THE RADIAL DEFOCUSING ENERGY-SUPERCRITICAL NONLINEAR WAVE EQUATION IN ALL SPACE DIMENSIONS

We consider the defocusing nonlinear wave equation u(tt) - Delta u + vertical bar u vertical bar(p)u = 0 with spherically-symmetric initial data in the regime 4/d-2 < p < 4/d-3 (which is energy-supercritical) and dimensions 3 <= d <= 6; we also consider d >= 7, but for a smaller range of p > 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.