Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
zbMATH Open document ID: 1061.35135
author: Frank Merle
author: P. Raphaël
published in: Geometric and Functional Analysis. GAFA
publication date: 2003-09-01T00:00:00Z
review text: The authors consider the Cauchy problem for the critical nonlinear Schrödinger equation \[ \begin{cases} iu_t = - \Delta u -| u | ^{4 / N}u,&(t,x) \in [0,\,T)\times \mathbb R^N, \\ u(0,x)= u_0(x), &u_0 :\mathbb R^N \to \mathbb C,\end{cases} \] with \(u_0\in H^1(\mathbb R^N)\), \(N\geq 1\). It is proved that the following estimation is true \[ | \nabla u(t) | _{L_2 } \leq C\left( \frac{\ln | \ln (T - t)| }{T-t} \right)^{1 / 2}. \]
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 1973268
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: blow-up estimate
zbMATH Keywords: Cauchy problem
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
full work available at URL: s00039-003-0424-9
OpenAlex ID: W1647523290
DOI: 10.1007/S00039-003-0424-9
description: scientific article; zbMATH DE number 1973268