Asymptotic analysis of two thermoelastic plates with dissipative histories

Abstract

In this paper we study the time decay of the solutions for the problems determined by two plates where the dissipation mechanisms are given by the history of the material. To be precise we consider the thermo-viscolestic plate with heat conduction of the Green-Naghdi type II and the thermoelastic plate when the heat equation is described by the history-dependent Moore-Gibson-Thompson equation. In both cases we prove the well-posedness of the problems by means of semigroup theory. In the first case we also prove that the solutions decay in an exponential way by means of Prüss characterizations of exponential stable semigroups. In the second case we prove that the solutions decay in a polynomial way with optimal rates of decay, which is proved by Tomilov-Borichev characterizations of polynomial stable semigroups.