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. © 2011 by The JohnsHopkinsUniversity Press.