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Shock Waves and Reaction—Diffusion Equations

  • Textbook
  • © 1983

Overview

Authors:
  1. Joel Smoller

    1. Department of Mathematics, University of Michigan, Ann Arbor, USA

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 258)

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Table of contents (24 chapters)

  1. Front Matter

    Pages i-xxi
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  2. Basic Linear Theory

    1. Front Matter

      Pages 1-1
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    2. Ill-Posed Problems

      • Joel Smoller
      Pages 3-12

    3. Characteristics and Initial-Value Problems

      • Joel Smoller
      Pages 13-16

    4. The One-Dimensional Wave Equation

      • Joel Smoller
      Pages 17-25

    5. Uniqueness and Energy Integrals

      • Joel Smoller
      Pages 26-32

    6. Holmgren’s Uniqueness Theorem

      • Joel Smoller
      Pages 33-38

    7. An Initial-Value Problem for a Hyperbolic Equation

      • Joel Smoller
      Pages 39-44

    8. Distribution Theory

      • Joel Smoller
      Pages 45-63

    9. Second-Order Linear Elliptic Equations

      • Joel Smoller
      Pages 64-77

    10. Second-Order Linear Parabolic Equations

      • Joel Smoller
      Pages 78-90

  3. Reaction—Diffusion Equations

    1. Front Matter

      Pages 91-91
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    2. Comparison Theorems and Monotonicity Methods

      • Joel Smoller
      Pages 93-105

    3. Linearization

      • Joel Smoller
      Pages 106-125

    4. Topological Methods

      • Joel Smoller
      Pages 126-166

    5. Bifurcation Theory

      • Joel Smoller
      Pages 167-191

    6. Systems of Reaction—Diffusion Equations

      • Joel Smoller
      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

      • Joel Smoller
      Pages 239-264

    3. The Single Conservation Law

      • Joel Smoller
      Pages 265-305
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Keywords

About this book

. . . 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.

Authors and Affiliations

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

    Joel Smoller

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