Shock Waves and Reaction-Diffusion Equations (Grundlehren der mathematischen Wissenschaften)

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9780387942599: Shock Waves and Reaction-Diffusion Equations (Grundlehren der mathematischen Wissenschaften)
Vom Verlag:

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.

Vom Verlag:

The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley. It presents the modern ideas in these fields in a way that is accessible to a wider audience than just mathematicians.
The book is divided into four main parts: linear theory, reaction-diffusion equations, shock-wave theory, and the Conley index. For the second edition numerous typographical errors and other mistakes have been corrected and a new chapter on recent results has been added. The new chapter contains discussions of the stability of traveling waves, symmetry-breaking bifurcations, compensated compactness, viscous profiles for shock waves, and general notions for construction traveling-wave solutions for systems of nonlinear equations.

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