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

Abstract

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