Shock Waves and Reaction-Diffusion Equations
by Joel Smoller

Table of Contents

Acknowledgment
Preface to the Second Edition
Preface to the First Edition
List of Frequently Used Symbols

PART I
Basic Linear Theory

CHAPTER 1
Ill-Posed Problems
§A. Some Examples
§B. Lewy's Example

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
§A. A Cursory View
§B. Fundamental Solutions
§C. Appendix

CHAPTER 8
Second-Order Linear Elliptic Equations
§A. The Strong Maximum Principle
§B. A-Priori Estimates
§C. Existence of Solutions
§D. Elliptic Regularity

CHAPTER 9
Second-Order Linear Parabolic Equations
§A. The Heat Equation
§B. Strong Maximum Principles

PART II
Reaction- Diffusion Equations

CHAPTER 10
Comparison Theorems and Monotonicity Methods
§A. Comparison Theorems for Nonlinear Equations
§B. Upper and Lower Solutions
§C. Applications

CHAPTER 11
Linearization
§A. Spectral Theory for Self-Adjoin! Operators
§B. Linearized Stability
§C. Appendix: The Krein-Rutman Theorem

CHAPTER 12
Topological Methods
§A. Degree Theory in Rⁿ
§B. The Leray-Schauder Degree
§C. An Introduction to Morse Theory
§D. A Rapid Course in Topology

CHAPTER 13
Bifurcation Theory
§A. The Implicit Function Theorem
§B. Stability of Bifurcating Solutions
§C. Some General Bifurcation Theorems
§D. Spontaneous Bifurcation; An Example

CHAPTER 14
Systems of Reaction-Diffusion Equations
§A. Local Existence of Solutions
§B. Invariant Regions
§C. A Comparison Theorem
§D. Decay to Spatially Homogeneous Solutions
§E. A Lyapunov Function for Contracting Rectangles
§F. Applications to the Equations of Mathematical Ecology

PART III
The Theory of Shock Waves

CHAPTER 15
Discontinuous Solutions of Conservation Laws
§A. Discontinuous Solutions
§B. Weak Solutions of Conservation Laws
§C. Evolutionary Systems
§D. The Shock Inequalities
§E. Irreversibility

CHAPTER 16
The Single Conservation Law
§A. Existence of an Entropy Solution
§B. Uniqueness of the Entropy Solution
§C. Asymptotic Behavior of the Entropy Solution
§D. The Riemann Problem for a Scalar Conservation Law

CHAPTER 17
The Riemann Problem for Systems of Conservation Laws
§A. The p-System
§B. Shocks and Simple Waves
§C. Solution of the General Riemann Problem

CHAPTER 18
Applications to Gas Dynamics
§A. The Shock Inequalities
§B. The Riemann Problem in Gas Dynamics
§C. Interaction of Shock Waves

CHAPTER 19
The Glimm Difference Scheme
§A. The Interaction Estimate
§B. The Difference Approximation
§C. Convergence

CHAPTER 20
Riemann Invariants, Entropy, and Uniqueness
§A. Riemann Invariants
§B. A Concept of Entropy
§C. Solutions with "Big" Data
§D. Instability of Rarefaction Shocks
§E. Oleinik's Uniqueness Theorem

CHAPTER 21
Quasi-Linear Parabolic Systems
§A. Gradient Systems
§B. Artificial Viscosity
§C. Isentropic Gas Dynamics

PART IV
The Conley Index

CHAPTER 22
The Conley Index
§A. An Impressionistic Overview
§B. Isolated Invariant Sets and Isolating Blocks
§C. The Homotopy Index

CHAPTER 23
Index Pairs and the Continuation Theorem
§A. Morse Decompositions and Index Pairs
§B. The Conley Index of an Isolated Invariant Set
§C. Continuation
§D. Some Further Remarks

CHAPTER 24
Travelling Waves
§A. The Structure of Weak Shock Waves
§B. The Structure of Magnetohydrodynamic Shock Waves
§C. Periodic Travelling Waves
§D. Stability of Steady-State Solutions
§E. Instability of Equilibrium Solutions of the Neumann Problem
§F. Appendix: A Criterion for Nondegeneracy

CHAPTER 25
Recent Results
Section I. Reaction-Diffusion Equations
§A. Stability of the Fitz-Hugh-Nagumo Travelling Pulse
§B. Symmetry-Breaking
§C. A Bifurcation Theorem
§D. Equivariant Conley Index
§E. Application to Semilinear Elliptic Equations
§E. Concluding Remarks
Section II. Theory of Shock Waves
§A. Compensated Compactness
§B. Stability of Shock Waves
§C. Miscellaneous Results
Section III. Conley Index Theory
§A. The Connection Index
§B. Conley's Connection Matrix
§C. Miscellaneous Results
Section IV. Stability of Travelling Waves-A Topological Approach
§A. Introduction
§B. The Search for σ_P(L)
§C. Applications to Fast-Slow Systems
§E. References

Bibliography
Author Index
Subject Index

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