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

We consider the Moore–Gibson–Thompson–Gurtin–Pipkin model u ttt + α u tt + β Δ 2 u t + γ Δ 2 u = - ϱ Δ θ θ t - ∫ 0 ∞ g (s) Δ θ (t - s) d s = ϱ Δ u tt + ϱ α Δ u t 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 α β > 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 α β = γ exponential stability never takes place. Even more, there exist particular kernels, called resonant, for which the semigroup exhibits periodic trajectories. [ABSTRACT FROM AUTHOR]