Long time behaviour of finite-energy radial solutions to energy subcritical wave equation in higher dimensions

TL;DR

This paper proves scattering behavior for finite-energy radial solutions of the defocusing energy subcritical wave equation in 4 to 6 dimensions, extending previous 3D results by analyzing the equation via reduction to 1D and characteristic lines.

Contribution

It generalizes scattering results for radial solutions from 3D to higher dimensions (4-6D) under finite energy and decay conditions, using a reduction to 1D and characteristic line methods.

Findings

Solutions scatter outside any light cone with finite energy.

Global scattering is achieved under certain decay conditions.

Extends 3D scattering results to higher dimensions.

Abstract

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the equation above to one-dimensional equation of $w$ and apply method of characteristic lines. This gives scattering of solutions outside any given light cone as long as the energy is finite. The scattering in the whole space can also be proved if we assume the energy decays at a certain rate as $x\rightarrow +\infty$. This generalize the 3-dimensional results in Shen[27] to higher dimensions.