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 Algebra i Analiz, 2013, Volume 25, Issue 2, Pages 162–192 (Mi aa1328)

This article is cited in 5 scientific papers (total in 5 papers)

Research Papers

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

C. Ortoleva, G. Perelman

Université Paris-Est Créteil, Créteil Cedex, France

Abstract: The energy critical focusing nonlinear Schrödinger equation $i\psi_t=-\Delta\psi-|\psi|^4\psi$ in $\mathbb R^3$ is considered; 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 $t\to+\infty$, where $\alpha(t)=\alpha_0\ln t$, $\lambda(t)=t^\nu$, $W(x)=(1+\frac13|x|^2)^{-1/2}$ is the ground state, and $\zeta^*$ is arbitrary small in $\dot H^1$.

Keywords: energy critical focusing nonlinear Schrödinger equation, Cauchy problem, ground state, blow up.

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English version:
St. Petersburg Mathematical Journal, 2014, 25:2, 271–294

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Received: 02.10.2012
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Citation: C. Ortoleva, G. Perelman, “Nondispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in $\mathbb R^3$”, Algebra i Analiz, 25:2 (2013), 162–192; St. Petersburg Math. J., 25:2 (2014), 271–294

Citation in format AMSBIB
\Bibitem{OrtPer13} \by C.~Ortoleva, G.~Perelman \paper Nondispersive vanishing and blow up at infinity for the energy critical nonlinear Schr\"odinger equation in~$\mathbb R^3$ \jour Algebra i Analiz \yr 2013 \vol 25 \issue 2 \pages 162--192 \mathnet{http://mi.mathnet.ru/aa1328} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3114854} \zmath{https://zbmath.org/?q=an:1303.35103} \elib{http://elibrary.ru/item.asp?id=20730202} \transl \jour St. Petersburg Math. J. \yr 2014 \vol 25 \issue 2 \pages 271--294 \crossref{https://doi.org/10.1090/S1061-0022-2014-01290-3} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000343074000008} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84924408463} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. R. Donninger, A. Zenginoğlu, “decay for the cubic wave equation”, Anal. PDE, 7:2 (2014), 461–495
2. J. Krieger, J. Nahas, “Instability of type II blow up for the quintic nonlinear wave equation on $\mathbb R^{3+1}$”, Bull. Soc. Math. France, 143:2 (2015), 339–355
3. J. Jendrej, “Bounds on the speed of type II blow-up for the energy critical wave equation in the radial case”, Int. Math. Res. Notices, 2016, no. 21, 6656–6688
4. J. Jendrej, “Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5”, J. Funct. Anal., 272:3 (2017), 866–917
5. J. Jendrej, “Construction of two-bubble solutions for the energy-critical NLS”, Anal. PDE, 10:8 (2017), 1923–1959
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