Necessary conditions and sufficient conditions for global existence in the nonlinear Schr\"odinger equation

TL;DR

This paper establishes precise criteria for when solutions to the supercritical nonlinear Schrödinger equation exist globally or blow up, extending previous results and refining blow-up conditions.

Contribution

It provides necessary and sufficient conditions for global existence and blow-up in the supercritical case, extending Weinstein's results and improving blow-up criteria.

Findings

Conditions coincide in the critical case

Extended Weinstein's results to supercritical case

Improved blow-up condition

Abstract

In this paper, we consider the nonlinear Schr\"odinger equation with the super critical power of nonlinearity in the attractive case. We give a sufficient condition and a necessary condition to obtain global or blowing up solutions. These conditions coincide in the critical case, thereby extending the results of Weinstein. Furthermore, we improve a blow-up condition.