Acta mathematica scientia, Series B >

2017 , Vol. 37 >Issue 4: 941 - 948

DOI: https://doi.org/10.1016/S0252-9602(17)30049-8

Articles

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

  • Lingyan YANG ,
  • Xiaoguang LI ,
  • Yonghong WU ,
  • Louis CACCETTA
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  • 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
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

Received date: 2016-06-28

  Revised date: 2016-12-28

  Online published: 2017-08-25

Supported by

This work was supported by the National Natural Science Foundation of China (11371267) and Sichuan Province Science Foundation for Youths (2012JQ0011).

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Abstract

For 2 < γ < min{4, n}, we consider the focusing Hartree equation iut + △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-scE[u]sc < M[Q]1-scE[Q]sc(sc=γ-2/2). In this paper, we consider the complementary case M[u]1-scE[u]sc ≥ M[Q]1-scE[Q]sc 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

Cite this article

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 . DOI: 10.1016/S0252-9602(17)30049-8

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