. . . 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.
Inhalt
I Basic Linear Theory.- 1 Ill-Posed Problems.- . Some Examples.- . Lewy's Example.- 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.- . A Cursory View.- . Fundamental Solutions.- . Appendix.- 8 Second-Order Linear Elliptic Equations.- . The Strong Maximum Principle.- . A-Priori Estimates.- . Existence of Solutions.- . Elliptic Regularity.- 9 Second-Order Linear Parabolic Equations.- . The Heat Equation.- . Strong Maximum Principles.- II Reaction-Diffusion Equations.- 10 Comparison Theorems and Monotonicity Methods.- . Comparison Theorems for Nonlinear Equations.- . Upper and Lower Solutions.- . Applications.- 11 Linearization.- . Spectral Theory for Self-Adjoint Operators.- . Linearized Stability.- . Appendix: The Krein-Rutman Theorem.- 12 Topological Methods.- . Degree Theory in Rn.- . The Leray-Schauder Degree.- . An Introduction to Morse Theory.- . A Rapid Course in Topology.- 13 Bifurcation Theory.- . The Implicit Function Theorem.- . Stability of Bifurcating Solutions.- . Some General Bifurcation Theorems.- . Spontaneous Bifurcation; An Example.- 14 Systems of Reaction-Diffusion Equations.- . Local Existence of Solutions.- . Invariant Regions.- . A Comparison Theorem.- . Decay to Spatially Homogeneous Solutions.- . A Lyapunov Function for Contracting Rectangles.- . Applications to the Equations of Mathematical Ecology.- III The Theory of Shock Waves.- 15 Discontinuous Solutions of Conservation Laws.- . Discontinuous Solutions.- . Weak Solutions of Conservation Laws.- . Evolutionary Systems.- . The Shock Inequalities.- . Irreversibility.- 16 The Single Conservation Law.- . Existence of an Entropy Solution.- . Uniqueness of the Entropy Solution.- . Asymptotic Behavior of the Entropy Solution.- . The Riemann Problem for a Scalar Conservation Law.- 17 The Riemann Problem for Systems of Conservation Laws.- . The p-System.- . Shocks and Simple Waves.- . Solution of the General Riemann Problem.- 18 Applications to Gas Dynamics.- . The Shock Inequalities.- . The Riemann Problem in Gas Dynamics.- . Interaction of Shock Waves.- 19 The Glimm Difference Scheme.- . The Interaction Estimate.- . The Difference Approximation.- . Convergence.- 20 Riemann Invariants, Entropy, and Uniqueness.- . Riemann Invariants.- . A Concept of Entropy.- . Solutions with "Big" Data.- . Instability of Rarefaction Shocks.- . Oleinik's Uniqueness Theorem.- 21 Quasi-Linear Parabolic Systems.- . Gradient Systems.- . Artificial Viscosity.- . Isentropic Gas Dynamics.- IV The Conley Index.- 22 The Conley Index.- . An Impressionistic Overview.- . Isolated Invariant Sets and Isolating Blocks.- . The Homotopy Index.- 23 Index Pairs and the Continuation Theorem.- . Morse Decompositions and Index Pairs.- . The Conley Index of an Isolated Invariant Set.- . Continuation.- . Some Further Remarks.- 24 Travelling Waves.- . The Structure of Weak Shock Waves.- . The Structure of Magnetohydrodynamic Shock Waves.- . Periodic Travelling Waves.- . Stability of Steady-State Solutions.- . Instability of Equilibrium Solutions of the Neumann Problem.- . Appendix: A Criterion for Nondegeneracy.- Author Index.