In this work, we consider the Timoshenko beam model with second sound. We introduce a new number χ 0 that characterizes the exponential decay. We prove that the corresponding semigroup associated to the system is exponentially stable if and only if χ 0 = 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.

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In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.

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