Stability analysis of a Timoshenko beam with local degenerate viscoelastic damping

We consider the Timoshenko beam with localized viscoelastic damping of Kelvin-Voigt type. The constitutive law has singularity at the interface of the damped and undamped region. More precisely, the damping coefficients are continuous, but their derivatives have singularities at the interface. We show that the system satisfies a sharper polynomial decay rate, which is consistent with the polynomial decay rate when the constitutive law is discontinuous and exponential stability when the law is of C¹-type. This is a big step toward the goal of obtaining eventually the optimal decay rate.