This article is cited in 8 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.

Received: 02.10.2012

Language: English

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

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