Stability results of some coupled wave systems with different kinds of localized damping

First, we consider a system of two wave equations coupled by velocities in one-dimensional space with one boundary fractional damping and we prove that the energy of our system decays polynomially with different rates.

Second, we investigate the stabilization of a locally coupled wave equations with only one internal viscoelastic damping of Kelvin-Voigt type and we prove that the energy of our system decays polynomially with rate 1/t.

Finally, we investigate the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non smooth coefficients and we establish the exponential stability of the solution if and only if the two waves have the same speed of propagation. In case of different speed propagation, we prove that the energy of our system decays polynomially with rate 1/t.