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Shock Waves and Reaction—Diffusion Equations

  • Book
  • © 1994

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 (25 chapters)

  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

      • 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

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.

Authors and Affiliations

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

    Joel Smoller

Bibliographic Information

  • Book Title: Shock Waves and Reaction—Diffusion Equations

  • Authors: Joel Smoller

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-1-4612-0873-0

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1994

  • Hardcover ISBN: 978-0-387-94259-9Published: 14 October 1994

  • Softcover ISBN: 978-1-4612-6929-8Published: 08 October 2012

  • eBook ISBN: 978-1-4612-0873-0Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 2

  • Number of Pages: XXIII, 634

  • Topics: Analysis

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