Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on $\mathbb{R}^{3}$

TL;DR

This paper proves that for radial solutions of a supercritical wave equation in three dimensions, either the critical Sobolev norm blows up at finite time or the solution exists globally and scatters, using a novel energy method.

Contribution

It establishes a dichotomy for radial solutions of supercritical wave equations, showing norm blow-up or scattering, with a new approach based on a generalized energy method.

Findings

Critical Sobolev norm diverges at finite maximal time for certain solutions.

Solutions with infinite maximal time scatter to linear solutions.

Introduces a generalized $L^p$-energy method for analyzing wave equations.

Abstract

We consider the wave equation in space dimension $3$, with an energy-supercritical nonlinearity which can be either focusing or defocusing. For any radial solution of the equation, with positive maximal time of existence $T$, we prove that one of the following holds: (i) the norm of the solution in the critical Sobolev space goes to infinity as $t$ goes to $T$, or (ii) $T$ is infinite and the solution scatters to a linear solution forward in time. We use a variant of the channel of energy method, relying on a generalized $L^p$-energy which is almost conserved by the flow of the radial linear wave equation.