Large global solutions for nonlinear Schrödinger equations I, mass-subcritical cases

Abstract

In this paper, we consider the nonlinear Schrödinger equation, $$i{\partial }_{t}u+\Delta u=\mu |u{|}^{p}u,(t,x)\in {\mathbb{R}}^{d+1},$$ with $\mu =\pm 1,p>0$. In this work, we consider the mass-subcritical cases, that is, $p\in (0,\frac{4}{d})$. We prove that under some restrictions on $d,p$, any radial initial data in the critical space ${\dot{H}}^{s_c}({\mathbb{R}}^d)$ with compact support, implies global well-posedness.

36 pages, to appear in AiM