Pointwise long time behavior for the mixed damped nonlinear wave equation in

Abstract

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