## Abstract

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 τ_{T} depends on the spatial position. We prove that when 2τ_{T}−τ_{q} is strictly positive the solutions are exponentially stable. When this property is satisfied in a proper sub-domain, but 2τ_{T}−τ_{q}≥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 2τ_{T}−τ_{q}=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 τ_{T} and τ_{ν} ^{⁎} depend on the spatial variable. We here consider the case when τ_{ν} ^{⁎}≥κ^{⁎}τ_{q} and τ_{T} 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.

Original language | English (US) |
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Pages (from-to) | 422-438 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 455 |

Issue number | 1 |

DOIs | |

State | Published - Nov 1 2017 |

### Bibliographical note

Funding 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.

Publisher Copyright:

© 2017 Elsevier Inc.

## Keywords

- Analyticity of solutions
- Dual-phase-lag heat equation
- Exponential stability
- Polynomial stability
- Three-phase-lag heat equation