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.

中文翻译：

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三相滞后热弹性板的规律性和稳定性

在本文中，我们考虑以下三相滞后热弹性板：$\begin{array}{r}\left\{\begin{array}{c}\rho \ddot{你}=-\mathrm{\Delta }\left(\mu \mathrm{\Delta }你-米\theta \right),\\ C(\dot{\theta }+{\tau }_q\ddot{\theta })=米\mathrm{\Delta }(\dot{你}+{\tau }_q\ddot{你})+\mathrm{\Delta }({ķ}^{*}\alpha +{\tau }_{\nu }^{*}\theta +ķ{\tau }_{\theta }\dot{\theta }).\end{array}\right.\end{array}$我们获得了相关的分析性和指数稳定性$C_0$-当系数满足时的半群${\tau }_{\nu }\ge {ķ}^{*}{\tau }_q$. 半群的解析性仍然成立${\tau }_{\nu }<{ķ}^{*}{\tau }_q$. 然而，Routh-Hurwitz 规则证明了解的不稳定性。