The Blow-up Dynamics for the $L^2$-Critical Hartree Equation with Harmonic Potential

Abstract

In this paper, we study the $L^2$-critical Hartree equation with harmonic potential which arises in quantum theory of large system of nonrelativistic bosonic atoms and molecules. Firstly, by using the variational characteristic of the nonlinear elliptic equation and the Hamilton conservations, we get the sharp threshold for global existence and blow-up of the Cauchy problem. Then, in terms of a change of variables, we first find the relation between the Hartree equation with and without harmonic potential. Furthermore, we prove the upper bound of blow-up rate in $\mathbb{R}^3$ as well as the mass concentration of blow-up solution for the Hartree equation with harmonic potential in $\mathbb{R}^N.$