We consider a wave equation in three space dimensions, with a power-like
nonlinearity which is either focusing or defocusing. The exponent is greater than 3
(conformally supercritical) and not equal to 5 (not energy-critical). We prove that for
any radial solution which does not scatter to a linear solution, an adapted
scale-invariant Sobolev norm goes to infinity at the maximal time of existence. The
proof uses a conserved generalized energy for the radial linear wave equation, new
Strichartz estimates adapted to this generalized energy, and a bound from below of
the generalized energy of any nonzero solution outside wave cones. It relies
heavily on the fact that the equation does not have any nontrivial stationary
solution. Our work yields a qualitative improvement on previous results on
energy-subcritical and energy-supercritical wave equations, with a unified
proof.