Scattering for radial bounded solutions of focusing supercritical wave equations in four dimensions

TL;DR

This paper proves that radial solutions of the focusing supercritical wave equation in four dimensions, which stay bounded in the critical Sobolev space, are global and asymptotically free, advancing understanding of wave behavior in supercritical regimes.

Contribution

It establishes the global existence and scattering for bounded radial solutions in the energy supercritical setting in four dimensions, a significant extension of previous results.

Findings

Radial solutions bounded in the critical Sobolev space are global.

Such solutions scatter to free waves as time approaches infinity.

The result applies specifically to the four-dimensional focusing supercritical wave equation.

Abstract

We consider the focusing wave equation with energy supercritical nonlinearity in dimension four. We prove that any radial solution that remains bounded in the critical Sobolev space is global and scatters to free waves as $t \to \pm \infty$.