Scattering and blowup beyond the mass-energy threshold for the cubic NLS with a potential
Volume 172 / 2023
Colloquium Mathematicum 172 (2023), 143-163
MSC: Primary 35P25; Secondary 35Q55.
DOI: 10.4064/cm8978-10-2022
Published online: 7 December 2022
Abstract
We consider the focusing cubic nonlinear Schrödinger equation with a potential, $i \partial _{t}u+(\Delta -V)u+|u|^{2}u=0$, with finite energy and finite variance initial data in dimension $3$. We investigate the scattering versus blow up dichotomy above the mass-energy threshold for finite variance solutions. Key ingredients in the proof include variational analysis, a virial argument and the concentration-compactness/rigidity method.
