Divergence of infinite-variance nonradial solutions to the 3d NLS equation

Abstract

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + \Delta u + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and by $Q(x)$ the ground state solution to $-Q+\Delta Q+|Q|^2Q=0$, we prove the following: if $M[u]E[u]<M[Q]E[Q]$ and $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}>\|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then either $u(t)$ blows-up in finite positive time or $u(t)$ exists globally for all positive time and there exists a sequence of times $t_n\to +\infty$ such that $\|\nabla u(t_n)\|_{L^2} \to \infty$. Similar statements hold for negative time.