The defocusing energy-supercritical nonlinear wave equation in three space dimensions

TL;DR

This paper proves that in the energy-supercritical defocusing nonlinear wave equation in three dimensions, solutions with bounded critical Sobolev norm are global and scatter, extending previous spherical symmetry results to a broader class.

Contribution

It establishes that for even powers p>4, solutions with bounded critical Sobolev norm do not blow up and scatter, generalizing prior spherical symmetry results.

Findings

Solutions with bounded critical Sobolev norm are global and scatter.

Blowup must be accompanied by unbounded Sobolev norm for even p>4.

Extends previous results beyond spherically symmetric solutions.

Abstract

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.