Scattering for Radial, Semi-Linear, Super-Critical Wave Equations with Bounded Critical Norm

In this paper we study the focusing cubic wave equation in 1 + 5 dimensions with radial initial data as well as the one-equivariant wave maps equation in 1+3 dimensions with the model target manifolds and . In both cases the scaling for the equation leaves the -norm of the solution invariant, which means that the equation is super-critical with respect to the conserved energy. Here we prove a conditional scattering result: if the critical norm of the solution stays bounded on its maximal time of existence, then the solution is global in time and scatters to free waves as . The methods in this paper also apply to all supercritical power-type nonlinearities for both the focusing and defocusing radial semi-linear equation in 1+5 dimensions, yielding analogous results.