Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications

TL;DR

This paper proves decay estimates for non-dispersive radial solutions to energy supercritical nonlinear wave equations and demonstrates scattering behavior under bounded Sobolev norms in the defocusing case.

Contribution

It establishes optimal decay estimates and shows scattering for radial solutions in the energy supercritical range, extending understanding of solution behavior.

Findings

Optimal pointwise decay estimates for radial solutions

Scattering results for solutions with bounded Sobolev norms

Extension of analysis to the full energy supercritical range

Abstract

In this paper we establish optimal pointwise decay estimates for non-dispersive (compact) radial solutions to non-linear wave equations in 3 dimensions, in the energy supercritical range. As an application, we show for the full energy supercritical range, in the defocusing case, that if the scale invariant Sobolev norm of a radial solution remains bounded in its maximal interval of existence, then the solution must exist for all times and scatter.