Shock waves and reaction-diffusion equations. 2nd ed. (English) Zbl 0807.35002

Smoller, Joel

Grundlehren der Mathematischen Wissenschaften. 258. New York: Springer- Verlag. xxii, 632 p. (1994).

In the second edition of this esteemed and useful exposition (for the first edition see [1983; Zbl 0508.35002]) the author added a new chapter which presents some recent developments in the theory.
Section I of the new chapter deals with reaction-diffusion systems. Results of C. Jones [Trans. Am. Math. Soc. 286, 431-469 (1984; Zbl 0567.35044)] on the stability of the travelling wave of the FitzHugh- Nagumo equations are described. Further, equivariant Conley index techniques are applied in order to prove theorems about symmetry breaking for \(O(n)\)-invariant semilinear elliptic boundary value problems.
Section II concerns shock-wave theory. First, it is shown how compensated compactness can be used to obtain global existence theorems for certain systems of pairs of conservation laws. Then results of T.-P. Liu [Mem. Am. Math. Soc. 328, 108 p. (1985; Zbl 0617.35058)] on nonlinear stability of travelling waves for systems of viscous conservation laws are discussed.
Section III deals with Conley’s connection index and connection matrix which are useful in constructing travelling waves for reaction-diffusion systems.
Finally, Section IV is devoted to results of R. Gardner and C. Jones [Trans. Am. Math. Soc. 327, No. 2, 465-524 (1991; Zbl 0755.35056)] on the point spectrum of linear operators which arise in stability problems for travelling waves of parabolic systems.

Reviewer: L.Recke (Berlin)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35L65	Hyperbolic conservation laws
35L67	Shocks and singularities for hyperbolic equations
35K57	Reaction-diffusion equations

Keywords:

Conley index; symmetry breaking; compensated compactness; travelling waves