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

Mao Zhang; Jingjing Pan; Jian Zhang

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

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KeyWord:Hartree equation, harmonic potential, blow-up rate, upper bound, mass concentration

Author Name	Affiliation
Mao Zhang	School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
Jingjing Pan	School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China
Jian Zhang	School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

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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 $\mathbb R^N$.