Decay of solutions for a mixture of thermoelastic solids with different temperatures

We study a system modeling thermomechanical deformations for mixtures of thermoelastic solids with two different temperatures, that is, when each component of the mixture has its own temperature. In particular, we investigate the asymptotic behavior of the related solutions. We prove the exponential stability of solutions for a generic class of materials. In case of the coupling matrix B being singular, we find that in general the corresponding semigroup is not exponentially stable. In this case we obtain that the corresponding solution decays polynomially as t−1/2t−1/2 in case of Neumann boundary condition. Additionally, we show that the rate of decay is optimal. For Dirichlet boundary condition, we prove that the rate of decay is t−1/6t−1/6. Finally, we demonstrate the impossibility of time-localization of solutions in case that two coefficients (related with the thermal conductivity constants) agree.