Shock Waves and Reaction—Diffusion Equations

Joel Smoller (auth.)0387907521, 9780387907529

The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley. It presents the modern ideas in these fields in a way that is accessible to a wider audience than just mathematicians.The book is divided into four main parts: linear theory, reaction-diffusion equations, shock-wave theory, and the Conley index. For the second edition numerous typographical errors and other mistakes have been corrected and a new chapter on recent results has been added. The new chapter contains discussions of the stability of traveling waves, symmetry-breaking bifurcations, compensated compactness, viscous profiles for shock waves, and general notions for construction traveling-wave solutions for systems of nonlinear equations.

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Table of contents :
Front Matter….Pages i-xxi
Front Matter….Pages 1-1
Ill-Posed Problems….Pages 3-12
Characteristics and Initial-Value Problems….Pages 13-16
The One-Dimensional Wave Equation….Pages 17-25
Uniqueness and Energy Integrals….Pages 26-32
Holmgren’s Uniqueness Theorem….Pages 33-38
An Initial-Value Problem for a Hyperbolic Equation….Pages 39-44
Distribution Theory….Pages 45-63
Second-Order Linear Elliptic Equations….Pages 64-77
Second-Order Linear Parabolic Equations….Pages 78-90
Front Matter….Pages 91-91
Comparison Theorems and Monotonicity Methods….Pages 93-105
Linearization….Pages 106-125
Topological Methods….Pages 126-166
Bifurcation Theory….Pages 167-191
Systems of Reaction—Diffusion Equations….Pages 192-236
Front Matter….Pages 237-237
Discontinuous Solutions of Conservation Laws….Pages 239-264
The Single Conservation Law….Pages 265-305
The Riemann Problem for Systems of Conservation Laws….Pages 306-336
Applications to Gas Dynamics….Pages 337-367
The Glimm Difference Scheme….Pages 368-390
Riemann Invariants, Entropy, and Uniqueness….Pages 391-425
Front Matter….Pages 237-237
Quasi-Linear Parabolic Systems….Pages 426-444
Front Matter….Pages 445-445
The Conley Index….Pages 447-477
Index Pairs and the Continuation Theorem….Pages 478-506
Travelling Waves….Pages 507-555
Back Matter….Pages 557-584