Exponential attractors for abstract equations with memory and applications to viscoelasticity

Valeria Danese; Pelin G. Geredeli; Vittorino Pata

doi:10.3934/dcds.2015.35.2881

Article Contents

2015, Volume 35, Issue 7: 2881-2904. Doi: 10.3934/dcds.2015.35.2881

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Exponential attractors for abstract equations with memory and applications to viscoelasticity

1.
Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano
2.
Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara
3.
Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Via Bonardi 9, Milano 20133

Received: May 31, 2014

Revised: September 30, 2014

Published: May 2017

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Abstract

We consider an abstract equation with memory of the form $$\partial_t x(t) + \int_{0}^\infty k(s) A x(t-s) ds + Bx(t)=0$$ where $A,B$ are operators acting on some Banach space, and the convolution kernel $k$ is a nonnegative convex summable function of unit mass. The system is translated into an ordinary differential equation on a Banach space accounting for the presence of memory, both in the so-called history space framework and in the minimal state one. The main theoretical result is a theorem providing sufficient conditions in order for the related solution semigroups to possess finite-dimensional exponential attractors. As an application, we prove the existence of exponential attractors for the integrodifferential equation $$\partial_{t t} u - h(0)\Delta u - \int_{0}^\infty h'(s) \Delta u(t-s) ds+ f(u) = g$$ arising in the theory of isothermal viscoelasticity, which is just a particular concrete realization of the abstract model, having defined the new kernel $h(s)=k(s)+1$.

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Mathematics Subject Classification: Primary: 35B41, 37L30, 45K05; Secondary: 74D99.

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