. (China) Hunt, L.R. (USA) Huseyin, K. (Canada) Infeld, E.S. (England) Ioffe, A.D. (Israel) Itoh, T. (Japan) Itoh, Y. (Japan) Ivanov, V. (Russia) Ize, A. (Brazil) Ize, J. (Mexico) Jackson, D.E. (USA) Jelinek, J. (USA) Jerome, J.W. (USA) Jimbo, S. (Japan) Johnson, R.A. (USA) Jolly, M. (USA) Jones, M. (USA) Joshi, M.C. (India) Jost, J. (Germany) Kaczynski, T. (Canada) Kahn, D. (France) Kanwal, R.P. (USA) Kaplan, D. (USA) Kartsatos, A.G. (USA) Katchakis, M.N. (USA) Kato, J. (Japan) Kaul, S.K. (Canada) Kawohl, B. (Germany) Kaymakealan, B. (Turkey) Keller, H. (USA) Kenmochi, N. (Japan) Kerkhoven, T. (USA) Kermani, S. (Iran) Kevrekidis, Y.G. (USA) Khamsi, M.A. (USA) Khrustalev, M.M. (Russia) Kikuchi, N. (Japan) Kim, A.V. (Russia) King, B.B. (USA) Kirschner, D. (USA)

List of Participants Kirk, W.A. (USA) Kitabayashi, K. (Japan) Klibanov, M.V. (USA) Koepfler, G. (USA) Koh, E.L. (Canada) Köksal, S. (USA) Kolmanovskii, V. (Russia) Komkov, V. (USA) Kong, Q. (Canada) Konig, D. (Germany) Korovin, S.K. (Russia) Korolevich, V.G. (Ukraine) Kovach, T. (USA) Kozlov, R.I. (Russia) Krabs W. (Germany) Krasnosel'skii, M.A. (Russia) Krause, U. (Germany) Krejci, P. (Czechoslovakia) Krishna, S.V. (India) Krishnamurthy, L. (USA) Krisztin, T. (Hungary) Krotov, V.F. (Russia) Kuczumow, T. (Poland) Kumar, P.R. (USA) Kuo, T.H. (Taiwan) Kuo, W. (USA) Kurihara, M. (Japan) Kurylev, Y. (USA) Kuzmina, L.K. (Russia) Kuznetsov, V.A. (Russia) Kuznetsov, P.K. (Russia) Kwak, M. (Korea) Kwong, M.K. (USA) Lacomblez, C. (France) Ladas, G. (USA) Ladde, G.S. (USA) Lalli, B.S. (Canada) Lan, K. (China) Landl, G. (Austria) Lasiecka, I. (USA) Lazer, A.C. (USA) Leach, M.J. (USA) Ledzewicz, U. (USA) Lee, C.S. (Malaysia) Leela, S. (USA) Lefton, L. (USA) Leiberman, G. (USA) Lemarechal, C. (France) Lenhart, S.M. (USA) Leonardi, S. (Italy) LeRoux, A.Y. (France) LeRoux, M.N. (France)

xliii

Lewis, A. (Canada) Lewis, R.T. (USA) Li, J. (USA) Li, T.-T. (China) Li, X. (China) Liao, X. (China) Librescu, L. (USA) Lin, E.B. (USA) Lin, L.J. (China) Lin, X.B. (USA) Lin, Y.L. (USA) Lin, Z. (USA) Litsyn, E. (Israel) Liu, J.W. (China) Liu, X. (Canada) Liu, Z. (USA) Liz, E. (Spain) Lo, J.T.-H. (USA) Londen, S.O. (Finland) Lopez-Gomez, J. (Spain) Lu, C. (USA) Lubkin, S. (USA) Lunardi, A. (Italy) Luo, X. (USA) Lupo, D. (Italy) Lynn, K. (USA) MacCamy, R.C. (USA) Mackey, M.C. (Canada) Maeda, H. (Japan) Majda, A. (USA) Majumdar, M. (USA) Makay, G. (USA) Malanowski, K. (Poland) Malek, S. (USA) Manasevich, R.F. (Chile) Mancini, G. (Italy) Marcellini, P. (Italy) Marie, V. (Yugoslavia) Marion, M. (France) Marshall, R. (USA) Martindale, J. (USA) Masuda, K. (Japan) Masuoka, T. (Japan) Matrosov, V.M. (Russia) Matrosova, N.I. (Russia) Matusu-Necasovä, S. (Czechoslovakia) Mawhin, J.L. (Belgium) Mazat, J.P. (France) Mäzen, S. (USA) McDonald, N. (Scotland) McGinnis, C.L. (USA) McKenna, P.J. (USA)

xliv McLaughlin, J.R. (USA) McRae, F. (USA) Medina, R. (Chile) Meerkov, S. (USA) Meir, A. (USA) Meisters, G.H. (USA) Melendez-Hevia, E. (Spain) Melikyan, A. (Russia) Mennicken, R. (Germany) Messere, E. (USA) Metzen, G. (USA) Meyers, R. (USA) Meyer-Spasche, R. (Germany) Micheli-Tzanakou, E. (USA) Mickens, R.E. (USA) Miele, A. (USA) Miersemann, E. (Germany) Milojevic, RS. (USA) Milton, J.G. (USA) Mitidieri, E. (Italy) Miyakawa, T. (Japan) Mizutani, Y. (Japan) Morales, C. (USA) Morante, S. (Italy) Mordukhovich, B. (USA) Mou, L. (USA) Moussa, N. (Egypt) Moustafa, M.S. (Egypt) Munger, F.S. (USA) Munoz de Ozak, M. (Colombia) Mursaleen, (India) Mustonen, V. (Finland) Myshkis, A.D. (Russia) Naito, K. (Japan) Nakao, M. (Japan) Nashed, M.Z. (USA) Naulin, R. (Chile) Naumkin, P.I. (Russia) Nauroschat, J. (Germany) Necas, J. (Czechoslovakia) Neuberger, J.W. (USA) Neumann, A.U. (Israel) Newman, R. (USA) Nicolaenko, B. (USA) Nicolosi, F. (Italy) Nieto, J.J. (Spain) Niezgodka, M. (Poland) Nirenberg, L. (USA) Nisbet, R.M. (USA) Nkashama, M.N. (Japan) Nosov, V.R. (Russia) Nossek, J.A. (Germany)

List of Participants Nossov, V.P. (Russia) Nowakowski, A. (Poland) Nussbaum, R.D. (USA) Oettli, W. (Germany) Ogorzalek, M. (Poland) Oharu, S. (Japan) Ohezin, S.P. (Russia) Okamoto, M. (Japan) Okhezin, V. (Russia) Olech, C. (Poland) Olinger, D.J. (USA) Olson, L. (USA) Onose, H. (Japan) Ο'Regan, D. (Ireland) Ortega, R. (Spain) Otani, M. (Japan) Othmer, H. (USA) Ou, Y.-R. (USA) Pagani, C.D. (Italy) Pahk, D.H. (Korea) Palanques-Maestre, A. (Spain) Palmer, K. (USA) Palmerio, A.F. (Brazil) Parisi, V. (Italy) Park, S. (Korea) Pathak, H.K. (India) Patula, W.T. (USA) Pavel, N. (USA) Payne, L. (USA) Pejsachowicz, J. (Italy) Penot, J.-P. (France) Perez-Munuzuri, V. (Spain) Pernarowski, M. (Canada) Pervozvansky, A.A. (Russia) Peterson, A. (USA) Peterson, J. (USA) Petryshyn, W.V. (USA) Pierre, M. (France) Pinto, M. (Chile) Ponce, G.A. (USA) Porteiro, J.L.F. (USA) Potter, A.J.B. (Scotland) Pyatnitskiy, Ye.S. (Russia) Qiao, H. (USA) Rabier, P.J. (USA) Rabinovitch, A. (Israel) Rabinowitz, P. (USA) Rajopadhye, S. (USA) Ramachandran, K.M. (USA) Raman, S. (USA) Rammaha, M. (USA) Rao, A.N.V. (USA)

List of Participants Rao, K.S. (USA) Rao, M.R.M. (India) Reddy, N. (USA) Reddy, Y.S.N. (USA) Redheffer, R. (USA) Rekab, K. (USA) Renardy, M. (USA) Resio, D.T. (USA) Reyna, L. (USA) Reynolds, J. (USA) Rheinboldt, W . (USA) Riahi, D.N. (USA) Ricceri, B. (Italy) Richardson, W . (USA) Rockafellar, R.T. (USA) Rodrigues, I. (USA) Rodrigues, J.F. (Portugal) Roseman, J. (Israel) Roska, T . (Hungary) Rossana, R.J. (USA) Rother, W. (Germany) Roxin, E.O. (USA) Roy, S. (USA) Ruan, S. (Canada) Rudin, L. (USA) Rueda, A. (Hungary) Rues, W . (Germany) Ruf, B.H. (Italy) Rumbos, A.J. (USA) Rumjanzev, V.V. (Russia) Russell, J.M. (USA) Ryan, E.P. (United Kingdom) Sabatier, P.C. (France) Saberi, A (USA) Sacks, P.E. (USA) Sambandham, M. (USA) Saranen, J. (Finland) Savageau, M.A. (USA) Savastuk, S.V. (Russia) Savchuk, V.P. (Russia) Schattier, Η. (USA) Schmidt, P.G. (USA) Schmidt, R. (Germany) Schmidt-Laine, CI. (France) Schneider, K.R. (Germany) Schultz, S.W. (USA) Schuricht, F. (Germany) Sefcik, J. (USA) Seidman, T. (USA) Seikkila, S. (Finland) Sell, G. (USA) Semazzi, F.H.M. (USA)

Sere, E. (France) Serre, D. (France) Serrin, J. (USA) Shadwick, W.F. (Canada) Shahin, M. (USA) Sharif, M. (Iran) Sherman, A. (USA) Shigasada, N. (Japan) Shiraishi, F. (Japan) Shivaji, R. (USA) Shivakumara, I.S. (India) Shyu, S.F. (USA) Sieveking, M. (Germany) Siljak, D. (USA) Simpson, L.B. (USA) Singh, K. (Hungary) Singh, M.C. (Canada) Singh, M.P. (India) Siriseno, H. (New Zealand) Sivasundaram, S. (USA) Sliwa, S. (USA) Smale, S. (USA) Sobolevskii, P.E. (Israel) Sokolowski, J. (Poland) Sorribas, A. (Spain) Souganidis, P.E. (USA) Spies, R.D. (USA) Spikes, P.W. (USA) Sprekels, J. (Germany) Srikanth, P.N. (India) Srzednicki, R. (Poland) Staffans, O.J. (Finland) Stakgold, I. (USA) Stanoyevitch, A. (USA) Stech, Η. (USA) Stetter, H.J. (Austria) Stivers, D.N. (USA) Strogatz, S. (USA) Stucki, J.W. (Switzerland) Sugiyama, S. (Japan) Sulcoski, M. (USA) Suresh, N. (USA) Sun, Y. (China) Sung, L.Y. (USA) Svobodny, T.P. (USA) Szulkin, A. (Sweden) Takahashi, W . (Japan) Tan, K.-K. (Canada) Tarafdar, E. (Canada) Tavernini, L. (USA) Taylor, J. (USA) Taylor, R.L. (USA)

xlvi Temam, R. (USA/Prance) Teo, K.L. (Australia) Tepedelenlioglu, N. (USA) Terman, D. (USA) Terracini, S. (Italy) Thao, T.H. (Algeria) Thibault, L. (France) Thron, C.D. (USA) Tintarev, K. (USA) Tolstonogov, A. (Russia) Torrejon, R.M. (USA) Treiman, J.S. (USA) Treu, G. (Italy) Triandaf, I. (USA) Trigiante, D. (Italy) Trinko, N.G. (Russia) Trouve, P. (Prance) Tsokos, C.P. (USA) Turi, J. (USA) Ubhaya, V. (USA) Uvah, J. (USA) Van de Fliert, B. (Netherlands) Van Horssen, W. (Netherlands) Vargas, C. (Mexico) Vassilyev, S.N. (Russia) Vasundhara Devi, J. (India) Vatsala, A.S. (USA) Velasco-Hernandez, J.X. (Mexico) Verri, M. (Italy) Villari, G. (Italy) Vincent, T.L. (USA) Violette, D. (Canada) Visintin, A. (Italy) Vladimirov, V.S. (Russia) Voit, E.O. (USA) Voronov, A.A. (Russia) Vosmansky, J. (Czechoslovakia) Vujicic, V.A. (Yugoslavia) Wang, C. (USA) Wang, H.C. (Taiwan) Wang, P. (USA) Wang, P.K.C. (USA) Wang, S. (USA) Wang, S. (China) Wang, T. (USA) Wang, Y. (Canada) Wang, Z.Q. (USA) Warchall, H. (USA) Ward Jr., J.R. (USA) Warga, J. (USA) Webb, G.F. (USA) Webb, J.R.L. (Great Britain)

List of Participants Wei, D. (USA) Weichert, D. (France) Weinan, E. (USA) Westerhoff, H.V. (Netherlands) Wheeler, M.F. (USA) Wheeler, R.L. (USA) Wimberly, C.R. (USA) Wong, K.H. (South Africa) Wortman, M.A. (USA) Wright, S. (USA) Wu, J. (Canada) Wu, W. (China) Xiao, Z. (China) Xu, J.-J. (Canada) Xu, L. (USA) Xu, X. (USA) Yadava, S.L. (India) Yagola, A.G. (Russia) Yamada, N. (Japan) Yang, H. (USA) Yang, Z.H. (China) Yao, D.D. (USA) Yin, H.M. (USA) Yorke, J. (USA) Yoshizawa, T. (Japan) You, Y. (USA) Yu, A. (Hong Kong) Yu, Q. (China) Yuan, X.Z. (Canada) Yuhasz, W.D. (USA) Zafer, A. (USA) Zaidman, S. (Canada) Zak, M. (USA) Zanolin, Fabio (Italy) Zeidan, V.M. (Canada) Zeidler, E. (Germany) Zennaro, M. (Italy) Zhang, B.G. (China) Zhang, B. (USA) Zhang, H. (USA) Zhang, S. (China) Zhou, Z. (China) Zuluaga, M. (Colombia)

Investigation of Oldrojt's viscoelastic model when viscous coefficient tends to zero P. E.

Sobolevskii*

Abstract. The local in time convergence of solutions is established. 1991 Mathematics Subject Classification: 76A10, 35L70.

Many years ago, L. Nirenberg said t h a t we can investigate the nonlinear problems by two approaches: contrary to its nonlinearity or t h a n k s to its nonlinearity. I remembered these words, when I investigated one viscoelastic model. This model (see [1]) can be defined by the reological relation κ0σ + Kida/dt

= η0ξ + ηιθξ/dt,

κ 1 σ(0, χ) = ηιξ(0, χ).

(1)

Here σ is the deviator of the stress tensor and ξ is the r a t e of t h e deformations tensor. Namely, £ is the 3 x 3 matrix with components = 1 /2(dvt/dx} + dvj/dxi). Further ν = (υι, v2,vs) is the rate of the fluid motion a n d kq, « ι , ηο, Vi a r e (positive) constants. If ?7ο«ι = Holl in (1), we shall obtain Newton's model. Relation (1) and the motion equation in Cauchy form leads us to the boundary value problem

dv/dt

+ ^^ ujt · dv/dxjt fc = l

— εΑν — j G(t,x)Avds {

— gradp = f,divv

= 0

( t > O . x e i l ) ; ν = 0 ( ί > 0 , ι £ 5Ω); υ ( 0 , ι ) = ι> 0 (ζ) (χ £ Ω); Jp(t,x)dx

= 0(t > 0).

(2)

Ω

To the seventy-fifth birthday of my dear teacher, S. G. Krein, who aroused my interest to problems of hydrodynamics.

P.E. Sobolevskii

2

Here ε = τ ? ι / « ι , G ( t , s ) = [170/«ι — ( κ ο'?ι)/κι] · exp[—«o/«;i(t — β)]; Ω is an open bounded domain of points χ = (χ 1,12,23) in R3 with boundary 3Ω and Ω = Ω U 3 Ω . T h e last condition in (2) is introduced for uniqueness of the pressure p. Problem (2) is the generalization of the first boundary value problem for NavierStokes system. For the first time problem (2) was investigated in the articles of A. P. Oskolkov and his pupil (see [2]). They applied Ladyzenskaja's methods (see [3]). These investigations were continued in the article of Yu. Ya. Agranovich, P. E . Sobolevskii [4]. It was proved that there exists a unique, local in time, solution of problem (2). Moreover the same result holds for a nonlinear viscoelastic model [5]. It means that for a viscoelastic model the same result is true as for Newton's model. An integral elastic member does not impede here, when the condition viscous coefficient

ε > 0

(3)

is fulfilled! This result is contrary to elasticity. But we want t o investigate the situation, when condition (3) is violated. In the Navier-Stokes case we come to the difficult Eulers' model. It turns out that in the viscoelastic case the elastic property helps us! This result is thanks to elasticity. Consider problem (2) for all ε G (0,£o] and an arbitrary function G(t,s). We shall assume that the problem data G(t,x), f(t, x),v°(x),dil are sufficiently smooth, and the concord conditions are fulfilled. Then for every fixed e G (0, e 0 ] there exists a number te > 0 such that problem (2) has a unique sufficiently smooth solution ( v , p ) = (ve,pe), which is defined on the time segment [0, te] (on [0, 0), υ(0, ·) = υ 0 (·) (4)

in Hilbert space H , which is equivalent to relation (2). Here Η is the closure in L2(Q) of all smooth solenoidal vector-functions in Ω finite near 5Ω, is an orthoprojector in Li(Cl) on H(H = [ j l ^ l ^ ] ) ) ^ = — f j ^ is a linear operator in £ 2 ( Ω ) with domain D(A) = W\(Ω) Π ο Μ ^ Ω ) Π Η. The concord conditions mean that tfv'it,

x)/(dty\t=0

= d V ( 0 , χ )/{dty

G D(A)

for i = 0 , 1 , . . . , i0, ε G (0, e 0 ].

(5)

T h e o r e m 1 : Let G(t,t)>G°>

0.

Then there exists a number t* > 0 such that the solution are defined on the time segment [0, 0 and all ε G (0,eo]. Estimate (11) means that estimate (8) is exact. Now we shall give a proof of theorem 1 for the case G(t, x) = 1. This proof is easy to generalize to an arbitrary smooth function G(t,s). e e Let ( v , p ) = (v ,p ) be a smooth solution of problem (2) for fixed ε € (Ο,εο], which is defined on the time segment [0, te]. We differentiate relation (4) on t three times and obtain d2v

TT

(dvt

d3v τ τ Λ Qt* Fh3 + 1 1 2 - , t=i

dv

/d2vk dv 2 l dt dx k pa2 \

d2v

\

,dv

dvk d2v dtdxk adt ΡΗΛτ,

+Vk

,

t~t

d3v \ 2 dt dx k \

df

4

P.E. Sobolevskii

d2v Ä

+ *

d t 4

^

έ

dv +

w

U ί

V

A

π

ä

+ e A

d t

3

+

a2v a*

+

dt

" 1 1

2

3

at

\\dv(t,x)ldxk\\l

j ( n )

+ 2ε

fc=l
,,*( 0 there is a subsequence ε" —*• 0 for which (Pe",Ve>>) converges to a "weak" solution (po, Vo) of problem (Po)· Furthermore, if (i) 7αχ in (2.10) is large enough (i.e., we have initial good enough field in the sense that there is sufficient exposure of a large enough area in the document), and (ii) αχ + η. However it has not been proved that σ has the form described in problem (2) above. Thus, basically, problems (1) and (2) have only very partially been solved so far. αι

20

Avner Friedman

3. The visible image We again consider only the two-dimensional problem and assume that the electric image studied in §2 is ideally sharp, that is const, 0

if

|x| < αϊ

otherwise

where Τ is the final time of development of the electric image; we shall denote this constant by — σ (σ is positive). To study the development of the visible image we assume for simplicity that at χ = ± α the electric flux vanishes. Thus, if we denote the electric potential by —tz, then u x ( ± a , y, t) = 0. The region under consideration is Ω = { - α < χ < α, -h

< y < 6} ;

y = 0 is the surface upon which the toner will be deposited. Since the toner is uniformly charged, Au = 1 in the set D(t) occupied by the toner. Outside the toner set Au = 0. As indicated in Figure 5, tt(x, b,t) = M, u(x, —h, t) = 0. Finally, the presence of the electric image is reflected in the condition u v ( x , 0 + , t ) — ti,(x,0—,t) = — σ

if

— αχ < χ < αϊ .

(3.1)

We seek solutions that are symmetric in x. Denote by Γ ( ί ) the part of dD(t) in {y > 0}. Then Γ ( ί ) is a free boundary, i.e., it is one of the unknowns of the problem. The continuity equation gives the following law: (3.2) where Ν = N(t) is the outward normal to D(t) and V„ is the velocity of points of Γ ( ί ) in the direction N(t). Figure 7 and (3.1) (3.2) summarize the toner problem. We shall refer to it also as problem ( T ) . Note that Au is bounded everywhere except near Λ — { ( x , 0 ) , —αϊ < χ < Qi}·

Nonlinear PDE problems in electrophotography

21

Figure 7

The physical parameters for the photocopier are such that h < b,

Μ < ah .

(3.3)

Suppose we seek Γ(ί) in the form Γ(