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

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.

The new version fixes an error (pointed out by Chengbo Wang) in an incorrect use of the Hardy-Littlewood-Sobolev inequality or radial functions, in the previous version of the paper. The main results in the paper remain unchanged. In addition,we have provided additional comments and corrected some misprints