Scattering for radial, bounded solutions of focusing supercritical wave equations

TL;DR

This paper proves that all bounded, radial solutions of a supercritical focusing wave equation in three dimensions are globally well-behaved and scatter, with implications for understanding blow-up behavior.

Contribution

It establishes scattering and global existence for bounded radial solutions in the supercritical regime, extending previous methods to this challenging setting.

Findings

Radial solutions bounded in the critical Sobolev space are globally defined.

Such solutions scatter to linear solutions as time goes to infinity.

Finite time blow-up solutions have unbounded Sobolev norm sequences.

Abstract

In this paper, we consider the wave equation in space dimension 3 with an energy-supercritical, focusing nonlinearity. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined and scatters to a linear solution. As a consequence, finite time blow-up solutions have critical Sobolev norm converging to infinity (along some sequence of times). The proof relies on the compactness/rigidity method, pointwise estimates on compact solutions obtained by the two last authors, and channels of energy arguments used by the authors in previous works on the energy-critical equation.