In this paper we investigate several qualitative properties of the solutions of the dual-phase-lag heat equation and the three-phase-lag heat equation. In the first case we assume that the parameter τT depends on the spatial position. We prove that when 2τT−τq is strictly positive the solutions are exponentially stable. When this property is satisfied in a proper sub-domain, but 2τT−τq≥0 for all the points in the case of the one-dimensional problem we also prove the exponential stability of solutions. A critical case corresponds to the situation when 2τT−τq=0 in the whole domain. It is known that the solutions are not exponentially stable. We here obtain the polynomial stability for this case. Last section of the paper is devoted to the three-phase-lag case when τT and τν ⁎ depend on the spatial variable. We here consider the case when τν ⁎≥κ⁎τq and τT is a positive constant, and obtain the analyticity of the semigroup of solutions. Exponential stability and impossibility of localization are consequences of the analyticity of the semigroup.
Bibliographical noteFunding Information:
R. Q. is supported by the Project ?An?lisis Matem?tico de las Ecuaciones en Derivada Parciales de la Termomec?nica? (MTM2013-42004-P) of the Spanish Ministry of Economy and Competitiveness. At the same time this work is part of the Project ?An?lisis Matem?tico de Problemas de la Termomec?nica? (MTM2016-74934-P) (AEI/FEDER, UE), which is currently submitted to the Spanish Ministry of Economy and Competitiveness. Y. W. is supported by China Scholarship Council.
© 2017 Elsevier Inc.
- Analyticity of solutions
- Dual-phase-lag heat equation
- Exponential stability
- Polynomial stability
- Three-phase-lag heat equation