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

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 $$E_{\kappa }(u_0,u_1)={\int }_{{\mathbb{R}}^d}(\mathrm{\mid }x{\mathrm{\mid }}^{\kappa }+1)(\frac{1}{2}{\mid \mathrm{\nabla }{u}_0(x)\mid }^2+\frac{1}{2}{\mid u_1(x)\mid }^2+\frac{1}{p+1}{\mid u_0(x)\mid }^{p+1})dx<+\mathrm{\infty }\mathrm{.}$$ Here $\kappa =\frac{(2-d)p+(d+2)}{p+1}$.