Global well-posedness and scattering for the defocusing energy supercritical NLS in high dimensions

Abstract

We consider the defocusing energy-supercritical nonlinear Schrödinger equation $i\partial_{t}u+Δu=|u|^p u$ in dimensions $d\ge5$. Killip-Visan [Comm. Partial Differential Equations, 2010] and Li-Li [Siam J. Math. Anal., 2022] proved that for $s_c:=\frac{d}{2}-\frac{2}{p}>1$, any solution that remains bounded in the critical Sobolev space $\dot H_x^{s_c}(\mathbb{R} ^d)$ must be global and scatter. In dimensions \(d \ge 8\), their results required either that \(p\) be even or that \(s_c < \frac{d+2-\sqrt{(d-2)^2-16}}{4}\). In this paper, we improve the upper bound on \(s_c\) to \(s_c<1+p\) by establishing some new nonlinear estimates. This allows us to cover all cases in which \(p\) lies in the local existence range.