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Continuous families of exponential attractors for singularly perturbed equations with memory

Gatti, S, Miranville, A, Pata, V and Zelik, S (2010) Continuous families of exponential attractors for singularly perturbed equations with memory P ROY SOC EDINB A, 140. pp. 329-366.

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Abstract

For a family of semigroups S-epsilon(t): H-epsilon -> H-epsilon depending on a perturbation parameter epsilon is an element of [0,1], where the perturbation is allowed to become singular at epsilon = 0, we establish a general theorem on the existence of exponential attractors epsilon(epsilon) satisfying a suitable Holder continuity property with respect to the symmetric Hausdorff distance at every epsilon is an element of [0,1]. The result is applied to the abstract evolution equations with memorypartial derivative(t)x(t) + integral(infinity)(0) k(epsilon)(s)B-0(x(t - s))ds + B1(x(t)) = 0, epsilon is an element of (0, 1],where k(epsilon)(s) = (1/epsilon)k(s/epsilon) is the resulting of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equationpartial derivative(t)x(t) + B-0(x(t)) + B-1(x(t)) = 0,formally obtained in the singular limit epsilon -> 0.

Item Type: Article
Authors :
AuthorsEmailORCID
Gatti, SUNSPECIFIEDUNSPECIFIED
Miranville, AUNSPECIFIEDUNSPECIFIED
Pata, VUNSPECIFIEDUNSPECIFIED
Zelik, SUNSPECIFIEDUNSPECIFIED
Date : February 2010
Identification Number : https://doi.org/10.1017/S0308210509000365
Uncontrolled Keywords : NONAUTONOMOUS DYNAMICAL-SYSTEMS, DAMPED WAVE-EQUATION, VISCOELASTICITY, CONSTRUCTION, RELAXATION, STABILITY, LIMIT
Depositing User : Symplectic Elements
Date Deposited : 28 Mar 2017 10:51
Last Modified : 28 Mar 2017 10:51
URI: http://epubs.surrey.ac.uk/id/eprint/806859

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