Kirchhoff型方程解的渐近行为

数学物理学报 ›› 2011, Vol. 31 ›› Issue (4): 1008-1021.

Kirchhoff型方程解的渐近行为

杨志坚¹|程建玲²

1.郑州大学数学系郑州 450001|2.郑州华信学院基础部河南新郑 451100

收稿日期:2009-07-06 修回日期:2011-03-15 出版日期:2011-08-25 发布日期:2011-08-25
基金资助:
国家自然科学基金(10971199)和河南省自然科学基金(092300410067)资助

Asymptotic Behavior of Solutions to the Kirchhoff Type Equation

YANG Zhi-Jian¹, CHENG Jian-Ling²

1.Department of Mathematics, Zhengzhou University, Zhengzhou |450001|2.Department of Basis, Zhengzhou Huaxin College, Henan Xinzheng 451100

Received:2009-07-06 Revised:2011-03-15 Online:2011-08-25 Published:2011-08-25
Supported by:
国家自然科学基金(10971199)和河南省自然科学基金(092300410067)资助

摘要/Abstract

摘要：

该文研究具强阻尼项的Kirchhoff型方程u_tt-M(||\nabla u||²)Δu-Δu_t+g(x, u)+h(u_t)=f(x)的初边值问题的解的长时间行为，其中M(s)=1+s^m/2, m≥1. 该文用两种方法证明上述问题对应的算子半群S(t)在相空间X=(H²(Ω)∩H₀¹(Ω)×H₀¹(Ω)中整体吸引子的存在性, 最后对抽象条件加以验证并给出具体实例.

关键词: 初边值问题, 无穷维动力系统, 整体解, 解的长时间行为, 整体吸引子

Abstract:

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) of the Kirchhoff type equation with strong damping u_tt-M(\||\nabla u||{²})Δu-Δu_t+g(x, u)+h(u_t)=f(x), with M(s)=1+s^m^/2, m≥1. With two different methords, it proves that the related continuous semigroup S(t) posseses in phase space X=(H²(Ω)∩H₀¹(Ω))×H₀¹(Ω) a global attractor. At the end of the paper, an example is shown, which indicates the existence of nonlinear functions g(x, u) and h(u_t).

Key words: Kirchhoff type equation, Initial boundary value problem, Infinite-dimensional dynamical system, Global solution, Longtime behavior of solutions, Global attractor

中图分类号:

35B40

杨志坚, 程建玲. Kirchhoff型方程解的渐近行为[J]. 数学物理学报, 2011, 31(4): 1008-1021.

YANG Zhi-Jian, CHENG Jian-Ling. Asymptotic Behavior of Solutions to the Kirchhoff Type Equation[J]. Acta mathematica scientia,Series A, 2011, 31(4): 1008-1021.

参考文献

[1] Nakao M, Zhijian Y. Global attractors for some quasi-linear wave equations with a strong dissipation. Advan Math Sci Appl, 2007, 17: 89--105

[2] Zhijian Y. Long time behavior of the Kirchhoff type equation with strong damping on R^N. J Differential Equations, 2007, 242: 269--286

[3] Xiaoming F, Shengfan Z. Kernel sections for non-autonomous strongly damped wave equations of non-degnerate Kirchhoff-type. Applied Mathematics and Computation, 2004, 158: 253--266

[4] Ono K. On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation. Math Meth Appl Sci, 1997, 20: 151--177

[5] Ghidaglia J M, Marzochi A. Longtime behaviour of strongly damped wave equations, global attractors and their dimension. SIAM J Math Anal, 1991, 22: 879--895

[6] Hosoya M, Yamada Y. On some nonlinear wave equations: global existence and energy decay of solutions. J Fac Sci Univ Tokyo Sect IA Math, 1991, 38: 239--250

[7] Ghisi M. Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term. J Differential Equations, 2006, 230: 128--139

[8] Limaco J, Clark H R, Medeiros L A. On damped Kirchhoff equation with variable coefficients. J Math Anal Appl, 2005, 307: 641--655

[9] Babin A V, Vishik M I. Attractors of Evolution Equations. Amsterdam: North-Holland, 1992

[10] Hale J K. Asymptotic Behavior of Dissipative Systems. Providence, RI: AMS, 1988

[11] Temam R. Infinite Dimensional Dynamical System in Mathematics and Physics. New York: Springer, 1997

[12] Chen F X, Guo B L, Wang P. Long time behavior of strongly damped nonlinear wave equations. J Differental Equatons 1998, 147: 231--241

[13] Karachalois N I, Stavrakakis N M. Existence of a global attractor for semilinear dissipative wave equation on R^N. J Differential Equations, 1999, 157: 183--205

[14] Shengfan Z. Attractors for strongly damped wave equations with critical exponents. Appl Math Lett, 2003, 16: 1307--1314

[15] Matsuyama T, Ikehata R. On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term. J Math Anal Appl, 1996, 204: 729--753

[16] Ono K. Global existence, decay and blow up of solutions for some mildly degenerate nonlinear Kirchhoff strings. J Differential Equations, 1997, 137: 273--301

[17] Ono K. Global existence, asymptotic behavior and global non-existence of solutons for damped non-linear wave equations of Kirchhoff type in the whole space. Math Meth Appl Sci, 2000, 23: 535--560

[18] Zhijian Y, Baoxia J. Global attractor for a class of Kirchhoff models. J Mathematical Physics 2009, 50: 1--29