Long time behavior for the visco-elastic damped wave equation in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^n_+$\end{document}</tex-math></inline-formula> and the boundary effect

In this paper, we investigate the existence and long time behavior of the solution for the nonlinear visco-elastic damped wave equation in \begin{document}$\mathbb{R}^n_+$\end{document} , provided that the initial data is sufficiently small. It is shown that for the long time, one can use the convected heat kernel to describe the hyperbolic wave transport structure and damped diffusive mechanism. The Green's function for the linear initial boundary value problem can be described in terms of the fundamental solution (for the full space problem) and reflected fundamental solution coupled with the boundary operator. Using the Duhamel's principle, we get the \begin{document}$ L^p $\end{document} decaying rate for the nonlinear solution \begin{document}$\partial_{{\bf x}}^{\alpha}u$\end{document} for \begin{document}$|\alpha|\le 1$\end{document} .