The radial defocusing energy-supercritical nonlinear wave equation in all space 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 $\frac{4}{d-2}<p<\frac{4}{d-3}$ (which is energy-supercritical) and dimensions $3\le d\le 6$; we also consider $d\ge 7$, but for a smaller range of $p>\frac{4}{d-2}$. 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.