Timoshenko systems with indefinite damping

Abstract

We consider the Timoshenko system in a bounded domain ( 0 , L ) ⊂ R 1 . The system has an indefinite damping mechanism, i.e. with a damping function a = a ( x ) possibly changing sign, present only in the equation for the rotation angle. We shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case, and provided a ¯ = ∫ 0 L a ( x ) d x > 0 and ‖ a − a ¯ ‖ L 2 < ϵ , for ϵ small enough. The decay rate will be described explicitly. In the arguments, we shall also give a new proof of exponential stability for the constant case a ≡ a ¯ . Moreover, we give a precise description of the decay rate and demonstrate that the system has the spectrum determined growth (SDG) property, i.e. the type of the induced semigroup coincides with the spectral bound for its generator.