Ergodic Theory and Dynamical Systems

Research Article

Exponential global attractors for semigroups in metric spaces with applications to differential equations

ALEXANDRE N. CARVALHOa1 and JAN W. CHOLEWAa2

a1 Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil (email: andcarva@icmc.usp.br)

a2 Institute of Mathematics, Silesian University, 40-007, Katowice, Poland (email: jcholewa@ux2.math.us.edu.pl)

Abstract

In this article semigroups in a general metric space V, which have pointwise exponentially attracting local unstable manifolds of compact invariant sets, are considered. We show that under a suitable set of assumptions these semigroups possess strong exponential dissipative properties. In particular, there exists a compact global attractor which exponentially attracts each bounded subset of V. Applications of abstract results to ordinary and partial differential equations are given.

(Received January 26 2010)

(Revised September 22 2010)

(Online publication March 15 2011)