Abstract. We consider the defocusing nonlinear wave equation utt − Δu + |u | p 4 4 u = 0 with spherically-symmetric initial data in the regime <p< d−2 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. 1
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