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

TL;DR

This paper investigates the behavior of solutions to the focusing nonlinear Hartree equation with a potential beyond the mass-energy threshold, establishing conditions for blow-up or global existence.

Contribution

It extends the blow-up versus global existence dichotomy to the Hartree equation with a potential beyond the threshold, using and adapting methods from prior work.

Findings

Established blow-up versus global existence criteria beyond the threshold

Extended the analysis to include potentials satisfying certain assumptions

Provided a comprehensive understanding of solution dynamics in this regime

Abstract

We study the dynamics of the focusing nonlinear Hartree equation with a Kato potential $$ i\partial_t u +\Delta u - Vu = -(|\cdot|^{-\gamma} \ast |u|^2)u, \quad x \in \mathbb{R}^d $$ 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.