Blow-up of cylindrically symmetric solutions for Fractional NLS

Abstract

In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional NLS, $$\text{i}{\partial }_{t}u=(-\Delta {)}^{s}u-|u{|}^{2\sigma }u\text{in}\mathbb{R}\times {\mathbb{R}}^N,$$ where $N\ge 2$, $1/2<s<1$ 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}.

9 pages