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On the continuity of pullback attractors for evolution processes

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Carvalho, Alexandre Nolasco de, Langa, José A. and Robinson, James C. (2009) On the continuity of pullback attractors for evolution processes. Nonlinear Analysis: Theory, Methods & Applications, Vol.71 (No.5-6). pp. 1812-1824. doi:10.1016/j.na.2009.01.016

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Official URL: http://dx.doi.org/10.1016/j.na.2009.01.016

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: Nonlinear Analysis: Theory, Methods & Applications
Publisher: Pergamon-Elsevier Science Ltd
ISSN: 0362-546X
Official Date: 1 September 2009
Dates:
DateEvent
1 September 2009Published
Volume: Vol.71
Number: No.5-6
Number of Pages: 13
Page Range: pp. 1812-1824
DOI: 10.1016/j.na.2009.01.016
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Funder: CNPq, FAPESP, MEC-Spain
Grant number: 305447/2005-0, 451761/2008-1, CAPES/DGU 267/2008, 2008/53094-4, MTM2005-01412

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