On the exponential stability of the Moore–Gibson–Thompson–Gurtin–Pipkin thermoviscoelastic plate

We consider the Moore–Gibson–Thompson–Gurtin–Pipkin model \( {\left\{ \begin{array}{ll} u_{ttt}+\alpha u_{tt} + \beta \Delta ^2 u_t + \gamma \Delta ^2 u =- \varrho \Delta \theta \\ \displaystyle \theta _t - \int _0^\infty g(s)\Delta \theta (t-s)ds = \varrho \Delta u_{tt} + \varrho \alpha \Delta u_t \end{array}\right. } \) $$\left\{\begin{array}{c}{u}_{\mathit{ttt}}+\alpha u_{\mathit{tt}}+\beta {\mathrm{\Delta }}^2{u}_t+\gamma {\mathrm{\Delta }}^{2}u=-\varrho \mathrm{\Delta }\theta \hfill \\ {\displaystyle {\theta }_t-{\int }_0^{\infty }g(s)\mathrm{\Delta }\theta (t-s)ds=\varrho \mathrm{\Delta }{u}_{\mathit{tt}}+\varrho \alpha \mathrm{\Delta }{u}_t}\hfill \end{array}\right)$$ where g is a positive, convex, and summable memory kernel. The system is shown to generate a strongly continuous semigroup, whose stability properties depend of the structural parameters \(\alpha ,\beta ,\gamma >0\) $\alpha ,\beta ,\gamma >0$ . In the subcritical regime \(\alpha \beta >\gamma \) $\alpha \beta >\gamma$ , we provide a necessary and sufficient condition on the memory kernel in order for exponential stability to occur. Such a condition is actually very general, allowing, for instance, any compactly supported g of the above kind. On the contrary, in the critical regime \(\alpha \beta =\gamma \) $\alpha \beta =\gamma$ exponential stability never takes place. Even more, there exist particular kernels, called resonant, for which the semigroup exhibits periodic trajectories.