Asymptotics of the Coleman-Gurtin model

Mickaël D. Chekroun; Francesco di Plinio; Nathan Glatt-Holtz; Vittorino Pata

doi:10.3934/dcdss.2011.4.351

Article Contents

2011, Volume 4, Issue 2: 351-369. Doi: 10.3934/dcdss.2011.4.351

This issue Previous Article Anisotropic phase field equations of arbitrary order Next Article Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes

Asymptotics of the Coleman-Gurtin model

1.
École Normale Supérieure - CERES-ERTI, Normale Supérieure - Ce 75231 Paris Cedex 05
2.
Indiana University Mathematics Department and The Institute of Scientiﬁc Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405
3.
Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405
4.
Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano

Received: January 2009

Revised: May 2009

Published: April 2011

Abstract / Introduction Related Papers Cited by

Abstract

This paper is concerned with the integrodifferential equation

$\partial_{t} u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\d s + \varphi(u)=f$

arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $\varphi$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.

Keywords:

Mathematics Subject Classification: 35B41, 35K05, 45K05, 47H20.

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