Shock Waves and Reaction—Diffusion Equations

  • Joel Smoller
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 258)

Table of contents

  1. Front Matter
    Pages i-xxiii
  2. Basic Linear Theory

    1. Front Matter
      Pages 1-1
    2. Joel Smoller
      Pages 3-12
    3. Joel Smoller
      Pages 17-25
    4. Joel Smoller
      Pages 26-32
    5. Joel Smoller
      Pages 33-38
    6. Joel Smoller
      Pages 45-63
    7. Joel Smoller
      Pages 64-77
    8. Joel Smoller
      Pages 78-90
  3. Reaction-Diffusion Equations

    1. Front Matter
      Pages 91-91
    2. Joel Smoller
      Pages 106-125
    3. Joel Smoller
      Pages 126-166
    4. Joel Smoller
      Pages 167-191
    5. Joel Smoller
      Pages 192-236
  4. The Theory of Shock Waves

    1. Front Matter
      Pages 237-237
    2. Joel Smoller
      Pages 265-305

About this book

Introduction

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

  • Joel Smoller
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-0873-0
  • Copyright Information Springer-Verlag New York, Inc / Northern Songs Limited 1994
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-6929-8
  • Online ISBN 978-1-4612-0873-0
  • Series Print ISSN 0072-7830
  • About this book