Energy distribution of solutions to defocusing semi-linear wave equation in higher dimensional space

Abstract

The topic of this paper is a semi-linear, defocusing wave equation $u_{t t}-\Delta u=-|u|^{p-1} u$ in sub-conformal case in the higher dimensional space whose initial data are radical and come with a finite energy. We prove some decay estimates of the the solutions if initial data decay at a certain rate as the spatial variable tends to infinity. A combination of this property with a method of characteristic lines give a scattering result if the initial data satisfy $$E_{\kappa}\left(u_{0}, u_{1}\right)=\int_{\mathbb{R}^{d}}\left(|x|^{\kappa}+1\right)\left(\frac{1}{2}\left|\nabla u_{0}(x)\right|^{2}+\frac{1}{2}\left|u_{1}(x)\right|^{2}+\frac{1}{p+1}\left|u_{0}(x)\right|^{p+1}\right) d x<+\infty.$$ Here $\kappa=\frac{(2-d)p+(d+2)}{p+1}$.