Scattering for the generalized Hartree equation with a potential

TL;DR

This paper proves scattering for the focusing generalized Hartree equation with a potential in three dimensions, using a simplified approach based on Tao's criteria and Morawetz estimates, extending previous methods to include potential effects.

Contribution

It introduces a novel mass-potential condition and employs a simplified scattering proof for the generalized Hartree equation with potential, extending prior frameworks.

Findings

Established scattering in the intercritical case for radial initial data.

Extended the mass-energy framework to include potential effects.

Provided a more straightforward proof compared to traditional methods.

Abstract

We consider the focusing generalized Hartree equation in $H^1(\R^3)$ with a potential, \begin{equation*} iu_t + \Delta u - V(x)u + (I_\gamma \ast |u|^p )|u|^{p-2} u=0, \end{equation*} where $I_\gamma = \frac{1}{|x|^{3-\gamma}}$, $p \geq 2$ and $\gamma < 3$. In this paper, we prove scattering for the generalized Hartree equation with a potential in the intercritical case assuming radial initial data. The novelty of our approach lies in the use of a general mass-potential condition, incorporating the potential V, which extends the standard mass-energy framework. To this end, we employ a simplified method inspired by Dodson and Murphy \cite{Dod-Mur}, based on Tao's scattering criteria and Morawetz estimates. This approach provides a more straightforward proof of scattering compared to the traditional concentration-compactness/rigidity method of Kenig and Merle \cite{KENIG}.