Differential and Integral Equations

Lower bounds for the $L^2$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation

Christophe Antonini

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Abstract

We consider the nonlinear Schrödinger equation with critical power $iu_t=-\Delta u-|u|^{4/N}u$, $t\geq 0$ and $x\in\mathbb T^N$ (the space-periodic case) in $H^1$. We consider a blow-up solution with minimal mass. We obtain in this context an optimal lower bound for the blow-up rate (that is, for $ | {\nabla u(t)}| _{L^2}$), and we observe that this lower bound equals the blow-up rate (which is explicitly known) of the minimal blow-up solutions in $\mathbb R^N$.

Article information

Source
Differential Integral Equations, Volume 15, Number 6 (2002), 749-768.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060815

Mathematical Reviews number (MathSciNet)
MR1893845

Zentralblatt MATH identifier
1016.35018

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B10: Periodic solutions 35B45: A priori estimates 35Q40: PDEs in connection with quantum mechanics

Citation

Antonini, Christophe. Lower bounds for the $L^2$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation. Differential Integral Equations 15 (2002), no. 6, 749--768. https://projecteuclid.org/euclid.die/1356060815


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