Blow-up dynamics for $L^2$-critical fractional Schr\"odinger equations

TL;DR

This paper investigates finite-time blow-up phenomena for $L^2$-critical fractional Schr"odinger equations with near-critical parameters, extending classical results to nonlocal operators with new analytical techniques.

Contribution

It extends known blow-up results to fractional Schr"odinger equations with nonlocal operators, providing detailed blow-up dynamics and handling the equation's complex structure.

Findings

Solutions blow up in finite time under negative energy and supercritical mass.

Provides a detailed description of the blow-up dynamics.

Extends classical blow-up analysis to nonlocal fractional Schr"odinger equations.

Abstract

In this paper, we will consider the $L^2$-critical fractional Schr\"odinger equation $iu_t-|D|^{\beta}u+|u|^{2\beta}u=0$ with initial data $u_0\in H^{\beta/2}(\mathbb{R})$ and $\beta$ close to $2$. We will show that the solution blows up in finite time if the initial data has negative energy and slightly supercritical mass. We will also give a specific description for the blow-up dynamics. This is an extension of the work of F. Merle and P. Rapha\"el for $L^2$-critical Schr\"odinger equations but the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new strategies are required.