The dynamics of the focusing NLH with a potential beyond the mass-energy threshold

Abstract

We study the dynamics of the focusing nonlinear Hartree equation with a Kato potential $$i{\partial }_{t}u+\hat{I}”u-Vu=-(|\cdot {|}^{-{\hat{I}}^3}\ast |u{|}^2)u,x\in {\mathbb{R}}^d$$ under some assumptions on the potential $V$. We prove the blow up versus global existence dichotomy for solutions beyond the threshold, based on the method from Duyckaerts-Roudenko [6]. Furthermore, our result compensates for the one of in [13] below that threshold.