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

Abstract

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $3\leq p<5$. 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 \[ \int_{{\mathbb R}^3} (|x|^\kappa+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}|u_0|^{p+1}\right) dx < +\infty. \] 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 \[ \int_{{\mathbb R}^3} |x|\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{4}|u_0|^{p+1}\right) dx < +\infty. \] These assumptions on the decay rate of initial data as $|x| \rightarrow \infty$ are weaker than previously known scattering results.