On the continuity of pullback attractors for evolution processes

Author(s)

Carvalho, Alexandre N. ^[1] ; Langa, Jose A. ^[2] ; Robinson, James C. ^[3]

Affiliation

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Computacao, BR-13560970 Sao Carlos, SP - Brazil

^[2] Univ Seville, Dept Ecuac Diferenciales & Anal Numer, Seville 41080 - Spain

^[3] Univ Warwick, Math Inst, Coventry CV4 7AL, W Midlands - England

• Document type: Journal article
• Source: NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS; v. 71, n. 5-6, p. 1812-1824, SEP 1 2009.
• Web of Science Citations: 23

Abstract

In this paper we give general results on the continuity of pullback attractors for nonlinear evolution processes. We then revisit results of {[}D. Li, P.E. Kloeden, Equi-attraction and the continuous dependence of pullback attractors on parameters, Stoch. Dyn. 4 (3) (2004) 373-384] which show that, under certain conditions, continuity is equivalent to uniformity of attraction over a range of parameters ({''}equi-attraction{''}): we are able to simplify their proofs and weaken the conditions required for this equivalence to hold. Generalizing a classical autonomous result {[}A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992] we give bounds on the rate of convergence of attractors when the family is uniformly exponentially attracting. To apply these results in a more concrete situation we show that a non-autonomous regular perturbation of a gradient-like system produces a family of pullback attractors that are uniformly exponentially attracting: these attractors are therefore continuous, and we can give an explicit bound on the distance between members of this family. (C) 2009 Elsevier Ltd. All rights reserved. (AU)