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On the continuity of global attractors


Authors: Luan T. Hoang, Eric J. Olson and James C. Robinson
Journal: Proc. Amer. Math. Soc. 143 (2015), 4389-4395
MSC (2010): Primary 35B41
DOI: https://doi.org/10.1090/proc/12598
Published electronically: April 6, 2015
MathSciNet review: 3373937
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Abstract: Let $ \Lambda $ be a complete metric space, and let $ \{S_\lambda (\cdot ):\ \lambda \in \Lambda \}$ be a parametrised family of semigroups with global attractors $ \mathscr {A}_\lambda $. We assume that there exists a fixed bounded set $ D$ such that $ \mathscr {A}_\lambda \subset D$ for every $ \lambda \in \Lambda $. By viewing the attractors as the limit as $ t\to \infty $ of the sets $ S_\lambda (t)D$, we give simple proofs of the equivalence of `equi-attraction' to continuity (when this convergence is uniform in $ \lambda $) and show that the attractors $ \mathscr {A}_\lambda $ are continuous in $ \lambda $ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise).


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Additional Information

Luan T. Hoang
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409-1042
Email: luan.hoang@ttu.edu

Eric J. Olson
Affiliation: Department of Mathematics/084, University of Nevada, Reno, Nevada 89557
Email: ejolson@unr.edu

James C. Robinson
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: j.c.robinson@warwick.ac.uk

DOI: https://doi.org/10.1090/proc/12598
Keywords: Global attractor, continuity, Baire one function, equi-attraction, Dini's Theorem
Received by editor(s): July 11, 2014
Received by editor(s) in revised form: July 15, 2014
Published electronically: April 6, 2015
Additional Notes: The third author was supported by an EPSRC Leadership Fellowship EP/G007470/1, which supported the time spent in Warwick by the first and second authors
Communicated by: Yingefi Yi
Article copyright: © Copyright 2015 American Mathematical Society