  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 $L^p$-energy which is almost conserved by the flow of the radial
linear wave equation.
