数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (4): 941-948.doi: 10.1016/S0252-9602(17)30049-8

• 论文 • 上一篇    下一篇

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

杨凌燕1, 李晓光1,2, 吴永洪3, Louis CACCETTA3   

  1. 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
  • 收稿日期:2016-06-28 修回日期:2016-12-28 出版日期:2017-08-25 发布日期:2017-08-25
  • 作者简介: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
  • 基金资助:

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

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

Lingyan YANG1, Xiaoguang LI1,2, Yonghong WU3, Louis CACCETTA3   

  1. 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).

摘要:

For 2 < γ < min{4, n}, we consider the focusing Hartree equation iut + △u + (|x| *|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| *|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]scM[Q]1-scE[Q]sc and obtain a criteria on blow-up and global existence for the Hartree equation (0.1).

关键词: Hartree equation, Threshold criteria, blow-up solution

Abstract:

For 2 < γ < min{4, n}, we consider the focusing Hartree equation iut + △u + (|x| *|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| *|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]scM[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