Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}_{+}^{n} $\end{document}</tex-math></inline-formula>

Abstract

In this paper, the asymptotic wave behavior of the solution for the nonlinear damped wave equation in \begin{document}$ \mathbb{R}^n_+ $\end{document} is investigated. We describe the double mechanism of the hyperbolic effect and the parabolic effect using the explicit functions. With the absorbing and radiative boundary condition, we show that the Green's function for the half space linear problem can be described in terms of the fundamental solution for the Cauchy problem and the reflected fundamental solution coupled with a boundary operator. Using the Duhamel's principle, we see that due to the fast decay property of the Green's function and the high nonlinearity, the pointwise decaying rate for the nonlinear solution and extra time decaying rate for its first order derivative are obtained.