Inhomogeneous NLS with partial harmonic confinement

We investigate the inhomogeneous nonlinear Schrödinger equation with partial harmonic confinement. First, we present a global well-posedness result for small data in the intercritical regime. Second, we obtain a threshold of global existence versus finite-time blow-up in the mass-critical regime. Finally, we prove the $L^2$ concentration of the mass-critical non-global solution with minimal mass. The challenge is to address the fact that the standard scale invariance is broken by the partial confinement. We use the associated ground state without potential in order to describe the threshold of global versus non-global existence of solutions.