# American Institute of Mathematical Sciences

2019, 27: 7-19. doi: 10.3934/era.2019007

## On the time decay in phase–lag thermoelasticity with two temperatures

 1 Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08222 Terrassa, Barcelona, Spain 2 Laboratoire de Mathématiques et Applications, Université de Poitiers, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

* Corresponding author: Alain Miranville

Received  June 2019 Revised  November 2019 Published  September 2019

Fund Project: The first and the third authors are supported by the project "Análisis Matemático de Problemas de la Termomecánica" (MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness.

The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions decay in a slow way. The second model that we consider is obtained taking first-order Taylor approximations for the inductive thermal displacement, the inductive temperature and the heat flux. The decay is, therefore, of exponential type.

Citation: Antonio Magaña, Alain Miranville, Ramón Quintanilla. On the time decay in phase–lag thermoelasticity with two temperatures. Electronic Research Archive, 2019, 27: 7-19. doi: 10.3934/era.2019007
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##### References:
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