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  • © 1994

Shock Waves and Reaction—Diffusion Equations

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

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  • ISBN: 978-1-4612-0873-0
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Table of contents (25 chapters)

  1. Front Matter

    Pages i-xxiii
  2. Basic Linear Theory

    1. Front Matter

      Pages 1-1
    2. Ill-Posed Problems

      • Joel Smoller
      Pages 3-12
    3. The One-Dimensional Wave Equation

      • Joel Smoller
      Pages 17-25
    4. Uniqueness and Energy Integrals

      • Joel Smoller
      Pages 26-32
    5. Holmgren’s Uniqueness Theorem

      • Joel Smoller
      Pages 33-38
    6. Distribution Theory

      • Joel Smoller
      Pages 45-63
    7. Second-Order Linear Elliptic Equations

      • Joel Smoller
      Pages 64-77
    8. Second-Order Linear Parabolic Equations

      • Joel Smoller
      Pages 78-90
  3. Reaction-Diffusion Equations

    1. Front Matter

      Pages 91-91
    2. Linearization

      • Joel Smoller
      Pages 106-125
    3. Topological Methods

      • Joel Smoller
      Pages 126-166
    4. Bifurcation Theory

      • Joel Smoller
      Pages 167-191
    5. Systems of Reaction — Diffusion Equations

      • Joel Smoller
      Pages 192-236
  4. The Theory of Shock Waves

    1. Front Matter

      Pages 237-237
    2. The Single Conservation Law

      • Joel Smoller
      Pages 265-305

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.

Keywords

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

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: 978-0-387-94259-9

  • Softcover ISBN: 978-1-4612-6929-8

  • eBook ISBN: 978-1-4612-0873-0

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 2

  • Number of Pages: XXIII, 634

  • Topics: Analysis

Buying options

eBook
USD 219.00
Price excludes VAT (USA)
  • ISBN: 978-1-4612-0873-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 279.99
Price excludes VAT (USA)
Hardcover Book
USD 279.99
Price excludes VAT (USA)