The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping
Abstract: The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.
Cite this paper: Lin, G. and Yang, Y. (2020) The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping. International Journal of Modern Nonlinear Theory and Application, 9, 63-80. doi: 10.4236/ijmnta.2020.94005.
References

[1]   Ladyzhenskaya, O.A. and Seregin, G.A. (1999) On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier—Stokes Equations. Journal of Mathematical Fluid Mechanics, 1, 356-387.
https://doi.org/10.1007/s000210050015

[2]   Ladyzhenskaya, O.A. (1988) Attractors of Nonlinear Evolution Problems with Dissipation. Journal of Soviet Mathematics, 40, 632-640.
https://doi.org/10.1007/BF01094189

[3]   Ladyzhenskaya, O.A. (1985) Finite-Dimensionality of Bounded Invariant Sets for Navier-Stokes Systems and Other Dissipative Systems. Journal of Soviet Mathematics, 28, 714-726. https://doi.org/10.1007/BF02112336

[4]   Temam, R. (1990) Inertial Manifolds. The Mathematical Intelligencer, 12, 68-74.
https://doi.org/10.1007/BF03024036

[5]   Foias, C., Sell, G.R. and Temam, R. (1988) Inertial Manifolds for Nonlinear Evolutionary Equations. Journal of Differential Equations, 73, 309-353.
https://doi.org/10.1016/0022-0396(88)90110-6

[6]   Sell, G.R. and You, Y.C. (1992) Inertial Manifolds: The Non-Self-Adjoint Case. Journal of Differential Equations, 96, 203-255.
https://doi.org/10.1016/0022-0396(92)90152-D

[7]   Guo, B.L. (2000) The Infinite Dimension System. National Defense Industry Press, Beijing.

[8]   Chueshov, I., Lasiecka, I. and Toundykov, D. (2009) Global Attractor for a Wave Equation with Nonlinear Localized Boundary Damping and a Source Term of Critical Exponent. Journal of Dynamics and Differential Equations, 21, Article No. 269.
https://doi.org/10.1007/s10884-009-9132-y

[9]   Yang, Z.J. and Wang, Y.Q. (2010) Global Attractor for the Kirchhoff Type Equation with a Strong Dissipation. Journal of Differential Equations, 249, 3258-3278.
https://doi.org/10.1016/j.jde.2010.09.024

[10]   Gao, Y.L., Sun, Y.T. and Lin, G.G. (2016) The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping. International Journal of Modern Nonlinear Theory and Application, 5, 185-202.
https://doi.org/10.4236/ijmnta.2016.54018

[11]   Lin, G.G. (2011) Nonlinear Evolution Equations. Yunnan University Press, Kunming.

[12]   Kirchhoff, G. (1883) Vorlesungen Uber Mechamic. Tauber, Leipzig.

[13]   Yang, Z.J. and Da, F. (2018) Stability of Attractors for the Kirchhoff Wave Equation with Strong Damping and Critical Nonlinearities. Journal of Mathematical Analysis and Applications, 469, 298-320.
https://doi.org/10.1016/j.jmaa.2018.09.012

[14]   De Brito, E.H. (1987) Nonlinear Initial Boundary Value Problems. Nonlinear Analysis: Theory, Methods & Applications, 11, 125-137.
https://doi.org/10.1016/0362-546X(87)90031-9

[15]   Matsuyama, T. and Ikerata, R. (1996) On Global Solutions and Energy Decay for the Wave Equations of Kirchhoff Type with Nonlinear Damping Terms. Journal of Mathematical Analysis and Applications, 204, 729-753.
https://doi.org/10.1006/jmaa.1996.0464

[16]   Ono, K. (1997) On Global Solutions and Blow-Up Solutions of Nonlinear Kirchhoff Strings with Nonlinear Dissipation. Journal of Mathematical Analysis and Applications 216, 321-342. https://doi.org/10.1006/jmaa.1997.5697

[17]   Yamada, Y. (1982) On Some Quasilinear Wave Equations with Dissipative Terms. Nagoya Mathematical Journal, 87, 17-39.
https://doi.org/10.1017/S0027763000019929

[18]   Bozhkov, Y. and Dimas, S. (2013) Group Classification and Conservation Laws for a Two-Dimensional Generalized Kuramot—Sivashinsky Equation. Journal of Nonlinear Analysis, 84, 117-135.
https://doi.org/10.1016/j.na.2013.02.010

[19]   Autuori, G., Pucci, P. and Salvatori, M.C. (2009) Asymptotic Stability for Nonlinear Kirchhoff Systems. Nonlinear Analysis: Real World Applications, 10, 889-909.
https://doi.org/10.1016/j.nonrwa.2007.11.011

[20]   Yang, Z.J. (2007) Longtime Behavior of the Kirchhoff Type Equation with Strong Damping on RN. Journal of Differential Equations, 242, 269-286.

[21]   Wu, S.T. and Tsai, L.Y. (2006) Blow-Up of Solutions for Some Nonlinear Wave Equations of Kirchhoff Type with Some Dissipation. Nonlinear Analysis: Theory, Methods & Applications, 65, 243-264.
https://doi.org/10.1016/j.na.2004.11.023

[22]   Chueshov, I. (2011) Long-Time Dynamics of Kirchhoff Wave Models with Strong Nonlinear Damping. Journal of Differential Equations, 252, 1229-1262.
https://doi.org/10.1016/j.jde.2011.08.022

[23]   Lin, G.G. and Li, Z.X. (2019) A Class of Higher Order Nonlinear Kirchhoff Equation Attractor Family and Its Dimension. Journal of Shandong University (Science Edition), 54, 1-11.

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