In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.

中文翻译：

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具有局部摩擦和开尔文-沃格特耗散机制的半线性波动方程的指数衰减

在本文中，我们关注具有两个局部阻尼的非均质介质Ω中的半线性粘弹性波动方程。第一个是粘弹性类型的，并且根据几何控制条件分布在边界的邻域ω周围。第二个是摩擦阻尼，我们认为它会损害控制的几何条件。我们表明，对于在有限能相空间的有界集合中获取的所有有限能初始数据，波动方程的能量均匀且指数地变为零。