Large global solutions for nonlinear Schr\"odinger equations I, mass-subcritical cases

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 $\mu=\pm1, p>0$. In this work, we consider the mass-subcritical cases, that is, $p\in (0,\frac4d)$. 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.