Sharp threshold of blow-up and scattering for the fractional Hartree equation

被引：36

作者：

Guo, Qing ^{[1
]}

Zhu, Shihui ^{[2
,3
]}

机构：

[1] Minzu Univ China, Coll Sci, Beijing 100081, Peoples R China

[2] Univ Elect Sci & Technol China, Sch Math Sci, Chengdu 611731, Sichuan, Peoples R China

[3] Sichuan Normal Univ, Dept Math, Chengdu 610066, Sichuan, Peoples R China

来源：

JOURNAL OF DIFFERENTIAL EQUATIONS | 2018年 / 264卷 / 04期

基金：

中国国家自然科学基金;

关键词：

Fractional Schrodinger equation; L-2-supercritical; Scattering; Blow-up; GLOBAL WELL-POSEDNESS; NONLINEAR SCHRODINGER-EQUATION; COMPUTING GROUND-STATES; NUMERICAL-METHODS; DYNAMICS; COMPACTNESS; MINIMIZERS; EXISTENCE;

D O I：

10.1016/j.jde.2017.11.001

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We consider the fractional Hartree equation in the L-2-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If M[u(0)](s-sc/sc) E[u(0)] < M[Q](s-sc/sc) E[Q] and M[u(0)](s-sc/sc) parallel to u(0)parallel to(2)(<(H)over dot>s) < M[Q](s-sc/sc) parallel to Q parallel to(2)(<(H)over dot>s), then the solution u(t) is globally well-posed and scatters; if M[u(0)](s-sc/sc) E[u(0)] < M[Q](s-sc/sc) E[Q] and M[u(0)](s-sc/sc) parallel to u(0)parallel to(2)(<(H)over dot>s) > M[Q](s-sc/sc)parallel to Q parallel to(2)(<(H)over dot>s), the solution u(t) blows up in finite time. This condition is sharp in the sense that the solitary wave solution e(it) Q(x) is global but not scattering, which satisfies the equality in the above conditions. Here, Qis the ground-state solution for the fractional Hartree equation. (C) 2017 Elsevier Inc. All rights reserved.

引用

页码：2802 / 2832

页数：31

共 40 条

[11] Positive solutions of the nonlinear Schrodinger equation with the fractional Laplacian
Felmer, Patricio
Quaas, Alexander
Tan, Jinggang
[J]. PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 2012, 142 (06) : 1237 - 1262
[12] Uniqueness of non-linear ground states for fractional Laplacians in
Frank, Rupert L.
Lenzmann, Enno
[J]. ACTA MATHEMATICA, 2013, 210 (02) : 261 - 318
[13] Blowup for Nonlinear wave equations describing Boson stars
Froehlich, Juerg
Lenzmann, Enno
[J]. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 2007, 60 (11) : 1691 - 1705
[14] Existence and stability of standing waves for nonlinear fractional Schroumldinger equations
Guo, Boling
Huang, Daiwen
[J]. JOURNAL OF MATHEMATICAL PHYSICS, 2012, 53 (08)
[15] Semiclassical states for weakly coupled fractional Schrodinger systems
Guo, Qing
He, Xiaoming
[J]. JOURNAL OF DIFFERENTIAL EQUATIONS, 2017, 263 (04) : 1986 - 2023
[16] Scattering for the focusing L2-supercritical and (H) over dot2-subcritical biharmonic NLS equations
Guo, Qing
[J]. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2016, 41 (02) : 185 - 207
[17] Decay estimates for a class of wave equations
Guo, Zihua
Peng, Lizhong
Wang, Baoxiang
[J]. JOURNAL OF FUNCTIONAL ANALYSIS, 2008, 254 (06) : 1642 - 1660
[18] Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrodinger and wave equations
Guo, Zihua
Wang, Yuzhao
[J]. JOURNAL D ANALYSE MATHEMATIQUE, 2014, 124 : 1 - 38
[19] Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity
Hajaiej, H.
[J]. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2013, 399 (01) : 17 - 26
[20] A sharp condition for scattering of the radial 3D cubic nonlinear Schrodinger equation
Holmer, Justin
Roudenko, Svetlana
[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2008, 282 (02) : 435 - 467

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