Dynamics of the Nonlinear Timoshenko System with Variable Delay

Abstract

This paper is concerned with the wellposedness of global solution and existence of global attractor to the nonlinear Timoshenko system subject to continuous variable time delay in the angular rotation of the beam filament. The waves are assumed to propagate under the same speed in the transversal and angular direction. A single mechanical damping is implemented to counter the destabilizing effect from the time delay term. By imposing appropriate assumptions on the damping term and sub-linear time delay term, we prove the existence of absorbing set and establish the quasi-stability of the gradient system generated from the solution to the system of equation. The quasi-stability property in turn implies the existence of finite dimensional global and exponential attractors that contain the unstable manifold formed from the set of equilibria.