Decay of solutions for a semilinear system of elastic waves in an exterior domain with damping near infinity

We show that the solutions of a semilinear system of elastic waves in an exterior domain with a localized damping near infinity decay in an algebraic rate to zero. We impose an additional condition on the Lamé coefficients. It seems that this restriction cannot be overcome by using the two-finite-speed propagation of the elastic model, since we do not assume compact support on the initial data and because the dissipation does not have compact support. The decay rates obtained for the total energy of the linear problem and the L²$L^2$-norm of the solution improve previous results. For the semilinear problem the decay rates in this paper seem to be the first contribution, mainly in the context of initial data without compact support and localized dissipation.