The defocusing energy-supercritical cubic nonlinear wave equation in dimension five

TL;DR

This paper proves that for the defocusing cubic nonlinear wave equation in five dimensions, a uniform critical norm bound guarantees global existence and scattering, completing the understanding in the energy-supercritical regime.

Contribution

It establishes the first non-radial global well-posedness and scattering result for the energy-supercritical cubic wave equation in five dimensions.

Findings

Solutions are globally well-posed under critical norm bounds.

Solutions scatter at infinity in both time directions.

The work completes the classification in the energy-supercritical regime.

Abstract

We consider the energy-supercritical nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time {\it a priori} bound on the critical norm implies that solutions exist globally in time and scatter at infinity in both time directions. Together with our earlier works in dimensions $d\geq 6$ with general data and dimension $d=5$ with radial data, the present work completes the study of global well-posedness and scattering in the energy-supercritical regime for the cubic nonlinearity under the assumption of uniform-in-time control over the critical norm.