Dynamics of locally damped Timoshenko systems

Abstract

In this paper, we study the long-time dynamics of a Timoshenko system modeling vibrations of beams with non-linear localized damping mechanisms acting on both displacement and angular rotation and subjected to non-linear source terms. Using recent quasi-stability methods, we prove the existence of smooth finite dimensional global attractor, which is characterized as unstable manifold of the set of stationary solutions. Moreover, the existence of exponential attractors is shown. These aspects were not previously considered for the Timoshenko system with localized damping.