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In this paper we study the focusing cubic wave equation in 1+5 dimensionswith radial initial data as well as the one-equivariant wave maps equation in1+3 dimensions with the model target manifolds $\mathbb{S}^3$ and$\mathbb{H}^3$. In both cases the scaling for the equation leaves the$\dot{H}^{\frac{3}{2}} \times \dot{H}^{\frac{1}{2}}$-norm of the solutioninvariant, which means that the equation is super-critical with respect to theconserved energy. Here we prove a conditional scattering result: If thecritical norm of the solution stays bounded on its maximal time of existence,then the solution is global in time and scatters to free waves both forwardsand backwards in infinite time. The methods in this paper also apply to allsupercritical power-type nonlinearities for both the focusing and defocusingradial semi-linear equation in 1+5 dimensions, yielding analogous results.

Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm