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

We consider the Moore–Gibson–Thompson–Gurtin–Pipkin model (Formula presented.) 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 α,β,γ>0. In the subcritical regime αβ>γ, 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 αβ=γ exponential stability never takes place. Even more, there exist particular kernels, called resonant, for which the semigroup exhibits periodic trajectories.