Stabilization of non-homogeneous elastic materials with voids

We study the asymptotic behavior of the solution of the non-homogeneous elastic system with voids and a thermal effect. We first prove the well-posedness of this system under some realistic assumptions on the coefficients. Since this system suffers of exponential stability (as shown in dimension 1 in Pamplona, Muñoz Rivera, and Quintanilla (2009) [18], ), our main results concern strong and polynomial stabilities again under some assumptions on the coefficients. These stabilities are obtained in a closed subspace of the natural Hilbert space. Hence we characterize its orthogonal and further show that in the whole space the energy tends strongly or polynomially to the energy of the projection of the initial datum on this orthogonal space. In this respect we extend and precise former results obtained in one dimension in Pamplona, Muñoz Rivera, and Quintanilla (2009) [18].