Large global solutions for nonlinear Schrödinger equations II, mass-supercritical, energy-subcritical cases

Marius Beceanu, Qingquan Deng, Avy Soffer, Yifei Wu

Published 2018-11-11Version 1

In this paper, we consider the nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= \mu|u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p>0, \mu=\pm1$. In this work, we consider the defocusing mass-supercritical, energy-subcritical cases, that is, $p\in (\frac4d,\frac4{d-2}),\mu=\pm1$. We prove that under some restrictions on $d,p$, any radial initial data in the rough space $H^{s_0}(\mathbb R^d), s_0<s_c$ with the support away from the origin, there exists an incoming/outgoing decomposition, such that the initial data in the outgoing part leads to the global well-posedness and scattering in the forward time; while the initial data in the incoming part leads to the global well-posedness and scattering in the backward time.