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

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

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

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. 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 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).

摘要/Abstract

摘要：

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).

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

Abstract:

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

杨凌燕, 李晓光, 吴永洪, Louis CACCETTA. GLOBAL WELL-POSEDNESS AND BLOW-UP FOR THE HARTREE EQUATION[J]. 数学物理学报(英文版), 2017, 37(4): 941-948.

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.

参考文献

[1] Cazenave T. Semilinear Schrödinger Equations. Courant Lecturre Notes in Mathematics, 10. American Mathematical Society, 2003
[2] Chen J Q, Guo B L. Strong instability of standing waves for a nonlocal Schrödinger equation. Phys D, 2007, 227:142-148
[3] Duyckaerts T, Roudenko S. Going beyound the threshold:scattering and blow-up in the focusing NLS equation. Commun Math Phys, 2015, 334:1573-1615
[4] Gaididei, Yu B, Rasmussen K O, Christiansen P L. Nonlinear excitations in two-dimensional molecular structures with impurities. Phys Rev E, 1995, 52:2951-2962
[5] Gao Y F, Wang Z Y. Scattering versus blow-up for the focusing L² supercritical Hartree equation. Z Angew Math Phy, 2014, 65:179-202
[6] Holmer J, Platte R, Roudenko S. Blow-up criteria for the 3D cubic nolinear Schrödinger equation. Nonlinearity, 2010, 23:977-1030
[7] Li D, Miao C X, Zhang X Y. The focusing energy-critical Hartree equation. J Diff Eq, 2009, 246:1139-1163
[8] Ma L, Lin Z. Classificatiion of positive solitary solutions of the nonlinear Choquard equation. Arch Rational Mech Anal, 2010, 195:455-467
[9] Lions P L. The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1984, 1:109-145, 223-284
[10] Miao C X, Xu G X, Zhao L F. Global well-posedness and scattering for the mass-critical Hartree equation for radial data. J Math Pures Appl, 2009, 91:49-79
[11] Miao C X, Xu G X, Zhao L F. The cauchy problem of the Hartree equation. J Partial Differ Equ, 2008, 21:22-44
[12] Pitaevskii L P. Vortex lines in an imperfect Bose gas. Sov Phys JETP, 1961, 13:451-454
[13] Weinstein M I. Nonlinear Schrödinger equations and sharp interpolation estimates. Commun Math Phys, 1983, 87:567-576
[14] Zhang J. Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations. Nonlinear Anal-TMA, 2002, 48:191-207
[15] Zhang J, Zhang L M, Chen J. A numerical simulation method of nonlinear Schrödinger equation. Acta Math Sci, 2007, 27A(6):1111-1117

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

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

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 6

Metrics

本文评价

推荐阅读 10

[1]	夏红强. GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE MASS-CRITICAL HARTREE EQUATION IN HIGH DIMENSIONS[J]. 数学物理学报(英文版), 2015, 35(1): 255-274.
[2]	朱世辉, 张健. LIMITING BEHAVIOR OF BLOW-UP SOLUTIONS OF THE NLSE WITH A STARK POTENTIAL[J]. 数学物理学报(英文版), 2012, 32(3): 1181-1192.
[3]	朱世辉, 张健. ON THE CONCENTRATION PROPERTIES FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH A STARK POTENTIAL[J]. 数学物理学报(英文版), 2011, 31(5): 1923-1938.
[4]	徐桂香, 苑佳. INTERACTION MORAWETZ ESTIMATE AND A SIMPLIFIED PROOF ON THE ENERGY SCATTERING FOR HARTREE EQUAITONS[J]. 数学物理学报(英文版), 2011, 31(1): 15-21.
[5]	陈琼蕾; 章志飞. GEOMETRIC OPTICS FOR 3D-HARTREE-TYPE EQUATION WITH COULOMB POTENTIAL[J]. 数学物理学报(英文版), 2006, 26(4): 646-654.
[6]	李用声, 陈庆益. ON INITIAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR SCHRÖDINGER EQUATIONS[J]. 数学物理学报(英文版), 1996, 16(4): 421-431.