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

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.