In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schr\"odinger equation, $$ i\partial_{ t}u+\Delta u= |u|^p u, \quad (t,x)\in \mathbb R^{ d+1}, $$ with $p\in (\frac4d,\frac4{ d-2})$. We prove that under some restrictions on $d,p$, any radial function in the rough space $H^{ s_0}(\mathbb R^d),\textit{ for some } s_0<s_c$ 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 forward in time; while the initial data in the incoming part leads to the global well-posedness and scattering backward in time. The proof is based on Phase-Space analysis of the nonlinear dynamics.
Categories: math.AP math-ph math.MP