Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system

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.

On the continuity of pullback attractors for evolution processes

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.

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Lower semicontinuity of attractors for non-autonomous dynamical systems

Dynamical Systems and Numerical Analysis. By A. Stuart & R. Humphries. Cambridge University Press, 1996. 685 pp. ISBN 0-521-49672-1. £40.

We study the stability of attractors under non-autonomous perturbations that are uniformly small in time. While in general the pullback attractors for the nonautonomous problems converge towards the autonomous attractor only in the Hausdorfi semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a ‘gradient-like’ structure (the union of the unstable manifolds of a flnite number of hyperbolic equilibria) implies convergence (i.e. also lower semicontinuity) provided that the local unstable manifolds perturb continuously. We go further when the underlying autonomous system is itself gradient-like, and show that all trajectories converge to one of the hyperbolic trajectories as t ! 1. In flnite-dimensional systems, in which we can reverse time and apply similar arguments to deduce that all bounded orbits converge to a hyperbolic trajectory as t ! i1, this implies that the ‘gradient-like’ structure of the attractor is also preserved under small non-autonomous perturbations: the pullback attractor is given as the union of the unstable manifolds of a flnite number of hyperbolic trajectories.

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James C. Robinson

Papers

The structure of attractors in non-autonomous perturbations of gradient-like systems