Exterior scattering of non-radial solutions to energy subcritical wave equations

TL;DR

This paper proves exterior scattering for non-radial solutions of the energy subcritical defocusing wave equation in dimensions 3 to 5, extending previous radial results to more general solutions.

Contribution

It establishes the first exterior scattering result for non-radial solutions in the energy subcritical wave equation setting.

Findings

Solutions with finite energy scatter outside large balls as time goes to infinity.

Generalizes radial scattering results to non-radial solutions.

Provides a framework for understanding wave behavior in higher dimensions.

Abstract

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in dimension $d \in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\leq d \leq 5$ and $1+6/d<p<1+4/(d-2)$. More precisely, given any solution with a finite energy, there exists a solution $u_L$ to the homogeneous linear wave equation, so that the following limit holds \[ \lim_{t\rightarrow +\infty} \int_{|x|>t+R} |\nabla_{x,t} u(x,t)- \nabla_{x,t} u_L(x,t)|^2 dx = 0 \] for any fixed real number $R$. This generalize the previously known exterior scattering result in the radial case.