Blow-up of cylindrically symmetric solutions for Fractional NLS

TL;DR

This paper investigates the conditions under which solutions to a fractional nonlinear Schrödinger equation blow up, specifically for cylindrically symmetric initial data, extending previous results from radial symmetry cases.

Contribution

It establishes a blow-up criterion for cylindrically symmetric solutions in the mass critical and supercritical regimes, expanding known results beyond radial symmetry.

Findings

Blow-up criterion for cylindrically symmetric solutions

Extension of blow-up results from radial to cylindrical symmetry

Applicable in mass critical and supercritical cases

Abstract

In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional NLS, $$ \textnormal{i} \, \partial_t u=(-\Delta)^s u-|u|^{2 \sigma} u \quad \text{in} \,\, \R \times \R^N, $$ where $N \geq 2$, $1/2 <s<1$ and $0<\sigma<2s/(N-2s)$. In the mass critical and supercritical cases, we establish a criterion for blow-up of solutions to the problem for cylindrically symmetric data. The results extend the known ones with respect to blow-up of solutions to the problem for radially symmetric data in \cite{BHL}.