Ergodic Theory and Dynamical Systems

Research Article

Lower semicontinuity of attractors for non-autonomous dynamical systems

ALEXANDRE N. CARVALHOa1, JOSÉ A. LANGAa2 and JAMES C. ROBINSONa3

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

a2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1036, 41080-Sevilla, Spain (email: langa@us.es)

a3 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: j.c.robinson@warwick.ac.uk)

Abstract

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

(Received May 26 2007)

(Revised September 05 2008)