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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]

We investigate the dissipative structure for the system of Moore–Gibson–Thompson (MGT) thermoelasticity in the whole space. To analyze the dissipative structure, it is very useful to rewrite the equations into a symmetric hyperbolic system and apply the so‐called stability condition. When we rewrite our system into the symmetric hyperbolic form in the multidimensional case, it is important to take the constraint conditions into account. Indeed, the stability condition with the constraint conditions guarantees the dissipative property for our system in some cases. In this paper, we introduce the stability condition with constraints for the general problem and apply this argument to the system of MGT thermoelasticity. Furthermore, we discuss the optimality of the decay estimates we obtain, together with the no regularity‐loss phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a model of poro-thermoelasticity with microtemperatures, where the behavior of the body is influenced by the history of both temperature and microtemperatures. Mathematically, this translates into a system of partial integro-differential equations. Under suitable condition on the tensors appearing in the model, we prove that the resulting system is well posed. In the one-dimensional case, the exponential decay of the energy is proved. [ABSTRACT FROM AUTHOR]

We study antiplane shear deformations for isotropic and homogeneous strain gradient mixtures of the Kelvin‐Voigt type in a cylinder. Our aim is to analyze the behaviour of the solutions with respect to the time variable when a dissipative structural mechanism is considered. We study three different cases, each at a time. For each case we prove existence and uniqueness of solutions. We obtain the exponential decay of the solutions in the hyperviscosity and viscosity cases. Exponential decay is also expected when the dissipation is generated by the relative velocity (in the generic case, although a particular combination of the constitutive parameters leads to slow decay). These results are proved with the help of the theory of semigroups. We study antiplane shear deformations for isotropic and homogeneous strain gradient mixtures of the Kelvin‐Voigt type in a cylinder. Our aim is to analyze the behaviour of the solutions with respect to the time variable when a dissipative structural mechanism is considered. We study three different cases, each at a time. For each case we prove existence and uniqueness of solutions. We obtain the exponential decay of the solutions in the hyperviscosity and viscosity cases.... [ABSTRACT FROM AUTHOR]