The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

TL;DR

This paper proves that for a certain supercritical nonlinear wave equation, solutions with bounded critical Sobolev norm are global and scatter, ruling out blowup in these cases across multiple dimensions.

Contribution

It establishes the global existence and scattering of solutions with bounded critical Sobolev norm for energy-supercritical nonlinear wave equations in all space dimensions.

Findings

Solutions with bounded critical Sobolev norm are global and scatter.

Blowup or failure to scatter implies unbounded Sobolev norm.

Results hold for dimensions 3 to 6 and part of higher dimensions.

Abstract

We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2}<p<\frac4{d-3}$ (which is energy-supercritical) and dimensions $3\leq d\leq 6$; we also consider $d\geq 7$, but for a smaller range of $p>\frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.