Energy Distribution of Radial Solutions to Energy Subcritical Wave Equation with an Application on Scattering Theory

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 ($3\leq p<5$) whose initial data are radial and come with a finite energy. We split the energy into inward and outward energies, then apply energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: "scattering energy" which concentrates around the light cone $|x|=|t|$ and moves to infinity at the light speed and "retarded energy" which is at a distance of at least $|t|^\beta$ behind when $|t|$ is large. Here $\beta$ is an arbitrary constant smaller than $\beta_0(p) = \frac{2(p-2)}{p+1}$. A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data $(u_0,u_1)$ than previously known results. More precisely, we assume \[ \int_{{\mathbb R}^3} (|x|^\kappa+1)\left(\frac{1}{2}|\nabla u_0|^2 + \frac{1}{2}|u_1|^2+\frac{1}{p+1}|u|^{p+1}\right) dx < +\infty. \] Here $\kappa>\kappa_0(p) =1-\beta_0(p) = \frac{5-p}{p+1}$ is a constant.