Exponential stability in linear viscoelasticity with almost flat memory kernels

Vittorino Pata

doi:10.3934/cpaa.2010.9.721

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2010, Volume 9, Issue 3: 721-730. Doi: 10.3934/cpaa.2010.9.721

This issue Previous Article Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system Next Article Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions

Exponential stability in linear viscoelasticity with almost flat memory kernels

Vittorino Pata^1,

1.
Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano

Received: April 2009

Revised: August 2009

Published: May 2010

Abstract / Introduction Related Papers Cited by

Abstract

This article is focused on the solution semigroup in the history space framework arising from an abstract version of the boundary value problem with memory

$\partial_{t t} u(t)-\Delta [u(t)+\int_0^\infty \mu(s)[u(t)-u(t-s)] ds ]=0,\quad u(t)_{|\partial\Omega}=0,$

modelling linear viscoelasticity. The exponential stability of the semigroup is discussed, establishing a necessary and sufficient condition involving the memory kernel $\mu$.

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Mathematics Subject Classification: Primary: 35B40, 45K05; Secondary: 45M10, 47D06.

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