Stability Results of an Elastic/Viscoelastic Transmission Problem of Locally Coupled Waves with Non Smooth Coefficients

We investigate the stabilization of a locally coupled wave equations with only one internal viscoelastic damping Kelvin-Voigt type (see System (1.2)-(1.4)). The main novelty in this paper is that both and coupling coefficients are non smooth (1.5)). First, using general criteria Arendt-Batty, combined an uniqueness result, we prove our system strongly stable. Next, spectrum approach, non-exponential (uniform) stability system. Finally, frequency domain piecewise multiplier technique construction new satisfying some ordinary differential equations, show energy solutions decays polynomially $t^{-1}$ .