The Timoshenko system with history and Cattaneo law

In this paper we study a fully hyperbolic thermoelastic Timoshenko system with past history where the thermal effects are given by Cattaneo’s law. We obtain exponential stability of solutions if and only if a new condition on the wave speed of propagation is verified. Otherwise, when that condition fails, we obtain polynomial stability of solutions. In this case optimal rates of decay are established.