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

Abstract:

The following energy critical focusing nonlinear Schrödinger equation in $\mathbb R^3$ is considered: $i\psi _t=-\Delta \psi -|\psi |^4\psi$; 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 $\alpha (t)=\alpha _0\ln t$, $\lambda (t)=t^{\nu }$, $W(x)=(1+\frac 13|x|^2)^{-1/2}$ is the ground state, and $\zeta ^*$ is arbitrary small in $\dot H^1$.