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

Abstract

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schrödinger equation, $$i{\partial }_{t}u+\Delta u=|u{|}^{p}u,(t,x)\in {\mathbb{R}}^{d+1},$$ with $p\in (\frac{4}{d},\frac{4}{d-2})$. We prove that under some restrictions on $d,p$, any radial function in the rough space $H^{s_0}({\mathbb{R}}^d),\text{for some }{s}_0<s_c$ 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.

61 pages, last version, published in Comm. Math. Phys