راه‌حل‌های موج مسافرتی سیستم‌های پارابولیک (ترجمه‌های مونوگراف‌های ریاضی، جلد ۱۴۰)

The theory of traveling waves described by parabolic equations and systems is a rapidly developing branch of modern mathematics. This book presents a general picture of current results about wave solutions of parabolic systems, their existence, stability, and bifurcations. The main part of the book contains original approaches developed by the authors. Among these are a description of the long-term behavior of the solutions by systems of waves; construction of rotations of vector fields for noncompact operators describing wave solutions; a proof of the existence of waves by the Leray-Schauder method; local, global, and nonlinear stability analyses for some classes of systems; and a determination of the wave velocity by the minimax method and the method of successive approximations. The authors show that wide classes of reaction-diffusion systems can be reduced to so-called monotone and locally monotone systems. This fundamental result allows them to apply the theory to combustion and chemical kinetics. With introductory material accessible to nonmathematicians and a nearly complete bibliography of about 500 references, this book is an excellent resource on the subject. Mathematicians studying systems of partial differential equations, reaction-diffusion systems; physicists interested in autowave processes, dissipative structures; combustion scientists and chemists interested in mathematical issues of chemical kinetics. Title 1 Copyright 2 Contents 3 Preface 6 Introduction 8 Part I. Stationary Waves 44 Chapter 1. Scalar Equation 45 Chapter 2. Leray-Schauder Degree 127 Chapter 3. Existence of Waves 158 Chapter 4. Structure of the Spectrum 191 Chapter 5. Stability and Approach to a Wave 221 Part II. Bifurcation of Waves 261 Chapter 6. Bifurcation of Nonstationary Modes of Wave Propagation 262 Chapter 7. Mathematical Proofs 275 Part III. Waves in Chemical Kinetics and Combustion 298 Chapter 8. Waves in Chemical Kinetics 299 Chapter 9. Combustion Waves with Complex Kinetics 336 Chapter 10. Estimates and Asymptotics of the Speed of Combustion Waves 376 Supplement 409 Bibliography 430 The theory of traveling waves described by parabolic equations and systems is a rapidly developing branch of modern mathematics. This book presents a general picture of the results about wave solutions of parabolic systems, their existence, stability, and bifurcations. It also contains original approaches developed by the authors. Aizik I. Volpert, Vitaly A. Volpert, Vladimir A. Volpert. Includes Bibliographical References (p. 427-448)