  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.
