Blow-up criteria for fractional nonlinear Schr\"odinger equation

TL;DR

This paper establishes new blow-up criteria for solutions to the focusing fractional nonlinear Schrödinger equation using localized virial estimates, extending previous results to non-radial initial data in both critical and supercritical cases.

Contribution

It introduces generalized blow-up criteria for non-radial solutions of fractional NLS, broadening the understanding beyond radial cases and previous work.

Findings

Derived blow-up criteria in $L^2$-critical and supercritical regimes

Extended blow-up results to non-radial initial data

Utilized localized virial estimates for analysis

Abstract

We consider the focusing fractional nonlinear Schr\"odinger equation \[ i\partial_t u - (-\Delta)^s u = -|u|^\alpha u, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $s \in (1/2,1)$ and $\alpha>0$. By using localized virial estimates, we establish general blow-up criteria for non-radial solutions to the equation. As consequences, we obtain blow-up criteria in both $L^2$-critical and $L^2$-supercritical cases which extend the results of Boulenger-Himmelsbach-Lenzmann [{\it Blowup for fractional NLS}, J. Funct. Anal. 271 (2016), 2569--2603] for non-radial initial data.