Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation

Guoguang Lin; Ying Jin

doi:doi:10.11648/j.ajam.20190701.14

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Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation

Guoguang Lin, Ying Jin

Published in American Journal of Applied Mathematics (Volume 7, Issue 1)

Received: 1 April 2019 Accepted: 10 May 2019 Published: 3 June 2019

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Abstract

In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.

Published in	American Journal of Applied Mathematics (Volume 7, Issue 1)
DOI	10.11648/j.ajam.20190701.14
Page(s)	21-29
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), 2024. Published by Science Publishing Group

Keywords

Kirchhoff Equation, The Existence and Uniqueness of Solutions, Global Attractor Family, Hausdorff Dimension, Fractal Dimension

References

[1]	Kirchhoff G. Vorlesungen uber Mechanik [M]. Teubner: Sluttgart, 1883.
[2]	Kosuke Ono. On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with strong dissipation [J]. Mathematical Methods in the Applied Sciences, 1997, 20(2):151-177.
[3]	Xiaoming Fan, Yaguang Wang. Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise [J]. Stochastic Analysis and Applications, 2007, 25(2):381-396.
[4]	Zhijian Yang, Yunqing Wang. Global attractor for the Kirchhoff type equation with a strong dissipation [J]. J. Differential Equations. 2010, 249: 3258-3278.
[5]	Yang Zhijian, Jianling Cheng. Asymptotic Behavior of Solutions of Kirchhoff Equations [J]. Journal of Mathematical Physics. 2011, 31A (4): 1008-1021.
[6]	M. M. Cavalcanti. Existence and exponential decay for a Kirchhoff-Carrier model with viscosity [J]. Journal of Mathematical Analysis and Applications, 1998, 226:40-60.
[7]	Mitsuhiro Nakao. Global attractors for nonlinear wave equations with nonlinear dissipative terms [J]. Journal of Differential Equations, 2006, 227(1):204-229.
[8]	Varga Kalantarov, Sergey Zelik. Finite-dimensional attractors for the quasi-linear strongly-damped wave equation [J]. Journal of Differential Equations, 2009, 247:1120-1155.
[9]	Guoguang Lin. Nonlinear Evolution Equation [M]. Kunming: Yunnan University Press, 2011:16-123.
[10]	Zhengde Dai. Dai Zhengde Papers Collection [C]. Kunming: Yunnan University Press, 2016.
[11]	Penghui Lv, Jingxin Lu and Guoguang Lin. Global attractor for a class of nonlinear generalized Kirchhoff models [J]. Journal of Advances in Mathematics, 2016, 12(08): 6452-6462.
[12]	Zhijian Yang, Xiao Li. Finite dimensional attractors for the Kirchhoff equation with a strong dissipation [J]. Journal of Mathematical Analysis and Applications, 2011, 375(2): 579-593.
[13]	Igor Chueshov. Longtime dynamics of Kirchhoff wave models with strong nonlinear damping [J]. Journal of Differential Equations, 2011, 252(2): 1229-1262.
[14]	Zhijian Yang, Pengyan Ding. Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN [J]. Journal of Mathematical Analysis and Applications, 2016, 434(2): 1826-1851.
[15]	Zhijian Yang, Pengyan Ding, Zhiming Liu. Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity [J]. Applied Mathematics letters. 2014, 33(1): 12-17.
[16]	Yuting Sun, Yunlong Gao, Guoguang Lin. The global attractor for the higher-order Kirchhoff-type equation with nonlinear strongly damped term [J]. International Journal of Modern Nonlinear Theory and Application. 2016, 5: 203-217.
[17]	Ling Chen, Wei Wang, Guoguang Lin. The global attractor and their Hausdorff and Fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation [J]. International Journal of Advances in Mathematics. 2016, 12(09): 6608-6621.
[18]	Penghui Lv, Jinxin Lu, Guoguang Lin. Global attractor a class of nonlinear generalized Kirchhoff models [J]. Journal of Advances in Mathematics. 2016, 12(08): 6452-6462.
[19]	Guoguang Lin, Sanmei Yang. Hausdorff dimension and Fractal dimension of the global attractor for the higher-order coupled Kirchhoff-type equations [J]. Journal of Applied Mathematics and Physics. 2017, 5: 2411-2424.
[20]	Penghui Lv. Long-time behavior of a class of generalized nonlinear Kirchhoff-Boussinesq equations [D]. Kunming: Yunnan University, 2016.

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Guoguang Lin, Ying Jin. (2019). Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation. American Journal of Applied Mathematics, 7(1), 21-29. https://doi.org/10.11648/j.ajam.20190701.14

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ACS Style

Guoguang Lin; Ying Jin. Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation. Am. J. Appl. Math. 2019, 7(1), 21-29. doi: 10.11648/j.ajam.20190701.14

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AMA Style

Guoguang Lin, Ying Jin. Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation. Am J Appl Math. 2019;7(1):21-29. doi: 10.11648/j.ajam.20190701.14

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@article{10.11648/j.ajam.20190701.14,
  author = {Guoguang Lin and Ying Jin},
  title = {Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {1},
  pages = {21-29},
  doi = {10.11648/j.ajam.20190701.14},
  url = {https://doi.org/10.11648/j.ajam.20190701.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190701.14},
  abstract = {In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.},
 year = {2019}
}

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TY  - JOUR
T1  - Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation
AU  - Guoguang Lin
AU  - Ying Jin
Y1  - 2019/06/03
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20190701.14
DO  - 10.11648/j.ajam.20190701.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 21
EP  - 29
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20190701.14
AB  - In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.
VL  - 7
IS  - 1
ER  -

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Author Information

Guoguang Lin

School of Mathematics and Statistics, Yunnan University, Kunming, China
Ying Jin

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

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APA Style

Guoguang Lin, Ying Jin. (2019). Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation. American Journal of Applied Mathematics, 7(1), 21-29. https://doi.org/10.11648/j.ajam.20190701.14

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ACS Style

Guoguang Lin; Ying Jin. Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation. Am. J. Appl. Math. 2019, 7(1), 21-29. doi: 10.11648/j.ajam.20190701.14

Copy | Download

AMA Style

Guoguang Lin, Ying Jin. Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation. Am J Appl Math. 2019;7(1):21-29. doi: 10.11648/j.ajam.20190701.14

Copy | Download

@article{10.11648/j.ajam.20190701.14,
  author = {Guoguang Lin and Ying Jin},
  title = {Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation},
  journal = {American Journal of Applied Mathematics},
  volume = {7},
  number = {1},
  pages = {21-29},
  doi = {10.11648/j.ajam.20190701.14},
  url = {https://doi.org/10.11648/j.ajam.20190701.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190701.14},
  abstract = {In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation
AU  - Guoguang Lin
AU  - Ying Jin
Y1  - 2019/06/03
PY  - 2019
N1  - https://doi.org/10.11648/j.ajam.20190701.14
DO  - 10.11648/j.ajam.20190701.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 21
EP  - 29
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20190701.14
AB  - In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.
VL  - 7
IS  - 1
ER  -

Copy | Download