Abstract The purpose of this note is to investigate the stabilization of a system of two wave equations coupled by velocities with only one localized damping. The main novelty in this note is that the waves are not necessarily propagating at same speed and the coupling coefficient is not assumed to be positive and small. Assume that the coupling region and the damping region intersect. We prove that our system is strongly stable without geometric conditions. We then study the energy decay rate by distinguishing two cases. The first one is when the waves propagate at the same speed. In this case, under appropriate geometric conditions, we establish an exponential energy decay estimate for usual initial data. For the other case, we first show that our system is not uniformly stable. Next, under the same geometric conditions, we establish a polynomial energy decay of type 1 t for smooth initial data. Finally, in one space dimension, using the real part of the asymptotic expansion of eigenvalues of the system, we prove that the obtained polynomial decay rate is optimal.

中文翻译：

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几何条件下两个耦合波动方程的N-d系统的局部间接镇定

摘要 本笔记的目的是研究由速度耦合的两个波动方程系统的稳定性，其中只有一个局部阻尼。本笔记的主要新颖之处在于波不一定以相同的速度传播，并且耦合系数不假定为正且很小。假设耦合区和阻尼区相交。我们证明了我们的系统在没有几何条件的情况下是非常稳定的。然后我们通过区分两种情况来研究能量衰减率。第一个是当波以相同的速度传播时。在这种情况下，在适当的几何条件下，我们为通常的初始数据建立了指数能量衰减估计。对于另一种情况，我们首先证明我们的系统不是一致稳定的。接下来，在相同的几何条件下，我们为平滑的初始数据建立了类型 1 t 的多项式能量衰减。最后，在一维空间中，利用系统特征值渐近展开的实部，证明得到的多项式衰减率是最优的。