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March  2010, 13(2): 375-391. doi: 10.3934/dcdsb.2010.13.375

On the time decay of solutions in porous-thermo-elasticity of type II

M. Carme Leseduarte 1, , Antonio Magaña 1, and  Ramón Quintanilla 1,

1. 

Departament de Matemàtica Aplicada 2, ETSEIAT–UPC, C. Colom 11, 08222 Terrassa, Barcelona, Spain, Spain, Spain

Received  November 2008 Revised  July 2009 Published  December 2009

In this paper we investigate the asymptotic behaviour of the solutions of the linear theory of thermo-porous-elasticity. That is, we consider the theory of elastic materials with voids when the heat conduction is of type II. We assume that the only dissipation mechanism is the porous dissipation. First we prove that, generically, the solutions are exponentially stable on time or, in other words, the decay of solutions can be controlled by a negative exponential for a generic class of materials. The reason lies in the fact that the temperature is strongly coupled with both the microscopic and macroscopic structures of the materials and plays the role of a "driving belt" between the dissipation at the microscopic structure and the macroscopic one. Later we note that the decay of solutions cannot be fast enough to make the solutions be zero in a finite period of time. Finally, we show that when the coupling term between the microscopic (or macroscopic) structure and the thermal variable vanishes, the solutions do not decay exponentially (generically).
Keywords: semigroup of contractions., slow decay, exponential stability, porous-thermo-elasticity of type II.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35R9.
Citation: M. Carme Leseduarte, Antonio Magaña, Ramón Quintanilla. On the time decay of solutions in porous-thermo-elasticity of type II. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 375-391. doi: 10.3934/dcdsb.2010.13.375
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