\(\log\)-\(\log\) blow up solutions blow up at exactly \(m\) points
type: Item
instance of: scholarly article
title: \(\log\)-\(\log\) blow up solutions blow up at exactly \(m\) points
zbMATH Open document ID: 1382.35263
author: Chenjie Fan
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2017-11-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1510.00961
review text: The author considers the Cauchy Problem for the mass-critical focusing nonlinear Schrödinger equation \[ i u_t = -\Delta u- | u|^{4/d} u \] for initial conditions in \( H^1(\mathbb{R}^d) \) with \( d =1,2 \). He proves that for arbitrary points \( x_1, \ldots x_m \) there exists a solution \( u \) that blows up in finite time \( T \) at exactly these points. There exists an exact asymptotic formula based on a pseudo-conformal transformation of the global soliton solution showing that this blow-up follows a log-log rule.
reviewed by: Guido Schneider
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6810623
zbMATH Keywords: NLS
zbMATH Keywords: log-log blow up
zbMATH Keywords: \(m\) points blow up
zbMATH Keywords: bootstrap
zbMATH Keywords: propagation of regularity
zbMATH Keywords: topological argument
MaRDI profile type: Publication
OpenAlex ID: W2962755157
arXiv ID: 1510.00961
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DOI: 10.1016/J.ANIHPC.2016.11.002
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description: scientific article