# On the blow up phenomenon for the mass critical focusing Hartree equation with inverse-square potential

## TL;DR

This paper investigates the blow-up behavior of solutions to the mass critical focusing Hartree equation with inverse-square potential, establishing global existence below a mass threshold and constructing finite-time blow-up solutions at the threshold.

## Contribution

It proves global existence for sub-threshold mass, constructs blow-up solutions at the threshold, and analyzes mass concentration phenomena, addressing challenges from non-translation invariance and non-local nonlinearity.

## Key findings

- Solutions are global if initial mass is below ground state mass.
- Finite-time blow-up solutions exist at the minimal mass threshold.
- Mass concentration occurs during blow-up.

## Abstract

In this paper, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties are the equation is \emph{not} space-translation invariant and the nonlinearity is non-local. We first prove that if the mass of the initial data is less than that of ground states, then the solution will be global. Although we don't know whether the ground state is unique, we can verify all the ground states have the same, minimal mass threshold. Then at the minimal mass threshold, we can construct the finite-time blow up solution, which is a pseudo-conformal transformation of the ground state, up to the symmetries of the equation. Finally, we establish an mass concentration phenomenon of the finite-time blow up solution to the equation.