Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

We consider the dynamic elasticity equations with alocally distributed damping of Kelvin-Voigt type in a bounded domain. The damping is localized in a suitableopen subset, of the domain under consideration, which satisfies the piecewise multipliers condition of Liu. Using multipliertechniques combined with the frequency domain method, we show that: i) the energy of this system decays polynomiallywhen the damping coefficient is only bounded measurable, ii) the energy of this system decays exponentially whenthe damping coefficient as well as its gradient are bounded measurable, and the damping coefficient further satisfies a structural condition. These results generalize and improve, at the same time,on an earlier result of Liu and Rao involving the wave equation with Kelvin-Voigt damping; those authors proved the exponential decay of the energyprovided that the damping region is a neighborhood of the whole boundary, and further restrictions are imposed on the damping coefficient.