On the continuity of pullback attractors for evolution processes

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.CNPq[305447/2005-0]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq[451761/2008-1]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq[CAPES/DGU 267/2008]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP[2008/53094-4]MEC-Spain[MTM2005-01412]MEC-Spai