The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation*
DOI:
https://doi.org/10.24297/jam.v12i9.133Keywords:
Higher order, Attractor, Kirchhoff, Hausdorff dimension, Fractal dimensionAbstract
We investigate the global well-posedness and the longtime dynamics of solutions for the higher-order Kirchhoff-typeequation with nonlinear strongly dissipation:
2
( ) ( )
m m
t t t
u  ï€ ï„ u  ï¦ ï D u ï ( ) ( ) ( )
m
ï€ ï„ u  g u  f x . Under of the proper
assume, the main results are that existence and uniqueness of the solution is proved by using priori estimate and Galerkin
method, the existence of the global attractor with finite-dimension, and estimation Hausdorff and fractal dimensions of the
global attractor.
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References
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Kirchhoff-carrier model with viscosity, J. Math. Anal. Appl. 226(1998) 20-40.
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Differential Equations 137(1997) 273-301.
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[4] Z. -J. Yang, Long-time behavior of the Kirchhoff type equation with strong damping in RN, J. Differential Equations
242(2007) 269-286.
[5] Z. -J. Yang, Y.-Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential
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[6] Z. Yang, X. Li, Finite dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl.
375(2011) 579-593.
[7] Z. -J. Yang, P. -Y. Ding and Z. –M, Liu. Global attractor for the Kirchhoff type equations with strong nonlinear damping
and supercritical nonlinearity. Applied Mathematics letters. 33(2014) . pp .12-17.
[8] Zhijian Yang, Pengyan Ding. Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity
on RN. J. Math. Anal. Appl 434(2016) 1826-1851.
[9] Zhijian Yang, Pengyan Ding, Lei Li. Longtime dynamics of the Kirchhoff equations with fractional damping and
supercritical nonlinearity. J. Math. Aual. Appl 442(2016) 485-510.
[10] Igor Chueshov. Longtime dynamics of Kirchhoff wave models with strong nonlinear damping. J. Differential Equations
252(2012) 1229-1262.
[11] Guigui Xu, Guoguang Lin. Global attractors and their dimensions estimation for the generalized Boussinesq equation.
China science and technology information May. 2011.
[12] Lin Guoguang, Nonlinear evolution equation, Yunnan University Press, 2011.
Kirchhoff-carrier model with viscosity, J. Math. Anal. Appl. 226(1998) 20-40.
[2] K. Ono, Global existence, decay, and blow up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J.
Differential Equations 137(1997) 273-301.
[3] K. Ono, 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. Methods. Appl. Sci. 20(1997) 151-177.
[4] Z. -J. Yang, Long-time behavior of the Kirchhoff type equation with strong damping in RN, J. Differential Equations
242(2007) 269-286.
[5] Z. -J. Yang, Y.-Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential
Equations 249(2010) 3258-3278.
[6] Z. Yang, X. Li, Finite dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl.
375(2011) 579-593.
[7] Z. -J. Yang, P. -Y. Ding and Z. –M, Liu. Global attractor for the Kirchhoff type equations with strong nonlinear damping
and supercritical nonlinearity. Applied Mathematics letters. 33(2014) . pp .12-17.
[8] Zhijian Yang, Pengyan Ding. Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity
on RN. J. Math. Anal. Appl 434(2016) 1826-1851.
[9] Zhijian Yang, Pengyan Ding, Lei Li. Longtime dynamics of the Kirchhoff equations with fractional damping and
supercritical nonlinearity. J. Math. Aual. Appl 442(2016) 485-510.
[10] Igor Chueshov. Longtime dynamics of Kirchhoff wave models with strong nonlinear damping. J. Differential Equations
252(2012) 1229-1262.
[11] Guigui Xu, Guoguang Lin. Global attractors and their dimensions estimation for the generalized Boussinesq equation.
China science and technology information May. 2011.
[12] Lin Guoguang, Nonlinear evolution equation, Yunnan University Press, 2011.
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Published
2016-10-30
How to Cite
Chen, L., Wang, W., & Lin, G. (2016). The global attractors and their Hausdorff and fractal dimensions estimation for the higher-order nonlinear Kirchhoff-type equation*. JOURNAL OF ADVANCES IN MATHEMATICS, 12(9), 6608–6621. https://doi.org/10.24297/jam.v12i9.133
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