Scattering for focusing supercritical wave equations in odd dimensions

TL;DR

This paper proves that in odd dimensions, radial solutions to supercritical focusing wave equations that stay bounded in the critical Sobolev space are globally well-behaved and resemble linear solutions over time.

Contribution

It establishes scattering for bounded radial solutions in energy supercritical focusing wave equations in odd dimensions, a significant extension of known results.

Findings

Radial solutions bounded in the critical Sobolev space are global.

Such solutions scatter to linear solutions.

The result applies to general odd dimensions.

Abstract

We consider the wave equation with an energy supercritical focusing nonlinearity in general odd dimensions. We prove that any radial solution that remains bounded in the critical Sobolev space is global and scatters to a linear solution.