Shock Waves and Reaction-Diffusion Equations - Joel Smoller

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Résumé :
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.BRThe 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.

Sommaire:
1 Ill-Posed Problems.- 2 Characteristics and Initial-Value Problems.- 3 The One-Dimensional Wave Equation.- 4 Uniqueness and Energy Integrals.- 5 Holmgren's Uniqueness Theorem.- 6 An Initial-Value Problem for a Hyperbolic Equation.- 7 Distribution Theory.- 8 Second-Order Linear Elliptic Equations.- 9 Second-Order Linear Parabolic Equations.- 10 Comparison Theorems and Monotonicity Methods.- 11 Linearization.- 12 Topological Methods.- 13 Bifurcation Theory.- 14 Systems of Reaction-Diffusion Equations.- 15 Discontinuous Solutions of Conservation Laws.- 16 The Single Conservation Law.- 17 The Riemann Problem for Systems of Conservation Laws.- 18 Applications to Gas Dynamics.- 19 The Glimm Difference Scheme.- 20 Riemann Invariants, Entropy, and Uniqueness.- 21 Quasi-Linear Parabolic Systems.- 22 The Conley Index.- 23 Index Pairs and the Continuation Theorem.- 24 Travelling Waves.- 25 Recent Results.- Author Index.