Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation

TL;DR

This paper proves global existence and scattering for the defocusing cubic nonlinear wave equation in high dimensions without radial symmetry, assuming a bounded critical Sobolev norm, extending previous methods to supercritical regimes.

Contribution

It establishes the first global well-posedness and scattering results for the energy-supercritical cubic wave equation in dimensions d≥6 without radial symmetry.

Findings

Solutions are global if the critical Sobolev norm remains bounded.

Solutions scatter under the a priori Sobolev bound.

Extends techniques from supercritical NLS and NLW to new regime.

Abstract

In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ in the energy-supercritical regime, in dimensions $d\geq 6$, with no radial assumption on the initial data. We prove that if a solution satisfies an a priori bound in the critical homogeneous Sobolev space throughout its maximal interval of existence, that is, $u\in L_t^\infty(\dot{H}_x^{s_c}\times\dot{H}_x^{s_c-1})$, then the solution is global and it scatters. Our analysis is based on the methods of the recent works of Kenig-Merle \cite{KenigMerleSupercritical} and Killip-Visan \cite{KillipVisanSupercriticalNLS,KillipVisanSupercriticalNLW3D} treating the energy-supercritical nonlinear Schr\"odinger and wave equations.