Large Global Solutions for Nonlinear Schrödinger Equations II, Mass-Supercritical, Energy-Subcritical Cases

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schrodinger equation, $$\begin{aligned} i\partial _{t}u+\Delta u= |u|^p u, \quad (t,x)\in {\mathbb {R}}^{d+1}, \end{aligned}$$ with $$p\in (\frac{4}{d},\frac{4}{d-2})$$ . We prove that under some restrictions on d, p, any radial function in rough space $$H^{s_0}({\mathbb {R}}^d),\textit{for } s_0<s_c$$ support away from origin, there exists an incoming/outgoing decomposition, such initial data outgoing part leads to global well-posedness and scattering forward time; while incoming backward time. The proof is based Phase-Space analysis of dynamics.