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

Zhuangyi Liu; Ramón Quintanilla

doi:10.3934/cpaa.2018011

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

Zhuangyi Liu, Ramón Quintanilla

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

This note is devoted to the study of the time decay of the onedimensional 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 < 2τ_θ [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that τ_q = 2τ_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.

Original language	English (US)
Pages (from-to)	177-190
Number of pages	14
Journal	Communications on Pure and Applied Analysis
Volume	17
Issue number	1
DOIs	https://doi.org/10.3934/cpaa.2018011
State	Published - Jan 2018

Bibliographical note

Funding Information:
2000 Mathematics Subject Classification. Primary: 35Q70, 35B40; Secondary: 74F05. Key words and phrases. Phase-lag, thermoelasticity, polynomial stability, exponential stability. 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 Pro-blemas de la Termomecánica“(MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness. ∗ Corresponding author: Ramón Quintanilla.

Keywords

Exponential stability
Phase-lag
Polynomial stability
Thermoelasticity

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10.3934/cpaa.2018011

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title = "Time decay in dual-phase-lag thermoelasticity: Critical case",

abstract = "This note is devoted to the study of the time decay of the onedimensional dual-phase-lag thermoelasticity. In this theory two delay parameters τq and τθ are proposed. It is known that the system is exponentially stable if τq < 2τθ [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that τq = 2τθ 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 τθ is spatially dependent.",

keywords = "Exponential stability, Phase-lag, Polynomial stability, Thermoelasticity",

author = "Zhuangyi Liu and Ram{\'o}n Quintanilla",

note = "Funding Information: 2000 Mathematics Subject Classification. Primary: 35Q70, 35B40; Secondary: 74F05. Key words and phrases. Phase-lag, thermoelasticity, polynomial stability, exponential stability. The second author R. Q. is supported by the Projects “An{\'a}lisis Matem{\'a}tico de las Ecuaciones en Derivada Parciales de la Termomec{\'a}nica“(MTM2013-42004-P), “An{\'a}lisis Matem{\'a}tico de Pro-blemas de la Termomec{\'a}nica“(MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness. ∗ Corresponding author: Ram{\'o}n Quintanilla.",

year = "2018",

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doi = "10.3934/cpaa.2018011",

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T1 - Time decay in dual-phase-lag thermoelasticity

T2 - Critical case

AU - Liu, Zhuangyi

AU - Quintanilla, Ramón

N1 - Funding Information: 2000 Mathematics Subject Classification. Primary: 35Q70, 35B40; Secondary: 74F05. Key words and phrases. Phase-lag, thermoelasticity, polynomial stability, exponential stability. 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 Pro-blemas de la Termomecánica“(MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness. ∗ Corresponding author: Ramón Quintanilla.

PY - 2018/1

Y1 - 2018/1

N2 - This note is devoted to the study of the time decay of the onedimensional dual-phase-lag thermoelasticity. In this theory two delay parameters τq and τθ are proposed. It is known that the system is exponentially stable if τq < 2τθ [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that τq = 2τθ 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 τθ is spatially dependent.

AB - This note is devoted to the study of the time decay of the onedimensional dual-phase-lag thermoelasticity. In this theory two delay parameters τq and τθ are proposed. It is known that the system is exponentially stable if τq < 2τθ [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that τq = 2τθ 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 τθ is spatially dependent.

KW - Exponential stability

KW - Phase-lag

KW - Polynomial stability

KW - Thermoelasticity

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JO - Communications on Pure and Applied Analysis

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

Abstract

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