GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION

doi:10.1016/S0252-9602(17)30049-8

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (4): 941-948.doi: 10.1016/S0252-9602(17)30049-8

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GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION

Lingyan YANG¹, Xiaoguang LI^1,2, Yonghong WU³, Louis CACCETTA³

1. Sichuan Normal University, Chengdu, 610066, China;
2. Wuhan Institute of Physics and Machematics Chinese Academy of Science, Wuhan 430071, China;
3. Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received:2016-06-28 Revised:2016-12-28 Online:2017-08-25 Published:2017-08-25
About author:Lingyan YANG,E-mail:kafuka15@126.com;Xiaoguang LI,lixgmath@gmail.com;Yonghong WU,E-mail:y.wu@curtin.edu.au;Louis CACCETTA,l.caccetta@curtin.edu.au
Supported by:
This work was supported by the National Natural Science Foundation of China (11371267) and Sichuan Province Science Foundation for Youths (2012JQ0011).

Abstract

Abstract:

For 2 < γ < min{4, n}, we consider the focusing Hartree equation iu_t + △u + (|x|^-γ *|u|²)u=0, x ∈ Rⁿ. (0.1) Let M[u] and E[u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of -△ Q + Q=(|x|^-γ *|Q|²)Q. Guo and Wang[Z. Angew. Math. Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of (0.1) if M[u]^1-s_cE[u]^s_c < M[Q]^1-s_cE[Q]^s_c(s_c=γ-2/2). In this paper, we consider the complementary case M[u]^1-s_cE[u]^s_c ≥ M[Q]^1-s_cE[Q]^s_c and obtain a criteria on blow-up and global existence for the Hartree equation (0.1).

Key words: Hartree equation, Threshold criteria, blow-up solution

Lingyan YANG, Xiaoguang LI, Yonghong WU, Louis CACCETTA. GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION[J].Acta mathematica scientia,Series B, 2017, 37(4): 941-948.

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GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION

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