The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

Download Download as PDF (Size:435KB)  HTML   XML  PP. 185-202  
DOI: 10.4236/ijmnta.2016.54018    286 Downloads   386 Views  

ABSTRACT

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

Cite this paper

Gao, Y. , Sun, Y. and Lin, 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. doi: 10.4236/ijmnta.2016.54018.

References

[1] Ghisi, M. and Gobbino, M. (2009) Spectral Gap Global Solutions for Degenerate Kirchhoff Equations. Nonlinear Analysis, 71, 4115-4124.
https://doi.org/10.1016/j.na.2009.02.090
[2] Yang, Z.J., Ding, P.Y. and Li, L. (2016) Longtime Dynamics of the Kirchhoff Equations with Fractional Damping and Supercritical Nonlinearity. Journal of Mathematical Analysis Application, 442, 485-510.
https://doi.org/10.1016/j.jmaa.2016.04.079
[3] Yang, Z.J., Ding, P.Y. and Liu, Z.M. (2014) Global Attractor for the Kirchhoff Type Equations with Strong Nonlinear Damping and Supercritical Nonlinearity. Applied Mathematics Letters, 33, 12-17.
https://doi.org/10.1016/j.aml.2014.02.014
[4] Li, F.C. (2004) Global Existence and Blow-Up of Solutions for a Higher-Order Kirchhoff-Type Equation with Nonlinear Dissipation. Applied Mathematics Letters, 17, 1409-1414.
https://doi.org/10.1016/j.am1.2003.07.014
[5] Li, Y. (2011) The Asymptotic Behavior of Solutions for a Nonlinear Higher Order Kirchhoff Type Equation. Journal of Southwest China Normal University, 36, 24-27.
[6] Teman, R. (1998) Infinite Dimensional Dynamics Systems in Mechanics and Physics. Springer, New York.
[7] Wu, J.Z. and Lin, G.G. (2009) The Global Attractor of the Bossinesq Equation with Damping Term and Its Dimension Estimation. Journal of Yunnan University, 31, 335-340.
[8] Yang, Z.J. (2007) Longtime Behavior of the Kirchhoff Type Equation with Strong Damping on RN. Journal of Differential Equations, 242, 269-286.
https://doi.org/10.1016/j.jde.2007.08.004
[9] Yang, Z.J. and Liu, Z.M. (2015) Exponential Attractor for the Kirchhoff Equations with Strong Nonlinear Damping and Supercritical Nonlinearity. Applied Mathematics Letters, 46, 127-132.
https://doi.org/10.1016/j.aml.2015.02.019
[10] Lin, G.G. (2011) Nonlinear Evolution Equation. Yunnan University Press, Kunming.

  
comments powered by Disqus

Copyright © 2017 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.