On the continuity of pullback attractors for evolution processes (Q1030074)

The authors provide general results on continuity properties for pullback attractors of nonlinear evolution processes. They revisit earlier results due to \textit{D.\ Li} and \textit{P. E.\ Kloeden} [Stoch.\ Dyn.\ 4, No. 3, 373--384 (2004; Zbl 1059.37015)], which show that, under certain conditions, continuity is equivalent to uniformity of attraction over a range of parameters (``equi-attraction''). The corresponding proofs are simplified and the conditions for the above equivalence are weakened. Moreover, an autonomous result of [\textit{A.V.\ Babin} and \textit{M. I.\ Vishik}, Attractors of evolution equations. Transl. from the Russian by A. V. Babin. Studies in Mathematics and its Applications. 25. (Amsterdam) etc.: North- Holland. (1992; Zbl 0778.58002)] is generalized to the nonautonomous situation: Bounds on the rate of convergence of attractors are given, when the family is uniformly exponentially attracting. The results are illustrated as follows: It is shown that a nonautonomous regular perturbation of a gradient-like system produces a family of pullback attractors which are uniformly exponentially attracting. These attractors are therefore continuous, and an explicit bound on the distance between members of this family is given.