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We investigate the stabilization of a multidimensional system of coupled waveequations with only one Kelvin Voigt damping. Using a unique continuationresult based on a Carleman estimate and a general criteria of Arendt Batty, weprove the strong stability of the system in the absence of the compactness ofthe resolvent without any geometric condition. Then, using a spectral analysis,we prove the non uniform stability of the system. Further, using frequencydomain approach combined with a multiplier technique, we establish somepolynomial stability results by considering different geometric conditions onthe coupling and damping domains. In addition, we establish two polynomialenergy decay rates of the system on a square domain where the damping and thecoupling are localized in a vertical strip.

A N-dimensional elastic\viscoelastic transmission problem with Kelvin-Voigt damping and non smooth coefficient at the interface