# American Institute of Mathematical Sciences

January  2018, 17(1): 177-190. doi: 10.3934/cpaa.2018011

## Time decay in dual-phase-lag thermoelasticity: Critical case

 1 Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-2496, USA 2 Department of Mathematics, UPC, Colom 11, 08222 Terrassa, Spain

* Corresponding authorRamón Quintanilla

Received  March 2017 Revised  June 2017 Published  September 2017

Fund Project: The second author R. Q. is supported by the Projects "Análisis Matemático de las Ecuaciones en Derivada Parciales de la Termomecánica"(MTM2013-42004-P), "Análisis Matemático de Problemas de la Termomecánica"(MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness

This note is devoted to the study of the time decay of the one-dimensional dual-phase-lag thermoelasticity. In this theory two delay parameters $τ_q$ and $τ_{θ}$ are proposed. It is known that the system is exponentially stable if $τ_q<2 τ_{θ}$ [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that $τ_q=2 τ_{θ}$ as well the optimality of this decay rate. Second, we prove that the exponential stability remains true even if the inequality only holds in a proper sub-interval of the spatial domain, when $τ_{θ}$ is spatially dependent.

Citation: Zhuangyi Liu, Ramón Quintanilla. Time decay in dual-phase-lag thermoelasticity: Critical case. Communications on Pure & Applied Analysis, 2018, 17 (1) : 177-190. doi: 10.3934/cpaa.2018011
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