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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 ISSN 0362-546X.
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Official URL: http://dx.doi.org/10.1016/j.na.2009.01.016
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: |
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| 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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