Blow up and scattering criteria above the threshold for the focusing inhomogeneous nonlinear Schrödinger equation

Abstract

We consider the inhomogeneous nonlinear Schrödinger equation (INLS) in ${\mathbb{R}}^N$, $N\ge 1$, $$i{\partial }_{t}u+\Delta u+|x{|}^{-b}|u{|}^{p-1}u=0,$$ with finite-variance initial data $u_0\in H^1({\mathbb{R}}^N)$. We extend the dichotomy between scattering and blow-up for solutions above the mass-energy threshold (and with arbitrarily large energy). We also show other two blow-up criteria, wich are valid in any mass-supercritical setting, given there is local well-posedness.