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 subcritical, 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 the 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. This condition is so weak that the initial data may be outside the critical Sobolev space of this equation. This phenomenon is not covered by previously known scattering theory, as far as the author knows.