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

TL;DR

This paper proves global well-posedness for radial initial data in the mass-subcritical nonlinear Schrödinger equation under certain conditions, expanding understanding of solution behavior in these regimes.

Contribution

It establishes global solutions for mass-subcritical NLS with radial data in the critical space, under specific dimensional and exponent restrictions.

Findings

Global well-posedness for radial data in mass-subcritical regimes

Conditions on dimension and nonlinearity exponent for solutions

Extension of solution theory in critical Sobolev spaces

Abstract

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.