半线性Klein-Gordon方程的高频周期解

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 484-500.

半线性Klein-Gordon方程的高频周期解

童常青¹,郑静^2,*()

¹ 杭州电子科技大学理学院数学系杭州 310018
² 杭州电子科技大学经济学院统计系杭州 310018

收稿日期:2017-01-09 出版日期:2019-06-26 发布日期:2019-06-27
通讯作者: 郑静 E-mail:tongchangqing@hdu.edu.cn
基金资助:
国家社科基金(17BTJ023)

Periodic Solutions of a Semi-Linear Klein-Gordon Equations with High Frequencies

Changqing Tong¹,Jing Zheng^2,*()

¹ Department of Mathematics, College of Science, Hangzhou Dianzi University, Hangzhou 310018
² Department of Statistics, College of Economy, Hangzhou Dianzi University, Hangzhou 310018

Received:2017-01-09 Online:2019-06-26 Published:2019-06-27
Contact: Jing Zheng E-mail:tongchangqing@hdu.edu.cn
Supported by:
the The National Social Science Fund of China(17BTJ023)

摘要/Abstract

摘要：

该文对一些半线性Klein-Gordon方程，证明了高频周期解的存在性.对非线性项只假设它的正则性为C^k，且没有非线性项非常小的假设.利用Nash-Moser迭代，在Sobolev空间中得到了周期解.

关键词: Klein-Gordon方程, 周期解, Nash-Moser迭代

Abstract:

In this paper, we prove the existence of periodic solutions with high frequencies of some semi-linear Klein-Gordon equations. We only assume the nonlinearities are C^k regular and without smallness. Using Nash-Moser iteration, we obtained some periodic solutions in Sobolev space.

Key words: Klein-Gordon equation, Periodic solutions, Nash-Moser Iteration

中图分类号:

O175.27

童常青,郑静. 半线性Klein-Gordon方程的高频周期解[J]. 数学物理学报, 2019, 39(3): 484-500.

Changqing Tong,Jing Zheng. Periodic Solutions of a Semi-Linear Klein-Gordon Equations with High Frequencies[J]. Acta mathematica scientia,Series A, 2019, 39(3): 484-500.

参考文献 19

1	Berti M , Bolle P . Periodic solutions of nonlinear wave equations with general nonlinearities. Comm Math Phys, 2003, 243: 315- 328 doi: 10.1007/s00220-003-0972-8
2	Berti M , Bolle P . Multiplicity of periodic solutions of nonlinear wave equations. Nonlinear Analysis TMA, 2004, 56: 1011- 1046 doi: 10.1016/j.na.2003.11.001
3	Berti M , Bolle P . Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Mathematical Journal, 2006, 134: 359- 419 doi: 10.1215/S0012-7094-06-13424-5
4	Berti M , Bolle P . Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions. Arch Rational Mech Anal, 2010, 195: 609- 642 doi: 10.1007/s00205-008-0211-8
5	Berti M , Bolle P , Procesi M . An abstract Nash-Moser theorem with parameters and applications to PDEs. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2010, 27: 377- 399 doi: 10.1016/j.anihpc.2009.11.010
6	Berti M , Bolle P . Quasi-periodic solutions with Sobolev regularity of NLS on $T^d$ and a multiplicative potential. J European Math Society, 2013, 15: 229- 286 doi: 10.4171/JEMS
7	Bourgain J . Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom Funct Anal, 1995, 5: 629- 639 doi: 10.1007/BF01902055
8	Bourgain J . Quasi-periodic solutions of Hamiltonian perturbations of 2-D linear Schrödinger equations. Ann Math, 1998, 148: 363- 439 doi: 10.2307/121001
9	Bourgain J . Green's Function Estimates for Lattice Schrödinger Operators and Applications. Princeton: Princeton University Press, 2005
10	Brézis H , Coron J-M , Rabinowitz P . Free vibrations for a semi-linear wave equation and a theorem of P. Rabinowitz. Comm Pure Appl Math, 1980, 33: 667- 684 doi: 10.1002/(ISSN)1097-0312
11	Brézis H . Periodic solutions of nonlinear vibrating strings and duality principles. Bull Amer Math Soc, 1983, 8: 409- 426 doi: 10.1090/S0273-0979-1983-15105-4
12	Craig W , Wayne C . Newton's method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math, 1993, 46: 1409- 1498 doi: 10.1002/(ISSN)1097-0312
13	Delort J M . Periodic solutions of nonlinear Schrödinger equations: A para-differential approach. Analysis and PDE, 2011, 4: 639- 676 doi: 10.2140/apde
14	Fokam J-M . Forced vibrations via Nash-Moser iteration. Comm Math Phys, 2008, 283: 285- 304 doi: 10.1007/s00220-008-0509-2
15	Kuksin S . Hamiltonian perturbabations of infinite-dimensional linear systems with imaginary spectrum. Funktsional Anal i Prilozhen, 1987, 21: 22- 37
16	Kuksin S . Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Berlin: Springer-Verlag, 1993
17	Rabinowitz P . Periodic solutions of nonlinear hyperbolic partial differential equations. Comm Pure Appl Math, 1967, 20: 145- 205 doi: 10.1002/(ISSN)1097-0312
18	Rabinowitz P . Free vibrations for a semi-linear wave equation. Comm Pure Appl Math, 1978, 31: 31- 68
19	Wayne E . Periodic and quasi-periodic solutions of nonlinear wave equations. Comm Math Phys, 1990, 127: 479- 528 doi: 10.1007/BF02104499

[1]	程志波,毕中华,姚绍文. 一类具有偏差变元的p-Laplacian Liénard型方程在吸引奇性条件下周期解的存在性[J]. 数学物理学报, 2019, 39(2): 277-285.
[2]	蓝桂杰,付盈洁,魏春金,张树文. 具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解[J]. 数学物理学报, 2018, 38(5): 984-1000.
[3]	姚绍文, 程志波. 一类带阻尼项的奇性Duffing方程周期解的存在性[J]. 数学物理学报, 2018, 38(3): 543-548.
[4]	梁志清, 曾夏萍, 周泽文, 庞国萍, 黄军华. 状态反馈脉冲控制Holling-Tanner系统周期解的存在性^*[J]. 数学物理学报, 2018, 38(1): 174-189.
[5]	李周红, 张丰硕, 曹进德, Alsaedi Ahmed, Alsaadi Fuad E. 时标上带有反馈控制的非自治两种群竞争系统的概周期解[J]. 数学物理学报, 2017, 37(4): 730-750.
[6]	巴娜, 田范基, 郑列. 关于k-Hessian方程C^2+α局部解的存在性[J]. 数学物理学报, 2017, 37(3): 499-509.
[7]	傅金波, 陈兰荪. 基于生态环境和反馈控制的多种群竞争系统的正周期解[J]. 数学物理学报, 2017, 37(3): 553-561.
[8]	段双双, 黄守军. Keller-Segel生物学方程组周期解的爆破[J]. 数学物理学报, 2017, 37(2): 326-341.
[9]	魏含玉, 夏铁成. 广义Broer-Kaup-Kupershmidt孤子方程的拟周期解[J]. 数学物理学报, 2016, 36(2): 317-327.
[10]	姜阿尼. 一类具振动循环损失率的造血模型的伪概周期解[J]. 数学物理学报, 2016, 36(1): 80-89.
[11]	王雪臣, 魏俊杰. 时滞在具有Ivlev型功能反应函数的捕食者-食饵扩散系统中的作用[J]. 数学物理学报, 2014, 34(2): 234-250.
[12]	李晓培. 驻波广义Fisher-Kolmogorov方程的广义同宿轨[J]. 数学物理学报, 2014, 34(1): 126-138.
[13]	叶辉, 蔡东汉. 周期性果蝇模型解的整体稳定性[J]. 数学物理学报, 2013, 33(6): 1013-1021.
[14]	张兴永. 一阶带线性部分Hamilton系统的周期解[J]. 数学物理学报, 2013, 33(5): 894-905.
[15]	孙盛, 王方磊, 安玉坤. 非线性电报方程组的双周期正解[J]. 数学物理学报, 2012, 32(6): 1050-1055.