The Blow-up Dynamics for the $L^2$-Critical Hartree Equation with Harmonic Potential (Journal of Nonlinear Modeling and Analysis, 2024)
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, weget 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 betweenthe Hartree equation with and without harmonic potential. Furthermore, weprove the upper bound of blow-up rate in $\mathbb{R}^3$ as well as the mass concentrationof blow-up solution for the Hartree equation with harmonic potential in $\mathbb{R}^N.$
Keywords: Hartree equation; harmonic potential; blow-up rate; upper bound; mass concentration.