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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Asymptotics of the Coleman-Gurtin model

Pages: 351 - 369, Volume 4, Issue 2, April 2011

doi:10.3934/dcdss.2011.4.351       Abstract        References        Full Text (488.2K)       Related Articles

Mickaël D. Chekroun - École Normale Supérieure - CERES-ERTI, Normale Supérieure - Ce 75231 Paris Cedex 05, France (email)
Francesco di Plinio - Indiana University Mathematics Department and The Institute of Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, United States (email)
Nathan Glatt-Holtz - Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, United States (email)
Vittorino Pata - Politecnico di Milano - Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, 20133 Milano, Italy (email)

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:  Heat conduction with memory, history space framework, solution semigroup, global attractor, exponential attractor.
Mathematics Subject Classification:  35B41, 35K05, 45K05, 47H20.

Received: January 2009;      Revised: May 2009;      Published: November 2010.

 References