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: March 31, 2009

Revised: July 31, 2009

Published: April 2010

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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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