An introduction to semilinear evolution equations. Transl. by Yvan Martel. Revised ed. (English) Zbl 0926.35049

Cazenave, Thierry; Haraux, Alain

Oxford Lecture Series in Mathematics and its Applications. 13. Oxford: Clarendon Press. xiv, 186 p. (1998).

This is the translation of an almost unchanged version of the French original [Ellipses, Paris (1990; Zbl 0786.35070)], supplemented by a chapter “Stability of stationary solutions”. In this chapter a version of the principle of linearized stability (“linearized exponential stability implies nonlinear exponential stability”) is developed for abstract semilinear evolution equations in Banach spaces. Moreover, for stationary solutions to boundary value problems for semilinear heat equations results are proved concerning local and global asymptotic stability as well as the connection of stability and positivity (for arbitrary space dimension).
The bibliography presents the state of the art of 1990. Even so, the book is highly recommended as a very clearly written, excellent textbook.

Reviewer: L.Recke (Berlin)

MSC:

35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35B40	Asymptotic behavior of solutions to PDEs
47H20	Semigroups of nonlinear operators
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
47D06	One-parameter semigroups and linear evolution equations
35L20	Initial-boundary value problems for second-order hyperbolic equations
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

principle of linearized stability; local and global asymptotic stability; connection of stability and positivity

Citations:

Zbl 0786.35070