激波和反應(yīng)擴(kuò)散方程

內(nèi)容概要

For this edition, a number of typographical errors and minor slip-ups have been corrected. In addition, following the persistent encouragement of Olga Oleinik, I have added a new chapter, Chapter 25, which I titled "Recent Results." This chapter is divided into four sections, and in these I have discussed what I consider to be some of the important developments which have come about since the writing of the first edition. Section I deals with reaction-diffusion equations, and in it are described both the work of C. Jones, on the stability of the travelling wave for the Fitz-Hugh-Nagumo equations, and symmetry-breaking bifurcations. Section II deals with some recent results in shock-wave theory. The main topics considered are L. Tartar's notion of compensated compactness, together with its application to pairs of conservation laws, and T.-P. Liu's work on the stability of viscous profiles for shock waves. In the next section, Conley's connection index and connection matrix are described; these general notions are useful in constructing travelling waves for systems of nonlinear equations. The final section, Section IV, is devoted to the very recent results of C. Jones and R. Gardner, whereby they construct a general theory enabling them to locate the point spectrum of a wide class of linear operators which arise in stability problems for travelling waves. Their theory is general enough to be applicable to many interesting reaction-diffusion systems.

書籍目錄

AcknowledgmentPreface to the Second Edition Preface to the First Edition List of Frequently Used Symbols PART Ⅰ Basic Linear Theory  CHAPTER 1 lll-Posed Problems  CHAPTER 2 Characteristics and Initial-Value Problems  CHAPTER 3 The One-Dimensional Wave Equation  CHAPTER 4 Uniqueness and Energy Integrals  CHAPTER 5 Holmgren's Uniqueness Theorem  CHAPTER 6 An Initial-Value Problem for a Hyperbolic Equation  CHAPTER 7 Distribution Theory  CHAPTER 8 Second-Order Linear Elliptic Equations  CHAPTER 9 Second-Order Linear Parabolic EquationsPART ⅡReaction-Diffusion Equations  CHAPTER 10 Comparison Theorems and Monotonicity Methods  CHAPTER 11 Linearization  CHAPTER 12 Topological Methods  CHAPTER 13 Bifurcation Theory  CHAPTER 14 Systems of Reaction-Diffusion EquationsPART Ⅲ The Theory of Shock Waves  CHAPTER 15 Discontinuous Solutions of Conservation Laws  CHAPTER 16 The Single Conservation Law  CHAPTER 17 The Riemann Problem for Systems ofConservation Laws  CHAPTER 18 Applications to Gas Dynamics  CHAPTER 19 The Glimm Difference Scheme  CHAPTER 20 Riemann Invariants. Entropy. and Uniqueness  CHAPTER 21 Quasi-Linear Parabolic SystemsPART Ⅳ The Conley Index  CHAPTER 22 The Conley Index  CHAPTER 23 Index Pairs and the Continuation Theorem  CHAPTER 24 Travelling Waves  CHAPTER 25 Recent ResultsBibliographyAuthor IndexSubject Index

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