Exponential decay in one‐dimensional Type II/III thermoelasticity with two porosities

Abstract

In this paper, we consider the theory of thermoelasticity with a double porosity structure in the context of the Green–Naghdi Types II and III heat conduction models. For the Type II, the problem is given by four hyperbolic equations, and it is conservative (there is no energy dissipation). We introduce in the system a couple of dissipation mechanisms in order to obtain the exponential decay of the solutions. To be precise, we introduce a pair of the following damping mechanisms: viscoelasticity, viscoporosities, and thermal dissipation. We prove that the system is exponentially stable in three different scenarios: viscoporosity in one structure jointly with thermal dissipation, viscoporosity in each structure, and viscoporosity in one structure jointly with viscoelasticity. However, if viscoelasticity and thermal dissipation are considered together, undamped solutions can be obtained