具阻尼项的Boussinesq型方程的长时间行为

数学物理学报 ›› 2016, Vol. 36 ›› Issue (6): 1196-1210.

具阻尼项的Boussinesq型方程的长时间行为

耿范¹, 李瑞斋¹, 葛翔宇²

1. 郑州大学西亚斯国际学院文理学院河南新郑 451150;
2. 中南财经政法大学统计与数学学院武汉 430073

收稿日期:2016-03-15 修回日期:2016-08-07 出版日期:2016-12-26 发布日期:2016-12-26
作者简介:耿范,gengfan_01@163.com;葛翔宇,xiangyu_ge@163.com
基金资助:
国家社会科学基金（10BJY104）资助

Long Time Behavior of Boussinesq Type Equation with Damping Term

Geng Fan¹, Li Ruizhai¹, Ge Xiangyu²

1. College of Arts and Sciences, Sias International College of Zhengzhou University, Henan Xinzheng 451150;
2. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073

Received:2016-03-15 Revised:2016-08-07 Online:2016-12-26 Published:2016-12-26
Supported by:
Supported by the Chinese National Social Science Foundation (10BJY104)

摘要/Abstract

摘要：

该文研究具阻尼项的Boussinesq型方程u_tt-Δu+Δ²u-Δu_t-Δg（u）=f（x）初边值问题的解的长时间行为.利用半群分解的方法证明了上述问题对应的无穷维动力系统在能量相空间E=V₂×H中整体吸引子的存在性和吸引子Hausdorff维数的有限性，其中对非线性项g（u）的抽象条件加以验证并给出具体实例.

关键词: Boussinesq型方程, 初边值问题, 无穷维动力系统, 整体吸引子, Hausdorff维数

Abstract:

In this paper, we study the long time behavior of the solution of the initial boundary value problem of Boussinesq type equation with damping term:u_tt-Δu+Δ²u-Δu_t-Δg(u)=f(x). The main result is that the existence of global attractor of the infinite dimensional dynamical system and the existence of the global attractor and the Hausdorff dimension of the attractor in the phase space E=V₂×H are proved by the method of semi group decomposition, The abstract ondition of nonlinear term g(u) is verified and given a concrete example.

Key words: Boussinesq type equation, Initial boundary value problem, Infinite dimensional dynamical system, Global attractor, Hausdorff dimension

中图分类号:

O175.29

耿范, 李瑞斋, 葛翔宇. 具阻尼项的Boussinesq型方程的长时间行为[J]. 数学物理学报, 2016, 36(6): 1196-1210.

Geng Fan, Li Ruizhai, Ge Xiangyu. Long Time Behavior of Boussinesq Type Equation with Damping Term[J]. Acta mathematica scientia,Series A, 2016, 36(6): 1196-1210.

参考文献

[1] Boussinesq J. Theorie des ondes et des temous qui se prepatent le long d'un canal rectangularly horizontal,en communicant au liquids contend dans ce canal des vitesses sunsiblemend parielles de la surface an fond. J Math Pures Appl, 1872, 17:55-108
[2] Bona J L, Sachs R L. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equations. Comm Math Phys, 1998, 118:15-29
[3] Sachs R L. On the blowup of certain solutions of the "good" Boussinesq equation. Appl Anal, 1990, 36:145-152
[4] Deift P, Tomei C, Trubowitz E. Inverse scattering and Boussinesq equation. Comm Pure Appl Math, 1982, 35:567-628
[5] Levine H A, Sleeman B D. A note on the non-existence of global solutions of initial bonudary value problems for the Boussinesq equation u_tt-u_xx-3u_xxxx=-12(u²)_xx. J Math Anal Appl, 1985, 107:206-210
[6] Yang Z J, Chen G W. Blowup of solutions for a class of generalized Boussinesq equations. Acta Math Sci, 1996, 16:31-40
[7] Yang Z J, Guo B L. Cauchy problem for the multi-dimensional Boussinesq-type equation. J Math Anal Appl, 2008, 340:64-80
[8] Varlamov V. On the cauchy problem for the damped Boussinesq-type equation. Differential Integral Equations, 1996, 9(3):619-634
[9] Yang X B, Zhao C D, Cao J. Dynamics of the discrete coupled nonlinear Schrodinger-Boussinesq equation. Appl Math Comp, 2013, 219:8508-8524
[10] Lai S Y, Wu Y H. The asymptotic solution of the cauchy problem for a generalized Boussinesq equation. Discrete Continuous Dynamical System, 2003, 3(3):401-408
[11] Hale J. Asympotic Behavior of Dissipative Sysrems. Providence RI:American Mathematical Society, 1988
[12] Teman R. Infinite Dimensional Dynamical System in Mathematics and Physics. New York:Springer, 1997
[13] Chen F X, Guo B L, Wang P. Long time behavior of strongly damped nonlinear wave equations. J Differential Equations, 1998, 147:231-241
[14] Ghidaglia J M, Marzocchi A. Longtime behavior of strongly damped wave equation global attractors and their dimension. SIAM J Math Anal, 1991, 22:879-895
[15] Nakao M,Yang Z J. Global attractors for some quasi-linear wave equations with a strong dissipation. Adv Math Sci Appl, 2007, 17:89-105
[16] Yang Z J. Global attractor for a nonlinear wave equation arising in elastic waveguide model. Nonlinear Analysis, 2009, 70:2132-2142
[17] Yang Z J. Global attractor and their Housdorff dimensions for a class of Kirchhoff models. J Math Physics, 2010, 51:032701
[18] 李清善,李红,宋长明,张熠然.具阻尼项的"坏"的Boussinesq型方程的Cauchy问题. 北京大学学报(自然科学版), 2008, 44(5):676-682 Li Q S, Li H, Song C M, Zhang Y R. Cauchy problem for the "bad" Boussinesq-type equation with damping term. Acta Scientiarum Naturalium Universitatis Pekinensis, 2008, 44(5):676-682
[19] Ghidaglia J M, Temam R. Attractors for damped nonlinear hyperbolic eqations. J Math Pure Appl, 1987, 66:273-319
[20] Yang Z J. Longtime dynamics of the damped Boussinesq equation. J Math Anal Appl, 2013, 399:180-190
[21] Varlamov V. Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation. Math Methods Appl Sci, 1996, 19:639-649
[22] Varlamov V. Long-time asymptotic of solutions of the second initial-boundary value problem for the damped Boussinesq equation. Abstr Appl Anal, 1998, 2:281-289
[23] 李春秋,赵才地,王玮明.非自治耗散Schroinger-Boussinesq方程组紧致核截面的存在性. 数学年刊(A辑), 2014, 35:307-322 Li C Q, Zhao C D, Wang W M. The existence of compact nuclear cross sections of the non autonomous dissipative schroinger-boussinesq equations. Annals of Mathematics (Ser A), 2014, 35:307-322
[24] Chen G W, Wang Y P, Wang S. Initial boundary value problem of the generalized cubic double dispersion equation. J Math Anal Appl, 2004, 299:563-577
[25] Wang S, Chen G W. Cauchy problem of the generalized double dispersion equation. Nonlinear Anal, 2006, 64:159-173