Optimal decay for coupled waves with Kelvin-Voigt damping

We consider a system of two coupled waves where one is conservative and the other is dissipative. When the dissipation is caused by an internal friction, Alabau et al. (2002), showed the polynomial decay of the associated semigroup with decay rate t−1∕2. In this paper, we substitute the friction damping by another one. This damping is of Kelvin-Voigt type, and for this system, we show that the semigroup loses speed and decays with the rate t−1∕4. Moreover, we prove that this decay rate is optimal.