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

我们仅使用Kelvin-Voigt型的一种内部粘弹性阻尼来研究局部耦合波动方程的稳定性（请参阅系统（1.2）-（1.4））。本文的主要新颖之处在于，阻尼系数和耦合系数都不是光滑的（参见（1.5））。首先，使用Arendt-Batty的一般准则并结合唯一性结果，我们证明了我们的系统非常稳定。接下来，使用频谱方法，我们证明了系统的非指数（均匀）稳定性。最后，使用频域方法，结合分段乘法器技术并构造满足某些常微分方程的新乘法器，我们证明系统的光滑解的能量以多项式衰减\（t ^ {-1} \）。