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

Abstract

We consider solutions $u(t)$ to the 3d NLS equation $i{\partial }_{t}u+\Delta u+|u{|}^{2}u=0$ such that $\Vert xu(t){\Vert }_{L^2}=\infty$ 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 $-Q+\Delta Q+|Q{|}^{2}Q=0$, we prove the following: if $M[u]E[u]<M[Q]E[Q]$ and $\Vert u_0{\Vert }_{L^2}\Vert \nabla u_0{\Vert }_{L^2}>\Vert Q{\Vert }_{L^2}\Vert \nabla Q{\Vert }_{L^2}$, 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 $\Vert \nabla u(t_n){\Vert }_{L^2}\to \infty$. Similar statements hold for negative time.

References in corpus (1)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Cited by in corpus (2)

• Scattering and Blow up for the Two Dimensional Focusing Quintic Nonlinear Schrödinger Equation
• Standing Ring Blowup Solutions for Cubic NLS