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Lower semicontinuity of attractors for non-autonomous dynamical systems

Published online by Cambridge University Press:  03 February 2009

ALEXANDRE N. CARVALHO ,
JOSÉ A. LANGA  and
JAMES C. ROBINSON
ALEXANDRE N. CARVALHO
Affiliation:
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: [email protected])
JOSÉ A. LANGA
Affiliation:
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1036, 41080-Sevilla, Spain (email: [email protected])
JAMES C. ROBINSON
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1765 - 1780
Copyright
Copyright © Cambridge University Press 2009

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