# Focusing nonlinear Hartree equation with inverse-square potential

## TL;DR

This paper studies the scattering behavior of radial solutions to a focusing nonlinear Hartree equation with inverse-square potential, addressing challenges from non-translation invariance and non-local nonlinearity.

## Contribution

It develops a scattering theory for the focusing Hartree equation with inverse-square potential, overcoming difficulties from non-locality and lack of translation invariance.

## Key findings

- Established scattering results for radial solutions in energy space
- Overcame weak dispersive estimates when potential parameter is negative
- Utilized virial-Morawetz estimates to control mass concentration

## Abstract

In this paper, we consider the scattering theory of the radial solution to focusing energy-subcritical Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$ using the method from \cite{Dodson2016}. The main difficulties are the equation is \emph{not} space-translation invariant and the nonlinearity is non-local. Using the radial Sobolev embedding and a virial-Morawetz type estimate we can exclude the concentration of mass near the origin. Besides, we can overcome the weak dispersive estimate when $a<0$, using the dispersive estimate established by \cite{zheng}.