Blow-up Dynamics for L2-Critical Fractional Schrödinger Equations. (25th May 2021)

• Record Type: Journal Article
• Authors: Lan, Yang
• Is Part Of: International mathematics research notices. Volume 2022:Number 18(2022)
• Page Start: 13753
• Page End: 13810
• Publication Date: 2021-05-25
• Subjects: Fractional Schrödinger equations -- L2-critical -- blow-up dynamics -- Primary 35Q55 -- Secondary 35B35 -- 35B40 -- 35B44; Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab086
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: In this paper, we consider the $L^2$ -critical fractional Schrödinger 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 show that if the initial data have negative energy and slightly supercritical mass, then the solution blows up in finite time. We also give a specific description for the blow-up dynamics. This is an extension of the works of F. Merle and P. Raphaël for $L^2$ -critical Schrödinger equations. However, the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new strategies are required.