On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation

Authors:
Frank Merle and Pierre Raphael

Journal:
J. Amer. Math. Soc. **19** (2006), 37-90

MSC (2000):
Primary 35Q55; Secondary 35Q51, 35B05

DOI:
https://doi.org/10.1090/S0894-0347-05-00499-6

Published electronically:
September 1, 2005

MathSciNet review:
2169042

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the $L^2$ critical nonlinear Schrödinger equation $iu_t=-\Delta u-|u|^{\frac {4}{N}}u$ with initial condition in the energy space $u(0,x)=u_0\in H^1$ and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in $L^2_{loc}$ a sharp and stable upper bound on the blow-up rate: $|\nabla u(t)|_{L^2}\leq C\left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}}$. In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, $\lim _{t\to T}\sqrt {T-t}|\nabla u(t)|_{L^2}=+\infty .$ In this paper, we prove the sharp lower bound \[ |\nabla u(t)|_{L^2}\geq C \left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}}\] by exhibiting the dispersive structure in the scaling invariant space $L^2$ for this log-log regime. In addition, we will extend to the pure energy space $H^1$ a dynamical characterization of the solitons among the zero energy solutions.

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Additional Information

**Frank Merle**

Affiliation:
Université de Cergy–Pontoise, Cergy-Pointoise, France, Institute for Advanced Studies and Centre National de la Recherche Scientifique

MR Author ID:
123710

**Pierre Raphael**

Affiliation:
Université de Cergy–Pontoise, Cergy-Pointoise, France, Institute for Advanced Studies and Centre National de la Recherche Scientifique

Keywords:
Critical Schrödinger equation,
finite time blowup,
blow-up rate,
log-log law.

Received by editor(s):
March 2, 2004

Published electronically:
September 1, 2005

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.