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 utt − u + |u|2u = 0 in the energy-supercritical regime, in dimensions d 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 ∈ L ∞ t ( ˙H sc x × ˙H sc−1 x ), then the solution is global and it scatters. Our analysis is based on the methods of the recent works of Kenig and Merle (2008) [21] and Killip and Visan (2010) [26,27] treating the energy-supercritical nonlinear Schrödinger and wave equations. © 2012 Elsevier Inc. All rights reserved.