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 S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{S}^3}$$\end{document} and H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{H}^3}$$\end{document}. In both cases the scaling for the equation leaves the H˙32×H˙12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{H}^{\frac{3}{2}} \times \dot{H}^{\frac{1}{2}}}$$\end{document}-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 t→±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t \to \pm \infty}$$\end{document}. 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.