## Differential and Integral Equations

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

Christophe Antonini

#### 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

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