非线性KGS系统解的唯一性和能量衰减

数学物理学报 ›› 2013, Vol. 33 ›› Issue (5): 842-849.

非线性KGS系统解的唯一性和能量衰减

石启宏

兰州大学数学与统计学院兰州 730000|河北金融学院基础学科部河北保定 071051

收稿日期:2012-04-18 修回日期:2013-05-16 出版日期:2013-10-25 发布日期:2013-10-25
基金资助:
河北省自然科学基金(A2013410007)和河北省高校自然科学基金(Z2011111)资助.

Uniqueness of the Solution and Energy Decay for Nonlinear KGS System

SHI Qi-Hong

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000;
Department of Basic Sciences, Hebei Finance University, Hebei Baoding 071051

Received:2012-04-18 Revised:2013-05-16 Online:2013-10-25 Published:2013-10-25
Supported by:
河北省自然科学基金(A2013410007)和河北省高校自然科学基金(Z2011111)资助.

摘要/Abstract

摘要：

作为前期研究工作的延续, 该文借助于积分型函数, 利用能量方法证明了在3+1维空间中具有阻尼项的非线性Klein-Gordon-Schr\"{o}dinger系统解的唯一性. 此外, 运用摄动方法, 获得了该系统能量的一致衰减估计.

关键词: 非线性, Klein-Gordon-Schr\"{o}dinger, 唯一性, 摄动能量, 一致衰减

Abstract:

As a continuation of previous work, in this paper, using energy method, we prove the uniqueness of the solution for nonlinear Klein-Gordon-Schr\"{o}dinger system with the aid of an integral-type function. On the other hand, by utlizing the purterbation method, we get the uniform decay of the system energy.

Key words: Nonlinear, Klein-Gordon-Schr\"{o}dinger, Uniqueness, Purterbation energy, Uniform decay

中图分类号:

35B25

石启宏. 非线性KGS系统解的唯一性和能量衰减[J]. 数学物理学报, 2013, 33(5): 842-849.

SHI Qi-Hong. Uniqueness of the Solution and Energy Decay for Nonlinear KGS System[J]. Acta mathematica scientia,Series A, 2013, 33(5): 842-849.

参考文献

[1] Yukawa H. On the interaction of elementary particles I. Proc Physico-Math Soc Japan, 1935, 17: 48--57

[2] Fukuda I, Tsutsumi M. On the Yukawa-coupled Klein-Gordon-Schr\"{o}dinger equations in three space dimensions. Proc Japan Acad, 1975, 51: 402--405

[3] Fukuda I, Tsutsumi M. On coupled Klein-Gordon-Schr\"{o}dinger equations I. Bull Sci Engrg Res Lab Waseda Univ, 1975, 69: 51--62

[4] Fukuda I, Tsutsumi M. On coupled Klein-Gordon-Schr\"{o}dinger equations II. J Math Anal Appl, 1978, 66: 358--378

[5] Biler P. Attractors for the systerm of Schr\"{o}dinger and klein-gordon equations with Yukawa coupling. SIAM J Math Anal, 1990, 21: 1190--1212

[6] Hsyashi N, Von Wahl W. On the global strong solutions of coupled Klein-Gordon-Schr\"{o}dinger equations. J Math Soc Japan, 1987, 39: 489--497

[7] Guo B, Miao C X. Asymptotic behavior of coupled Klein-Gordon-Schr\"{o}dinger equations. Sci China Ser, 1995, 25A: 705--714

[8] Ohta M. Stability of stationary states for the coupled Klein-Gordon-Schr\"{o}dinger equations. Non Anal, 1996, 27(4): 455--461

[9] Guo B, Li Y. Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schr\"{o}dinger equations.
J Math Anal Appl, 2003, 282: 256--265

[10] Lu K, Wang B. Global attractors for the Klein-Gordon-Schr\"{o}dinger equation in unbounded domains. J Diff Equ, 2001, 270: 281--316

[11] Ozawa T, Tsutsumi Y. Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schr\"{o}dinger equations.
Adv Stud Pure Math, 1994, 23: 295--305

[12] Park J, Kim J. Maximal attractors for the Klein-Gordon-Schr\"{o}dinger equation in unbounded domain. Acta Appl Math, 2009, 108: 197--213

[13] Kikuchi H, Ohta M. Stability of standing waves for the Klein-Gordon-Schr\"{o}inger system. J Math Anal Appl, 2010, 365: 109--114

[14] Nakanishi K, Schlag W. Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation. J Differ Equ, 2011, 250: 2299--2333

[15] Cavalcanti M M, Cavalcanti V N D. Global existence and uniform decay for the coupled Klein-Gordon-Schr\"{o}dinger equations. NoDEA, 2000, 7: 285--307

[16] Miao C, Xu G. Global solutions of the Klein-Gordon-Schr\"{o}dinger system with rough data in R²⁺¹. J Differ Equ, 2006, 227: 365--405

[17] Shi Q, Wang S, Wang C. On the existence of weak solutions for nonlinear Schr\"{o}dinger-Klein-Gordon equations.
Acta Mathematica Scientia (Chinese), 2011, 31A: 1122--1132

[18] Bisognin V, Cavalcanti M M, Domingos Cavalcanti V N, Soriano J. Uniform decay for the coupled Klein-Gordon-Schr\"{o}dinger equations with locally distributed damping. Nonlinear Differ Equ Appl, 2008, 15: 91--113