## Differential and Integral Equations

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

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

#### 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