Research ArticleOpen AccessGoogle Scholar indexed
Estimate on the Dimension of Global Attractor for Nonlinear Higher-Order Coupled Kirchhoff Type Equations
Department of Mathematics, Yunnan University, Kunming, China
Department of Mathematics, Yunnan University, Kunming, China
- 1 Department of Mathematics, Yunnan University, Kunming, China
- 2 Department of Mathematics, Yunnan University, Kunming, China
Advances in Pure Mathematics·Volume 08 (2018)·Pages 11–24·Published 9 January 2018·DOI10.4236/apm.2018.81002
Copy link · social · email
Abstract
In this paper, we investigate the finite dimensions of the global attractor for nonlinear higher-order coupled Kirchhoff type equations with strong linear damping in Hilbert spaces <i>E</i><SUB>0</SUB> and <i>E</i><SUB>1</SUB> . Under the appropriate assumptions, we acquire a precise estimate of the upper bound for its Hausdorff and Fractal dimensions.
KeywordsHigher-OrderKirchhoff-Type EquationsGlobal AttractorHausdorff DimensionFractal Dimension
- Lin, G.G. and Hu, L.J. (2017) The Attractor for a Class of Higher-Order Coupled Kirchhoff Type Equations with Strong Linear Damping. European Journal of Mathematics and Computer Science, 4, 63-77.
- Teman, R. (1988) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York. https://doi.org/10.1007/978-1-4684-0313-8
- Huang, Y. (1998) Estimates of the Hausdorff Dimension and Fractal Dimension of Global Attractors for Nonlinear Wave Equations with Damping. Acta Mathematics Applicatae Sinica, 21, 257-266.
- Zhou, S. (1999) On Dimension of Global Attractor for Damped Nonlinear Wave Equations. Proceedings of the American Mathematical Society, 40, 1432-1438.
- Zhou, S.F. (2000) Attractor and Dimension for Strongly Damped Nonlinear Wave Equation. Acta Mathematicae Applicatea Sinica, 16, 266-273. https://doi.org/10.1007/BF02679891
- Yang, Z.J. (2007) Longtime Behavior of the Kirchhoff Type Equation with Strong Damping on. Journal of Differential Equations, 242, 269-286. https://doi.org/10.1016/j.jde.2007.08.004
- Yang, Z.J. (2010) Global Attractors and Their Hausdorff Dimensions for a Class of Kirchhoff Models. Journal of Mathematical Physics, 51, 231.
- Fan, X.M. and Zhou, S.F. (2004) Kernel Sections for Non-Autonomous Strongly Damped Wave Equations of Non-Degenerate Kifchhoff-Type. Applied Mathematics and Computation, 158, 253-266. https://doi.org/10.1016/j.amc.2003.08.147
- Lin, G.G. and Gao, Y.L. (2017) The Global and Exponential Attractors for the Higher-Order Kirchhoff-Type Equation with Strong Linear Damping. Journal of Mathematics Research, 9, 145-167. https://doi.org/10.5539/jmr.v9n4p145