Shock Waves and Reaction—Diffusion Equations

1. Front Matter

   Pages i-xxiii

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2. Basic Linear Theory

   1. Front Matter

      Pages 1-1

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   2. Ill-Posed Problems

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      Pages 3-12

   3. Characteristics and Initial-Value Problems

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      Pages 13-16

   4. The One-Dimensional Wave Equation

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      Pages 17-25

   5. Uniqueness and Energy Integrals

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      Pages 26-32

   6. Holmgren’s Uniqueness Theorem

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      Pages 33-38

   7. An Initial-Value Problem for a Hyperbolic Equation

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      Pages 39-44

   8. Distribution Theory

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      Pages 45-63

   9. Second-Order Linear Elliptic Equations

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      Pages 64-77

   10. Second-Order Linear Parabolic Equations

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       Pages 78-90

3. Reaction-Diffusion Equations

   1. Front Matter

      Pages 91-91

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   2. Comparison Theorems and Monotonicity Methods

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      Pages 93-105

   3. Linearization

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      Pages 106-125

   4. Topological Methods

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      Pages 126-166

   5. Bifurcation Theory

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      Pages 167-191

   6. Systems of Reaction — Diffusion Equations

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      Pages 192-236

4. The Theory of Shock Waves

   1. Front Matter

      Pages 237-237

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   2. Discontinuous Solutions of Conservation Laws

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      Pages 239-264

   3. The Single Conservation Law

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      Pages 265-305

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 con­ structing travelling waves for systems of nonlinear equations. The final sec­ tion, 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 applica­ ble to many interesting reaction-diffusion systems.

• Reaction-Diffusion Equations
• bifurcation
• compactness
• distribution
• integral
• stability

• Department of Mathematics, University of Michigan, Ann Arbor, USA

  Joel Smoller

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