Stabilization of a locally damped thermoelastic system

We show that the solutions of a thermoelastic system with a localized nonlinear distributed damping decay locally with an algebraic rate to zero, that is, given an arbitrary R > 0, the total energy E(t) satisfies for t > 0: E(t) < C (1 + t)-γ for regular initial data such that E(0) < R, where C and γ are positive constants. In the two-dimensional case, we obtain an exponential decay rate when the nonlinear dissipation behaves linearly close to the origin.