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A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation

Received: 4 November 2020     Accepted: 16 November 2020     Published: 24 November 2020
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Abstract

In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.

Published in American Journal of Applied Mathematics (Volume 8, Issue 6)
DOI 10.11648/j.ajam.20200806.12
Page(s) 300-310
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Kirchhoff-Type Equation, Prior Estimation, Galerkin Method, A family of Global Attractors, Hausdorff Dimension, Fractal Dimension

References
[1] Kirchhoff G. Vorlesungen uber mathematiche Physik [M]. stuttgart: Leipzig, B. G. Teubner, 1883.
[2] Guoguang Lin. Nonlinear evolution equation [M]. Kunming: Yunnan University Press, 2011.
[3] Xilai Dai, Guoguang Lin. Existence and uniqueness of solutions to strongly damped wave equations [J]. Journal of Yunnan University: Natural Science Edition, 2011, 33 (S2): 327-332.
[4] Narasimha K. Nonlinear vibration of an elastic string [J]. Sound Vibration, 1968, 8: 134-146.
[5] Boling Guo. Nonlinear evolution equation [M]. Shanghai: Shanghai Science and Technology Press, 1995.
[6] Zhengde Dai. Zhengde Dai Papers Collection [C]. Kunming: Yunnan University Press, 2016.
[7] Mitsuhiro N. An attractor for a nonlinear dissipative wave equation of Kirchhoff type [J]. Journal of Mathematical Analysis and Applications, 2009, 353 (2): 652-659.
[8] Zaiyun Zhang, Zhenhai Liu, Xiujin Miao. Estimate on the Dimension of Global Attractor for Nonlinear Dissipative Kirchhoff Equation [J]. Acta Applicandae Mathematicae, 2010, 110 (1): 271-282.
[9] Igor C. Longtime dynamics of Kirchhoff wave models with strong nonlinear Damping [J]. Journal of Differen- tial Equations, 2012, 252 (2): 1229-1262.
[10] Yunlong Gao, Yuting Sun, Guoguang Lin. The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higer-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping [J]. International Journal of Modern Nonlinear Theory and Application 2016, 5 (4): 185-202.
[11] Matsuyama T, Ikerata R. On the global solutions and energy decay for the wave equations of Kirchhoff-type with nonlinear damping terms [J]. Journal of Mathematical Analysis and Applications, 1996, 204 (3): 729-753.
[12] Kajitani K, Satoh A. On extension of solutions of Kirchhoff equations [J]. Journal of the Mathematical Society of Japan, 2004, 56 (2): 405-416.
[13] Lei Wan, Jinbao Dang, Guoguang Lin. Global attractor and dimension estimation of Fractional nonlinear Schrodinge equation [J]. Journal of Yunnan University: Natural Science Edition, 2010, 32 (2): 130-135.
[14] Ling Chen, Wei Wang, Guoguang Lin. The global attractors and the Hausdoff and fractal demensions estimation for the higher-order nonlinear Kirchhoff-type Equation [J]. Journal of Advances in Mathematics. 2016, 12 (09): 6608-6621.
[15] Guoguang Lin, Sanmei Yang. Hausdoff and Fractal demensions of the global attractor for the higher-order coupled Kirchhoff-type Equation [J]. Journal of Applied Mathematics and Physics. 2017, 5: 2411-2424.
Cite This Article
  • APA Style

    Guoguang Lin, Yuhang Chen. (2020). A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. American Journal of Applied Mathematics, 8(6), 300-310. https://doi.org/10.11648/j.ajam.20200806.12

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    ACS Style

    Guoguang Lin; Yuhang Chen. A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. Am. J. Appl. Math. 2020, 8(6), 300-310. doi: 10.11648/j.ajam.20200806.12

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    AMA Style

    Guoguang Lin, Yuhang Chen. A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation. Am J Appl Math. 2020;8(6):300-310. doi: 10.11648/j.ajam.20200806.12

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  • @article{10.11648/j.ajam.20200806.12,
      author = {Guoguang Lin and Yuhang Chen},
      title = {A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation},
      journal = {American Journal of Applied Mathematics},
      volume = {8},
      number = {6},
      pages = {300-310},
      doi = {10.11648/j.ajam.20200806.12},
      url = {https://doi.org/10.11648/j.ajam.20200806.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200806.12},
      abstract = {In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - A Family of Global Attractors for the Higher-order Kirchhoff-type Equations and Its Dimension Estimation
    AU  - Guoguang Lin
    AU  - Yuhang Chen
    Y1  - 2020/11/24
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    DO  - 10.11648/j.ajam.20200806.12
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    EP  - 310
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20200806.12
    AB  - In this paper, we study the long-time behavior of solutions for a class of initial boundary value problems of higher order Kirchhoff –type equations, and make appropriate assumptions about the Kirchhoff stress term. We use the uniform prior estimation and Galerkin method to prove the existence and uniqueness of the solution of the equation, when the order m and the order q meet certain conditions. Then, we use the prior estimation to get the bounded absorption set, it is further proved that using the Rellich-Kondrachov compact embedding theorem, the solution semigroup generated by the equation has a family of global attractor. Then the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Frechet differentiable. Finally, it proves that the Hausdorff dimension and Fractal dimension of a family of global attractors are finite.
    VL  - 8
    IS  - 6
    ER  - 

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Author Information
  • School of Mathematics and Statistics, Yunnan University, Kunming, China

  • School of Mathematics and Statistics, Yunnan University, Kunming, China

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