# Shock Waves and Reaction—Diffusion Equations

Springer Science & Business Media, 6 déc. 2012 - 581 pages
. . . the progress of physics will to a large extent depend on the progress of nonlinear mathe matics, of methods to solve nonlinear equations . . . and therefore we can learn by comparing different nonlinear problems. WERNER HEISENBERG I undertook to write this book for two reasons. First, I wanted to make easily available the basics of both the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by C. Conley. These important subjects seem difficult to learn since the results are scattered throughout the research journals. 1 Second, I feel that there is a need to present the modern methods and ideas in these fields to a wider audience than just mathe maticians. Thus, the book has some rather sophisticated aspects to it, as well as certain textbook aspects. The latter serve to explain, somewhat, the reason that a book with the title Shock Waves and Reaction-Diffusion Equations has the first nine chapters devoted to linear partial differential equations. More precisely, I have found from my classroom experience that it is far easier to grasp the subtleties of nonlinear partial differential equations after one has an understanding of the basic notions in the linear theory. This book is divided into four main parts: linear theory, reaction diffusion equations, shock wave theory, and the Conley index, in that order. Thus, the text begins with a discussion of ill-posed problems.

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### Table des matičres

 CHAPTER 13 CHAPTER 4 26 An InitialValue Problem for a Hyperbolic Equation 39 B Fundamental Solutions 52 CHAPTER 8 64 CHAPTER 9 78 CHAPTER 10 92 Linearization 106
 D The Riemann Problem for a Scalar Conservation Law 301 B Shocks and Simple Waves 320 C Solution of the General Riemann Problem 335 C Interaction of Shock Waves 358 B The Difference Approximation 377 CHAPTER 20 391 D Instability of Rarefaction Shocks 414 B Artificial Viscosity 431

 The KreinRutman Theorem 122 B The LeraySchauder Degree 139 D A Rapid Course in Topology 156 SB Stability of Bifurcating Solutions 176 CHAPTER 14 192 C A Comparison Theorem 213 E A Lyapunov Function for Contracting Rectangles 227 CHAPTER 15 239 C Evolutionary Systems 254 B Uniqueness of the Entropy Solution 281
 CHAPTER 22 448 B Isolated Invariant Sets and Isolating Blocks 461 CHAPTER 23 478 C Continuation 494 CHAPTER 24 507 C Periodic Travelling Waves 521 E Instability of Equilibrium Solutions of the Neumann Problem 542 Bibliography 557 Author Index 571 570 Droits d'auteur