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

In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{2}u=0$ in the energy-supercritical regime, in dimensions $d\ge 6$, 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.