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**A stability technique for evolution partial differential equations. A dynamical systems approach.**
*(English)*
Zbl 1065.35002

Progress in Nonlinear Differential Equations and their Applications 56. Boston, MA: Birkhäuser (ISBN 0-8176-4146-7/hbk). xix, 377 p. (2004).

This book introduces a new method for the study of the asymptotic behavior of solutions to evolution partial differential equations; much of the text is dedicated to the application of this method to a wide class of nonlinear diffusion equations. The underlying theory hinges a new stability result, formulated in the abstract setting of infinite-dimensional dynamical systems, which states that under certain hypotheses, the \(\omega\)-limit set of a perturbed dynamical system is stable under arbitrary asymptotically small perturbations.

The stability theorem is examined in detail in the first chapter, followed by a review of basic results and methods – many original to the authors – for the solution of nonlinear diffusion equations. Further chapters provide a self-contained analysis of specific equations, with carefully constructed theorems, proofs, and references. In addition to the derivation of interesting limiting behaviors, the book features a variety of estimation techniques for solutions of semi- and quasilinear parabolic equations.

The stability theorem is examined in detail in the first chapter, followed by a review of basic results and methods – many original to the authors – for the solution of nonlinear diffusion equations. Further chapters provide a self-contained analysis of specific equations, with carefully constructed theorems, proofs, and references. In addition to the derivation of interesting limiting behaviors, the book features a variety of estimation techniques for solutions of semi- and quasilinear parabolic equations.

Reviewer: Andrey E. Shishkov (Donetsk)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

37L15 | Stability problems for infinite-dimensional dissipative dynamical systems |

35K55 | Nonlinear parabolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35B41 | Attractors |