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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Nondispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in $ \mathbb{R}^3$

Authors: C. Ortoleva and G. Perelman
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
Journal: St. Petersburg Math. J. 25 (2014), 271-294
MSC (2010): Primary 35Q55
Published electronically: March 12, 2014
MathSciNet review: 3114854
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Abstract: The following energy critical focusing nonlinear Schrödinger equation in $ \mathbb{R}^3$ is considered: $ i\psi _t=-\Delta \psi -\vert\psi \vert^4\psi $; it is proved that, for any $ \nu $ and $ \alpha _0$ sufficiently small, there exist radial finite energy solutions of the form $ \psi (x,t)= e^{i\alpha (t)}\lambda ^{1/2}(t) W(\lambda (t)x)+e^{i\Delta t}\zeta ^*+o_{\dot H^1} (1)$ as $ t\rightarrow +\infty $, where $ \alpha (t)=\alpha _0\ln t$, $ \lambda (t)=t^{\nu }$, $ W(x)=(1+\frac 13\vert x\vert^2)^{-1/2}$ is the ground state, and $ \zeta ^*$ is arbitrary small in $ \dot H^1$.

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Additional Information

C. Ortoleva
Affiliation: Université Paris-Est Créteil, Créteil Cedex, France

G. Perelman
Affiliation: Université Paris-Est Créteil, Créteil Cedex, France

Keywords: Energy critical focusing nonlinear Schr\"odinger equation, Cauchy problem, ground state, blow up
Received by editor(s): October 2, 2012
Published electronically: March 12, 2014
Dedicated: Dedicated to the memory of Vladimir Savelievich Buslaev
Article copyright: © Copyright 2014 American Mathematical Society

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