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

TL;DR

This paper proves that solutions to certain super-critical wave equations in 1+3 and 1+5 dimensions scatter if their critical norms remain bounded, extending results to various nonlinearities.

Contribution

It establishes a conditional scattering criterion for super-critical wave equations, applicable to multiple models and nonlinearities, based on boundedness of the critical norm.

Findings

Solutions scatter if the critical norm remains bounded.

Results apply to focusing and defocusing nonlinearities.

Method extends to all supercritical power-type nonlinearities.

Abstract

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 $\mathbb{S}^3$ 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 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 both forwards and backwards in infinite time. 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.