In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porouselasticity problem when several damping effects are present. We show that viscoelasticity and temperature produce slow decay in time, and the same result is obtained when the porous viscosity is combined with microtemperatures. However, when the viscoelasticity is coupled with porous damping or with microtemperatures the decay is controlled by a negative exponential.

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In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation when the motion of microvoids is assumed to be quasistatic. This question has been recently studied in the general dynamical case. Thus, the natural question is to know if the assumption of quasi-static motion for the microvoids implies significant differences in the behavior of the solutions from the results obtained in the general dynamical case. It is worth noting that this assumption involves a qualitative change in the system of equations to be analyzed because it arises from the combination of a parabolic equation with an hyperbolic one, rather different from the well-known system of the thermo-elastic problem. First, we study the coupling of elasticity with porosity and we show that if only porous dissipation is present, the decay of solutions is slow, but if viscoelasticity is added, then the solutions decay exponentially. After that, we introduce thermal effects in the system and we show that while temperature brings exponential stability to the solutions, microtemperature does not.

In this paper we consider the type III thermoelastic theory with microtemperatures. We study the time decay of the solutions and we prove that, under suitable conditions for the constitutive tensors, the solutions decay exponentially. This fact is in somehow shocking because it diers from the behavior of the solutions in the classical model of thermoelasticity with microtemperatures.

In this work we study modifications of the non-classical models of thermoelasticity, the one proposed Green and Lindsay and the one stated by Lord and Shulman, to two-temperature setting. We prove uniqueness results for the solutions of the systems of equations that model both theories for isotropic material. We also provide growth estimates for the solutions.