On the regularity and stability of three-phase-lag thermoelastic plate

Abstract

In this paper we consider the following three-phase-lag thermoelastic plate: $\begin{array}{r}\left\{\begin{array}{c}\rho \ddot{u}=-\mathrm{\Delta }\left(\mu \mathrm{\Delta }u-m\theta \right),\\ c(\dot{\theta }+{\tau }_q\ddot{\theta })=m\mathrm{\Delta }(\dot{u}+{\tau }_q\ddot{u})+\mathrm{\Delta }(k^{\ast }\alpha +{\tau }_{\nu }^{\ast }\theta +k{\tau }_{\theta }\dot{\theta }).\end{array}\right.\end{array}$We obtain analyticity and exponential stability for the associated $C_0$-semigroup when the coefficients satisfy ${\tau }_{\nu }\ge k^{\ast }{\tau }_q$. The analyticity of the semigroup still holds when ${\tau }_{\nu }<k^{\ast }{\tau }_q$. However, the instability of solution was proved by the Routh–Hurwitz rule.

Keywords:

• Three-phase-lag thermoelastic plate
• analyticity
• exponential stability

MSCs 2000:

• 35Q35
• 35Q30
• 35L65
• 76N10

Acknowledgments

The work of R. Quintanilla was supported by projects ‘Análisis Matemáticode Problemas de la Termomecánica’ (MTM2016-74934-P) (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness, and ‘Análisis Matemático Aplicado a la Termomecánica’ (Ref. PID2019- 105118GB-I00), (AEI/FEDER, UE) of the Spanish Ministry of Science, Innovation and Universities.

Disclosure statement

No potential conflict of interest was reported by the authors.