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《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (12): 1-11.doi: 10.6040/j.issn.1671-9352.0.2018.709

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一类高阶非线性Kirchhoff方程吸引子族及其维数

林国广,李卓茜   

  1. 云南大学数学与统计学院, 云南 昆明 650500
  • 发布日期:2019-12-11
  • 作者简介:林国广(1964— ),男,博士,教授,研究方向为非线性偏微分方程. E-mail:gglin@ynu.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(11561076)

Attractor family and dimension for a class of high-order nonlinear Kirchhoff equations

LIN Guo-guang, LI Zhuo-xi   

  1. School of Mathematics and Statistics, Yunnan University, Kunming 650500, Yunnan, China
  • Published:2019-12-11

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摘要： 研究一类带有非线性非局部源项和强阻尼项的高阶Kirchhoff方程的初边值问题。对非线性非局部源项、Kirchhoff应力项进行适当地假设。首先利用Galerkin有限元方法和先验估计证明方程整体解的存在性和唯一性;再由先验估计得到有界吸收集,从而获得高阶非线性Kirchhoff方程的整体吸引子族;将方程线性化并证明解半群的Frechet可微性,进一步证明线性化问题体积元的衰减性,最后证明整体吸引子族的Hausdorff维数及Fractal维数是有限的。

关键词: 高阶Kirchhoff方程, Galerkin有限元方法, 整体吸引子族, Hausdorff维数, Fractal维数

Abstract: The initial boundary value problem for a class of high-order Kirchhoff equations with nonlinear nonlocal source terms and strong damping terms is studied. For the nonlinear nonlocal source term and the Kirchhoff stress term, the existence and uniqueness of the global solution of the equation are firstly proved by Galerkin finite element method and a prior estimate. Then the bounded absorption set is obtained by a prior estimate, so the global attractor family of high-order nonlinear Kirchhoff equation is obtained. By linearizing the equation and proving the Frechet differentiable of the solution semigroup, it further proves the decay of the volume element of the linearization problem. Finally, the Hausdorff dimension and Fractal dimension of the global attractor family are proved to be finite.

Key words: high-order Kirchhoff equation, Galerkin finite element method, global attractor family, Hausdorff dimension, Fractal dimension

中图分类号: 

  • O175.29
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