On the thermoelasticity with two porosities: asymptotic behaviour

In this paper we consider the one-dimensional version of thermoelasticity with two porous structures and porous dissipation on one or both of them. We first give an existence and uniqueness result by means of semigroup theory. Exponential decay of the solutions is obtained when porous dissipation is assumed for each porous structure. Later, we consider dissipation only on one of the porous structures and we prove that, under appropriate conditions on the coefficients, there exists undamped solutions. Therefore, asymptotic stability cannot be expected in general. However, we are able to give suitable sufficient conditions for the constitutive coefficients to guarantee the exponential decay of the solutions.