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

C. Ortoleva ; G. Perelman

Algebra i analiz, Tome 25 (2013) no. 2, pp. 162-192.

Résumé

The energy critical focusing nonlinear Schrödinger equation $i{\psi }_t=-\mathrm{\Delta }\psi -|\psi {|}^4\psi$ 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 $\psi (x,t)=e^{i\alpha (t)}{\lambda }^{1/2}(t)W(\lambda (t)x)+e^{i\mathrm{\Delta }t}{\zeta }^{\ast }+o_{{\dot{H}}^1}(1)$ as $t\to +\mathrm{\infty }$, where $\alpha (t)={\alpha }_0\ln t$, $\lambda (t)=t^{\nu }$, $W(x)=(1+\frac{1}{3}|x{|}^2{)}^{-1/2}$ is the ground state, and ${\zeta }^{\ast }$ is arbitrary small in ${\dot{H}}^1$.

Détail

Mots-clés : energy critical focusing nonlinear Schrödinger equation, Cauchy problem, ground state, blow up.

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