On Modeling and Uniform Stability of a Partially Dissipative Viscoelastic Timoshenko System

In this paper we first explore the deduction of the mathematical model for some viscoelastic Timoshenko systems. As a consequence, a new partially dissipative viscoelastic Timoshenko system arises with damping mechanism acting only on the shear force. Then, we prove uniform decay rates for this new system with the help of a modern observability inequality, where the assumption of equal speeds of wave propagation is regarded as a sufficient condition. Moreover, we prove that equal wave speeds is also a necessary condition to establish uniform decay rates.