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The dimension of attractors of nonautonomous partial differential equations

Published online by Cambridge University Press:  17 February 2009

T. Caraballo ,
J. A. Langa  and
J. Valero
T. Caraballo
Affiliation:
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain; e-mail: caraball@us.es.
J. A. Langa
Affiliation:
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain; e-mail: langa@numer.us.es.
J. Valero
Affiliation:
Universidad Cardenal Herrera CEU, Comisario 3, 03203 Elche, Alicante, Spain; e-mail: valer.el@ceu.es.
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Abstract

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The concept of nonautonomous (or cocycle) attractors has become a proper tool for the study of the asymptotic behaviour of general nonautonomous partial differential equations. This is a time-dependent family of compact sets, invariant for the associated process and attracting “from –∞”. In general, the concept is rather different to the classical global attractor for autonomous dynamical systems. We prove a general result on the finite fractal dimensionality of each compact set of this family. In this way, we generalise some previous results of Chepyzhov and Vishik. Our results are also applied to differential equations with a nonlinear term having polynomial growth at most.

Type
Research Article
Information
The ANZIAM Journal , Volume 45 , Issue 2 , October 2003 , pp. 207 - 222
Copyright
Copyright © Australian Mathematical Society 2003

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