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

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.

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 |