Stability of thermoelastic diffusion problem of type III in spaces of different dimensions

In this article, we study the stability issues of a thermoelastic diffusion problem of type III in spaces of different dimensions. First we prove that the one-dimensional problem is exponentially stable without adding damping terms. Then we prove the lack of exponential stability in the two-dimensional space. Moreover, a frictional damping for the elastic component leads to the exponential stability in three-dimensional space. As the form of the damping term depends on the nature of the material, we will distinguish the case of isotropic materials and the case of anisotropic materials.