非线性 KGS 系统强解的范数增长

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 1142-1159.

非线性 KGS 系统强解的范数增长

崔爽(), 石启宏^*()

兰州理工大学数学系兰州 730050

收稿日期:2024-10-23 修回日期:2025-09-30 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 石启宏 E-mail:3063756368@qq.com;shiqh03@163.com
作者简介:崔爽, E-mail:3063756368@qq.com
基金资助:
国家自然科学基金(12561040);国家自然科学基金(12061040)

On the Norm Growth of Strong Solutions for Nonlinear KGS System

Shuang Cui(), Qihong Shi^*()

Department of Mathematics, Lanzhou University of Technology, Lanzhou 730050

Received:2024-10-23 Revised:2025-09-30 Online:2026-06-26 Published:2026-06-16
Contact: Qihong Shi E-mail:3063756368@qq.com;shiqh03@163.com
Supported by:
NSFC(12561040);NSFC(12061040)

摘要/Abstract

摘要：

该文研究低维欧氏空间 $\mathbb{R}^N(N\leq3)$ 中非线性 Klein-Gordon-Schrödinger (KGS) 系统的初边值问题. 通过引入正则化系统, 利用解序列的有界性和收敛性, 证明了非线性 KGS 系统在 $H^2\times H^2\times H^1$ 空间上全局强解的存在唯一性. 进一步, 通过构建修正能量, 作者得到了解在该空间中的范数估计. 需要强调的是证明方法不依赖于 Brezis-Gallouet 技巧以及紧性讨论.

关键词: Klein-Gordon-Schr?dinger 系统, 全局强解, 存在唯一性, Sobolev 范数增长.

Abstract:

This paper is concerned with the initial-boundary value problem for the nonlinear Klein-Gordon-Schrödinger (KGS) system in $\mathbb{R}^N(N\leq3)$. By introducing a regularized system and utilizing the boundedness and convergence of the solutions sequence, we prove the existence and uniqueness of global strong solutions to the nonlinear KGS system in the space $H^2 \times H^2 \times H^1$, and obtain the norm estimates for the solutions in the space $H^2 \times H^2 \times H^1$. The proof is independent of the Brezis-Gallouet technique and the compactness argument.

Key words: Klein-Gordon-Schr?dinger system, giobal strong solutions, existence and uniqueness, Sobolev norm growth

中图分类号:

O175.29

崔爽, 石启宏. 非线性 KGS 系统强解的范数增长[J]. 数学物理学报, 2026, 46(3): 1142-1159.

Shuang Cui, Qihong Shi. On the Norm Growth of Strong Solutions for Nonlinear KGS System[J]. Acta mathematica scientia,Series A, 2026, 46(3): 1142-1159.

参考文献 22

[1]	Yukawa H. On the interaction of elementary particles. I. Proc Physico-Math Soc Japan, 1935, 17 : 48-57
[2]	Baillon J B, Chadam J M. The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations. North-Holland Mathematics Studies, 1978, 30 : 37-44
[3]	Fukuda I, Tsutsumt M. On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions. Proc Japan Acad, 1975, 51 (6): 402-405
[4]	Fukuda I. Coupled Klein-Gordon-Schrödinger equations, II. J Math Anal Appl, 1978, 66 (2): 358-378 doi: 10.1016/0022-247X(78)90239-1
[5]	Li Y, Guo B. Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations. J Math Anal Appl, 2003, 282 (1): 256-265 doi: 10.1016/S0022-247X(03)00152-5
[6]	Biler P. Attractors for the system of Schrödinger and Klein-CGordon equations with Yukawa coupling. SIAM J Math Anal, 1990, 21 (5): 1190-1212 doi: 10.1137/0521065
[7]	Hayashi N. On the global strong solutions of coupled Klein-Gordon-Schrödinger equations. J Math Soc Japan, 1987, 39 (3): 489-497
[8]	Miao C, Xu G. Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbb{R}^{2+ 1}$. J Diff Equ, 2006, 227 (2): 365-405 doi: 10.1016/j.jde.2005.10.012
[9]	Cavalcanti M M, Cavalcanti V N D. Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations. Nonlinear Differ Equ Appl, 2000, 7 (3): 285-307 doi: 10.1007/PL00001426
[10]	Shi Q H, Li W T, Wang S. Kato-type estimates for NLS equation in a scalar field and unique solvability of NKGS system in energy space. J Diff Equ, 2014, 256 (10): 3440-3462 doi: 10.1016/j.jde.2014.01.035
[11]	Pecher H. Well-posedness results for a generalized Klein-Gordon-Schrödinger system. J Math Phys, 2019, 60 (10): Art 101510
[12]	Wang B. Classical Global solutions for Non-linear Klein-Gordon-Schrödinger equations. Math Methods Appl Sci, 1997, 20 (7): 599-616 doi: 10.1002/(ISSN)1099-1476
[13]	Thirouin J. On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle. Ann Inst H Poincare Anal Non Lineaire, 2017, 34 (2): 509-531 doi: 10.4171/aihpc
[14]	Zhong S. The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds. J Diff Equ, 2008, 245 (2): 359-376 doi: 10.1016/j.jde.2008.03.008
[15]	Guardia M, Kaloshin V. Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. J Eur Math Soc, 2015, 17 (1): 71-149 doi: 10.4171/jems
[16]	Shi Q H. Global boundedness for the nonlinear Klein-Gordon-Schrödinger system with power nonlinearity. Differ Integral Equ, 2023, 36 (9/10): 837-858
[17]	Hayashi N, Ozawa T. Smoothing effect for some Schrödinger equations. J Funct Anal, 1989, 85 (2): 307-348 doi: 10.1016/0022-1236(89)90039-6
[18]	Ozawa T, Tomioka K. Zakharov system in two space dimensions. Nonlinear Analysis, 2022, 214 : Art 112532
[19]	Ozawa T, Tomioka K. Schrödinger-improved Boussinesq system in two space dimensions. Journal of Evolution Equations, 2022, 22 (2): Art 35
[20]	Cazenave T, Haraux A. An Introduction to Semilinear Evolution Equations. New York: Oxford University Press, 1998
[21]	Ogawa T, Ozawa T. Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem. J Math Anal Appl, 1991, 155 (2): 531-540 doi: 10.1016/0022-247X(91)90017-T
[22]	Cazenave T. Semilinear Schrödinger Equations. New York: Courant Institute of Mathematical Sciences, 2003