The radial mass-subcritical NLS in negative order Sobolev spaces

TL;DR

This paper investigates the behavior of radial solutions to the mass-subcritical nonlinear Schrödinger equation in higher dimensions, establishing conditions for global existence, scattering, and the existence of threshold solutions with compact flow.

Contribution

It proves that bounded solutions in the critical Sobolev space are global and scatter in the defocusing case, and constructs threshold solutions with compact flow in the focusing case.

Findings

Bounded solutions in the critical Sobolev space are global and scatter in the defocusing case.

Existence of threshold solutions with compact flow in the focusing case.

Results apply to radial initial data in dimensions d ≥ 3.

Abstract

We consider the mass-subcritical NLS in dimensions $d\geq 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.