The stability number of the Timoshenko system with second sound

In this work, we consider the Timoshenko beam model with second sound. We introduce a new number χ0 that characterizes the exponential decay. We prove that the corresponding semigroup associated to the system is exponentially stable if and only if χ0=0. Otherwise there is a lack of exponential stability. In this case we prove that the semigroup decays as t−1/2. Moreover we show that the rate is optimal.