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

TL;DR

This paper analyzes the energy distribution of radial solutions to a semi-linear, energy sub-critical defocusing wave equation in three dimensions, revealing asymptotic energy behavior and establishing scattering results under weaker initial data conditions.

Contribution

It introduces a novel energy distribution analysis for radial solutions and proves scattering under less restrictive initial data assumptions.

Findings

Energy splits into 'scattering' and 'retarded' parts asymptotically.

Energy concentrates near the light cone and moves outward at light speed.

Scattering is proven under weaker initial data conditions than previous results.

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.