In this paper, we investigate the long time behavior of the solution for the nonlinear wave equation with frictional and visco-elastic damping terms in $ \mathbb{R}^n_+ $. It is shown that for the long time, the frictional damped effect is dominated. The Green's functions for the linear initial boundary value problem can be described in terms of the fundamental solutions for the full space problem and reflected fundamental solutions coupled with the boundary operator. Using the Duhamel's principle, we get the pointwise long time behavior of the solution $ \partial_{{\bf{x}}}^{\alpha}u $ for $ |\alpha|\le 1 $.

中文翻译：

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混合阻尼非线性波动方程在点上的逐点长时间性。 \ begin {document} $ \ mathbb {R} ^ n_ + $ \ end {document}

在本文中，我们研究了带有摩擦和粘弹性阻尼项的非线性波动方程解在\ \ mathbb {R} ^ n_ + $中的长时间行为。结果表明，长期以来，摩擦阻尼作用是主要的。线性初始边界值问题的格林函数可以用全空间问题的基本解和反映的基本解以及边界算符来描述。使用Duhamel原理，对于$ | \ alpha | \ le 1 $，我们得到了解$ \ partial _ {{\ bf {x}}} ^ {\ alpha} u $的逐点长时间行为。