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Continuous families of exponential attractors for singularly perturbed equations with memory

Published online by Cambridge University Press:  30 March 2010

Stefania Gatti ,
Alain Miranville ,
Vittorino Pata  and
Sergey Zelik
Stefania Gatti
Affiliation:
Dipartimento di Matematica, Università di Modena e Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy ([email protected])
Alain Miranville
Affiliation:
Laboratoire de Mathématiques et Applications, Université de Poitiers, SP2MI, UMR CNRS 6086, Boulevard Marie et Pierre Curie, Téléport 2, 86962 Chasseneuil Futuroscope Cedex, France ([email protected])
Vittorino Pata
Affiliation:
Dipartimento di Matematica ‘F. Brioschi’, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy ([email protected])
Sergey Zelik
Affiliation:
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK ([email protected])
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Abstract

For a family of semigroups Sε(t) : ℌε → ℌε depending on a perturbation parameter ε ∈ [0, 1], where the perturbation is allowed to become singular at ε = 0, we establish a general theorem on the existence of exponential attractors εε satisfying a suitable Hölder continuity property with respect to the symmetric Hausdorff distance at every ε ∈ [0, 1]. The result is applied to the abstract evolution equations with memory

where kε(s) = (1/ε)k(s/ε) is the rescaling of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equation

formally obtained in the singular limit ε → 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 140 , Issue 2 , April 2010 , pp. 329 - 366
Copyright
Copyright © Royal Society of Edinburgh 2010

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