Stability and regularity of solution to the Timoshenko beam equation with local Kelvin-Voigt damping

We consider the Timoshenko beam equation with locally distributed Kelvin-Voigt damping; i.e., the damping; is effective only in a part of the spatial domain for both shear stress and bending moment. We prove eventual differentiability, exponential and polynomial stability of the associated semigroup under some smoothness condition on the damping coefficient functions, particularly, at the interface of the damped and undamped region.