Shock Waves and Reaction-Diffusion Equations.

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 constructing travelling waves for systems of nonlinear equations.The final section, 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 applicable to many interesting reaction-diffusion systems.Many people have been kind in assisting me with this revision; in particular, special thanks are due to Bob Gardner, Xabier Garaizer, David Hoff, Chen Jing, Tai-Ping Liu, Blake Temple, Rick Wasserman, Zhou-Ping Xin, and especially to Vijay.Pant for his careful reading of the manuscript.In addition, I received many letters from mathematicians all over the world, who either corrected some of my errors, or showed me "different proofs."I wish to thank all of them for their interest.In closing, I want to dedicate this edition to the memory of both Charles Conley and Edward Conway.I This is not quite true; there are some good survey articles on shock waves (e.g., [Lx 5]) but these do not contain many proofs.Also in the theory of reaction -diffusion equations, there are the books [Pi] and [Mu], but they both seem to me to be research monographs.