Decay rates for the three-dimensional linear system of thermoelasticity

We consider the two and three-dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as t --> infinity. First of all, by a decoupling method, we reduce the problem to an observability inequality for the Lame system in linear elasticity and more precisely to whether the total energy of the solutions can be estimat ed in terms of the energy concentrated on its longitudinal component. We sh ow that when the domain is convex, the decay rate is never uniform. In fact , the lack of uniform decay holds in a more general class of domains in whi ch there exist rays of geometric optics of arbitrarily large length that ar e always reflected perpendicularly or almost tangentially on the boundary. We also show that, in three space dimensions, the lack of uniform decay may also be due to a critical polarization of the energy on the transversal co mponent of the displacement. In two space dimensions we prove a sufficient (and almost necessary) condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays, under the further a ssumption that the boundary of the domain does not have contacts of infinit e order with its tangents. We also give an example, due to D. HULIN, in whi ch these geometric properties hold. In three space dimensions we indicate ( without proof) how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold. In t wo space dimensions we prove that smooth solutions decay polynomially in th e energy space to a finite-dimensional subspace of solutions except when th e domain is a ball or an annulus. Finally we discuss some closely related c ontrollability and spectral issues.