Inward/outward Energy Theory of Non-radial Solutions to 3D Semi-linear Wave Equation

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u=-\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in the 3-dimensional space with $3\le p<5$. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if $3<p<5$ and $\kappa >\frac{5-p}{2}$, then the solution scatters as long as the initial data $(u_0,u_1)$ satisfy $${\int }_{{\mathbb{R}}^3}(\mathrm{\mid }x{\mathrm{\mid }}^{\kappa }+1)(\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}^2+\frac{1}{2}\mathrm{\mid }{u}_1{\mathrm{\mid }}^2+\frac{1}{p+1}\mathrm{\mid }{u}_0{\mathrm{\mid }}^{p+1})dx<+\mathrm{\infty }\mathrm{.}$$ If $p=3$, we can also prove the scattering result if initial data $(u_0,u_1)$ are contained in the critical Sobolev space and satisfy the inequality $${\int }_{{\mathbb{R}}^3}\mathrm{\mid }x\mathrm{\mid }(\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}^2+\frac{1}{2}\mathrm{\mid }{u}_1{\mathrm{\mid }}^2+\frac{1}{4}\mathrm{\mid }{u}_0{\mathrm{\mid }}^{p+1})dx<+\mathrm{\infty }\mathrm{.}$$ These assumptions on the decay rate of initial data as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ are weaker than previously known scattering results.