In this paper, the authors consider the focusing and defocusing nonlinear wave equations \(u_{tt}-\Delta u \pm |u|^{p-1}u=0\) in \(\mathbb{R}^{1+3}\) with radial, scaling-invariant initial data, when \(p>5\). The problem is scale-invariant in the homogeneous Sobolev space \(\dot H^{s_c}\) with \(s_c=3/2-2/(p-1)\). The case \(p>5\), i.e., \(s_c>1\), is called energy super-critical. As mentioned in the paper: “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.”