带部分调和势的非齐次非线性 Schrödinger 方程的爆破解

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 1350-1372.

带部分调和势的非齐次非线性 Schrödinger 方程的爆破解

简慧(),龚敏(),王莉^*()

华东交通大学理学院南昌 330013

收稿日期:2020-09-30 修回日期:2023-03-24 出版日期:2023-10-26 发布日期:2023-08-09
通讯作者: 王莉 E-mail:jianhui0711141@163.com;gluminous@163.com;wangli.423@163.com
作者简介:简慧，Email: jianhui0711141@163.com;|龚敏，Email: gluminous@163.com
基金资助:
国家自然科学基金(11761032);国家自然科学基金(12161038);江西省自然科学基金(20212BAB211006);江西省自然科学基金(20202BABL211004);江西省教育厅科技项目(GJJ212204)

On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement

Jian Hui(),Gong Min(),Wang Li^*()

School of Science, East China Jiaotong University, Nanchang 330013

Received:2020-09-30 Revised:2023-03-24 Online:2023-10-26 Published:2023-08-09
Contact: Li Wang E-mail:jianhui0711141@163.com;gluminous@163.com;wangli.423@163.com
Supported by:
NSFC(11761032);NSFC(12161038);Jiangxi Provincial Natural Science Foundation(20212BAB211006);Jiangxi Provincial Natural Science Foundation(20202BABL211004);Science and Technology Project of Education Department of Jiangxi Province(GJJ212204)

摘要/Abstract

摘要：

该文致力于研究带部分调和势的非齐次非线性 Schrödinger 方程的 Cauchy 问题. 该方程是玻色-爱因斯坦凝聚中的一个重要模型.结合非线性椭圆方程基态解的变分特征及质量和能量守恒, 首先得到了该问题整体解的存在性, 并利用尺度变换技巧证明了该方程在一些特殊初值情形下存在爆破解. 其次讨论了爆破解的 $L^{2}$ 集中现象.最后利用与上述基态解相关的变分结论研究了 $L^{2}$ 最小质量爆破解的动力学性质, 即具有最小质量的爆破解的极限 profile、精细质量集中和爆破速率. 该文将 Zhang[34] 的全局存在性和爆破结果推广到带非齐次非线性项的情形, 并将 Pan 和 Zhang[23] 的部分结果改进到空间维数 $N\geq2$ 且非线性项为非齐次的情形.

关键词: 非齐次非线性 Schr?dinger 方程, 部分调和势, 爆破, 质量集中, 极限 profile

Abstract:

This paper is devoted to the Cauchy problem of inhomogeneous nonlinear Schrödinger equation in the presence of a partial confinement, which is an important model in Bose-Einstein condensates. Combining the variational characterization of the ground state of a nonlinear elliptic equation and the conservations of mass and energy, we first obtain a global solution and show the existence of blow-up solutions for some special initial data by scaling techniques. Then, we study the $L^2$-concentration phenomenon for the blow-up solutions. Finally, we apply the variational arguments connected to the above ground state to investigate the dynamics of $L^2$-minimal blow-up solutions, i.e., the limiting profile, mass-concentration and blow-up rate of the blow-up solutions with minimal mass. We extend the global existence and blow-up results of Zhang[34] to the case of inhomogeneous nonlinearities and improve partial results of Pan and Zhang[23] to space dimensions $N\geq2$ in the inhomogeneous case.

Key words: Inhomogeneous nonlinear Schr?dinger equation, Partial confinement, Blow-up, Mass-concentration, Limiting

中图分类号:

O175.23

简慧, 龚敏, 王莉. 带部分调和势的非齐次非线性 Schrödinger 方程的爆破解[J]. 数学物理学报, 2023, 43(5): 1350-1372.

Jian Hui, Gong Min, Wang Li. On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1350-1372.

参考文献 37

[1]	Antonelli P, Carles R, Silva J D. Scattering for nonlinear Schrödinger equation under partial harmonic confinement. Comm Math Phys, 2015, 334 (1): 367-396 doi: 10.1007/s00220-014-2166-y
[2]	Bao W Z, Cai Y Y. Mathematics theory and numerical methods for Bose-Einstein condensation. Kinet Relat Models, 2013, 6(1): 1-135 doi: 10.3934/krm.2013.6.1
[3]	Bellazzini J, Boussaid N, Jeanjean L, Visciglia N. Existence and stability of standing waves for supercritical NLS with a partial confinement. Comm Math Phys, 2017, 353(1): 229-251 doi: 10.1007/s00220-017-2866-1
[4]	Cao D M, Han P G. Inhomogeneous critical nonlinear Schrödinger equations with a harmonic potential. J Math Phys, 2010, 51(4): 043505 doi: 10.1063/1.3371246
[5]	Cazenave T. Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics vol. 10. Providence: American Mathematical Society, 2003
[6]	Dalfovo F, Giorgini S, Pitaevskii L P, Stringari S. Theory of Bose-Einstein condensation in trapped gases. Rev Modern Phys, 1999, 71(3): 463-512 doi: 10.1103/RevModPhys.71.463
[7]	Feng B H. On the blow-up solutions for the nonlinear Schrödinger equations with combined power-type nonlinearities. J Evol Equ, 2018, 18(1): 203-220 doi: 10.1007/s00028-017-0397-z
[8]	Fukuizumi R, Ohta M. Stability of standing waves for nonlinear Schrödinger equations with potentials. Differential Integral Equations, 2003, 16(1): 111-128
[9]	Glassey R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J Math Phys, 1977, 18(9): 1794-1797 doi: 10.1063/1.523491
[10]	Gou T X. Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement. J Math Phys, 2018, 59(7): 071508 doi: 10.1063/1.5028208
[11]	Hmidi T, Keraani S. Blowup theory for the critical nonlinear Schrödinger equations revisited. Int Math Res Notices, 2005, 46: 2815-2828
[12]	Jia H F, Li G B, Luo X. Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement. Discrete Contin Dyn Syst, 2020, 40(5): 2739-2766 doi: 10.3934/dcds.2020148
[13]	Kwong M K. Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbf{R}^N$. Arch Ration Mech Anal, 1989, 105: 243-266 doi: 10.1007/BF00251502
[14]	Li X G, Zhang J. Limit behavior of blow-up solutions for critical nonlinear Schrödinger equation with harmonic potential. Differential Integral Equations, 2006, 19(7): 761-771
[15]	Liu Q, Zhou Y Q, Zhang J, Zhang W N. Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential. Applied Mathematics and Computation, 2006, 177(2): 482-487 doi: 10.1016/j.amc.2005.11.024
[16]	Liu Z X. On a class of inhomogeneous, energy-critical, focusing, nonlinear Schrödinger equations. Acta Math Sci, 2013, 33B(6): 1522-1530
[17]	Merle F. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math J, 1993, 69(2): 427-454
[18]	Merle F. Nonexistence of minimal blow-up solutions of equations i$u_t = -\Delta u-k(x)\|u\|^{\frac{4}{N}}u$ in $\mathbf{R}^N$. Ann Inst Henri Poincare, 1996, 64(1): 33-85
[19]	Merle F, Raphaël P. The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann Math, 2005, 161(1): 157-222 doi: 10.4007/annals
[20]	Ogawa T, Tsutsumi Y. Blow-up of $H^1$ solution for the nonlinear Schrödinger equation. J Differential Equations, 1991, 92(2): 317-330 doi: 10.1016/0022-0396(91)90052-B
[21]	Oh Y G. Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials. J Differential Equations, 1989, 81(2): 255-274 doi: 10.1016/0022-0396(89)90123-X
[22]	Ohta M. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Commun Pure Appl Anal, 2018, 17(4): 1671-1680
[23]	Ohta M. Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential. Funkcial Ekvac, 2018, 61(1): 135-143 doi: 10.1619/fesi.61.135
[24]	Pan J J, Zhang J. Mass concentration for nonlinear Schrödinger equation with partial confinement. J Math Anal Appl, 2020, 481(2): 123484 doi: 10.1016/j.jmaa.2019.123484
[25]	Pang P Y H, Tang H Y, Wang Y D. Blow-up solutions of inhomogeneous nonlinear Schrödinger equations. Calculus Var Partial Differ Equ, 2006, 26(2): 137-169 doi: 10.1007/s00526-005-0362-5
[26]	Pitaevskii L, Stringari S. Bose-Einstein Condensation. International Series of Monographs on Physics, 116. Oxford: Oxford University Press, 2003
[27]	Shu J, Zhang J. Sharp criterion of global existence for a class of nonlinear Schrödinger equation with critical exponent. Applied Mathematics and Computation, 2006, 182(2) : 1482-1487 doi: 10.1016/j.amc.2006.05.036
[28]	Shu J, Zhang J. Sharp criterion of global existence for nonlinear Schrödinger equation with a harmonic potential. Acta Mathematica Sinica, English Series, 2009, 25(4): 537-544 doi: 10.1007/s10114-009-7473-4
[29]	Strauss W A. Existence of solitary waves in higher dimensions. Comm Math Phys, 1977, 55: 149-162 doi: 10.1007/BF01626517
[30]	Sulem C, Sulem P. The Nonlinear Schrödinger Equation:Self-Focusing and Wave Collapse. New York: Springer-Verlag, 1999
[31]	Weinstein M I. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm Math Phys, 1983, 87: 567-576 doi: 10.1007/BF01208265
[32]	Weinstein M I. On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Comm Partial Differential Equations, 1986, 11: 545-565 doi: 10.1080/03605308608820435
[33]	Zhang J. Stability of attractive Bose-Einstein condensates. J Stat Phys, 2000, 101: 731-746 doi: 10.1023/A:1026437923987
[34]	Zhang J. Sharp threshold for blow-up and global existence in nonlinear Schrödinger equations under a harmonic potential. Comm Partial Differential Equations, 2005, 30(10): 1429-1443 doi: 10.1080/03605300500299539
[35]	Zhang J. Sharp threshold of global existence for nonlinear Schrödinger equation with partial confinement. Nonlinear Anal, 2020, 196: 111832 doi: 10.1016/j.na.2020.111832
[36]	Zhang M Y, Ahmed M S. Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential. Adv Nonlinear Anal, 2019, 9(1): 882-894
[37]	Zhu S H, Zhang J, Li X G. Limiting profile of blow-up solutions for the Gross-Pitaevskii equation. Science China Mathematics, 2009, 52(5): 1017-1030 doi: 10.1007/s11425-008-0140-x