Superlinear parabolic problems. Blow-up, global existence and steady states.

*(English)*Zbl 1128.35003
Birkhäuser Advanced Texts. Basler Lehrbücher. Basel: Birkhäuser (ISBN 978-3-7643-8441-8/hbk). xi, 584 p. (2007).

The book provides an up-to-date and self-contained account of many of the most important results and methods in the theory of superlinear parabolic and elliptic equations and systems. The authors succeeded in giving a readable presentation of both classical and current results in a research area that attracts a lot of attention. Because of the very high level of exposition, the book should be accessible to a large audience including graduate and postgraduate students and researchers in the field of partial differential equations.

The choice of subjects is well balanced. Of course, it reflects the interests of the authors but it gives a good illustration of several topics in superlinear parabolic equations and systems such as a priori bounds, blow-up and Fujita-type results, for example. The equations and systems under consideration are semilinear and special attention is devoted to problems involving gradient or non-local terms. In many cases the authors do not rewrite existing proofs of the results that they present but give their own simplified or improved versions. Also, detailed proofs of some results which are “folklore” but not proved in detail in the existing literature can be found here.

The choice of subjects is well balanced. Of course, it reflects the interests of the authors but it gives a good illustration of several topics in superlinear parabolic equations and systems such as a priori bounds, blow-up and Fujita-type results, for example. The equations and systems under consideration are semilinear and special attention is devoted to problems involving gradient or non-local terms. In many cases the authors do not rewrite existing proofs of the results that they present but give their own simplified or improved versions. Also, detailed proofs of some results which are “folklore” but not proved in detail in the existing literature can be found here.

Reviewer: Marek Fila (Bratislava)