Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on R³

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We consider the wave equation in space dimension 3, with an energy-supercritical nonlinearity which can be either focusing or defocusing. For any radial solution of the equation, with positive maximal time of existence T, we prove that one of the following holds: (i) the norm of the solution in the critical Sobolev space goes to infinity as t goes to T, or (ii) T is infinite and the solution scatters to a linear solution forward in time. We use a variant of the channel of energy method, relying on a generalized Lᵖ-energy which is almost conserved by the flow of the radial linear wave equation.