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Attractors for semigroups and evolution equations. (English) Zbl 0755.47049
Lezioni Lincee. Cambridge etc.: Cambridge University Press,. xi, 73 p. (1991).
In these lecture notes the asymptotic behavior of a semigroup of nonlinear operators on a metric space is studied. The main objective is to establish the existence of a compact minimal global attractor (or $B$- attractor) and to estimate its Hausdorff and fractal dimension. In the first part of the book the theory is developed for semigroups of compact operators and asymptotically compact semigroups. Concrete semi-linear evolution equations are studied in the second part. A short section is devoted to the Navier-Stokes equation. Here a recent estimate (due to the author) of the fractal dimension and the number of determining modes is presented. The principal applications concern hyperbolic equations. For them, the entire program presented in the first part, is developed. The book permits, on a few pages, to get a quick access to some important topics of dynamic systems including, in particular, the detailed treatment of interesting concrete models.

##### MSC:
 47H20 Semigroups of nonlinear operators 35-99 Partial differential equations (PDE) (MSC2000) 35B40 Asymptotic behavior of solutions of PDE 47-02 Research monographs (operator theory) 58-02 Research monographs (global analysis) 35-02 Research monographs (partial differential equations) 47D06 One-parameter semigroups and linear evolution equations