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

Shuang Ji, Jing Lu and Fanfei Meng

Mathematische Nachrichten, 2025, vol. 298, issue 11, 3444-3459

Abstract: In this paper, we study the dynamics of the focusing nonlinear Hartree equation with a Kato potential i∂tu+Δu−Vu=−(|·|−γ*|u|2)u,x∈Rd$$\begin{equation*} \mathrm{i}{\partial}_{t}u+\mathrm{\Delta}u-\textit{Vu}=-\operatorname{(}|\cdot {|}^{-\gamma}\ast |u{|}^{2})u,\hspace*{0.202em}x\in {\mathbb{R}}^{d} \end{equation*}$$under some assumptions on the potential V$V$. We prove the blow up versus global existence dichotomy for solutions beyond the threshold, based on the method from Duyckaerts–Roudenko in [6]. Our result extends the work of [13], which studied the blow up and scattering theory for solutions of (NLHV)$(\text{NLH}_{\text{V}})$ below that threshold.

Date: 2025
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