Research ArticleOpen AccessGoogle Scholar indexed
Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises
School of Mathematics and Information, Xinyang College, Xinyang, China
School of Mathematics and Information, Xinyang College, Xinyang, China
School of Mathematics and Statistics, Yunnan University, Kunming, China
- 1 School of Mathematics and Information, Xinyang College, Xinyang, China
- 2 School of Mathematics and Information, Xinyang College, Xinyang, China
- 3 School of Mathematics and Statistics, Yunnan University, Kunming, China
International Journal of Modern Nonlinear Theory and Application·Volume 06 (2017)·Pages 134–147·Published 27 November 2017·DOI10.4236/ijmnta.2017.64012
Copy link · social · email
Abstract
In this paper, we investigate the existence of random attractor for the random dynamical system generated by the Kirchhoff-type suspension bridge equations with strong damping and white noises. We first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the existence of the global attractors of the equation.
KeywordsKirchhoff-Type Suspension Bridge EquationsRandom AttractorsRandom Dynamical System
- Lazer, A.C. and McKenna, P.J. (1990) Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connection with Nonlinear Analysis. SIAM Review, 32, 537-578. https://doi.org/10.1137/1032120
- Qin, C.L. and Lin, G.G. (2010) Random Attractor for Weakly Damped Stochastic Kirchhoff Equation. Journal of Yunnan University (Natural Sciences Edition), 32, 101-108.
- Qin, C.L. and Du, J.J. (2016) Random Attractor for Strongly Damped Stochastic Kirchhoff Equation. Journal of Science of Teachers College and University, 36, 6-11.
- Ma, Q.Z. (2005) Asymptotic Behavior of the Solution for the Floating Beam. Journal of Northwest Normal University (Natural Science), 41, 4-7. (in Chinese)
- Zhong, C.K., Ma, Q.Z. and Sun, C.Y. (2007) Existence of Strong Solutions and Global Attractors for the Suspension Bridge Equations. Nonlinear Analysis, 67, 442-454.
- Park, J.Y. and Kang, J.R. (2011) Uniform Attractor for Non-Autonomous Suspension Bridge Equations with Localized Damping. Mathematical Methods in the Applied Sciences, 34, 487-496.
- Ma, Q.Z., Wang, S.P. and Chen, X.B. (2011) Uniform Attractors for the Coupled Suspension Bridge Equations. Applied Mathematics and Computation, 217, 6604-6615.
- Crauel, H. and Flandoli, F. (1994) Attractors for Random Dynamical Systems. Probability Theory and Related Fields, 100, 365-393. https://doi.org/10.1007/BF01193705
- Crauel, H., Debussche, A. and Flandoli, F. (1997) Random Attractors. Journal of Dynamics and Differential Equations, 9, 307-314. https://doi.org/10.1007/BF02219225
- Arnold, L. (1998) Random Dynamical Systems. Springer, New York. https://doi.org/10.1007/978-3-662-12878-7
- Zhou, S.F., Yin, F.Q. and Ouyang, Z.G. (2008) Random Attractor for Damped Nonlinear Wave Equation with White Noise. SIAM Journal on Applied Dynamical Systems, 4, 883-903. ttps://doi.org/10.1137/050623097
- Fan, X.M. (2008) Random Attractor for a Damped Stochastic Wave Equation with Multiplicative Noise. International Journal of Mathematics, 19, 421-437. https://doi.org/10.1142/S0129167X08004741
- Pazy, A. (1983) Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York. https://doi.org/10.1007/978-1-4612-5561-1