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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
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Abstract: We consider the $L^2$ critical nonlinear Schrödinger equation $iu_t=-\Delta u-\vert u\vert^{\frac{4}{N}}u$ 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: $\vert\nabla u(t)\vert _{L^2}\leq C\left(\frac{\log\vert\log(T-t)\vert}{T-t}\right)^{\frac{1}{2}}$.

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, $\lim_{t\to T}\sqrt{T-t}\vert\nabla u(t)\vert _{L^2}=+\infty.$

In this paper, we prove the sharp lower bound

\begin{displaymath}\vert\nabla u(t)\vert _{L^2}\geq C \left(\frac{\log\vert\log(T-t)\vert}{T-t}\right)^{\frac{1}{2}}\end{displaymath}

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

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

DOI: https://doi.org/10.1090/S0894-0347-05-00499-6
Keywords: Critical Schr\"odinger 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.

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