In this paper, we study the semilinear beam equation with a locally distributed nonlinear damping on a smooth bounded domain. We first construct approximate solutions, and we show that the aforementioned approximate solutions decay uniformly in the weak phase space by using an observability inequality associated to the linear problem and a unique continuation property. Then, we prove the global existence as well as the uniform decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively.

中文翻译：

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具有局部分布非线性阻尼的梁方程的均匀衰减率估计

在本文中，我们研究了在光滑有界域上具有局部分布非线性阻尼的半线性梁方程。我们首先构造近似解，并通过使用与线性问题相关的可观察性不等式和独特的连续性来证明上述近似解在弱相空间中均匀衰减。然后，我们分别通过传递到极限和使用弱下半连续性论证来证明原始模型的全局存在性和解的均匀衰减。