Research ArticleOpen AccessGoogle Scholar indexed
Existence of Random Attractor Family for a Class of Nonlinear Higher-Order Kirchhoff Equations
School of Mathematics and Statistics, Yunnan University, Kunming, China
School of Mathematics and Statistics, Yunnan University, Kunming, China
- 1 School of Mathematics and Statistics, Yunnan University, Kunming, China
- 2 School of Mathematics and Statistics, Yunnan University, Kunming, China
International Journal of Modern Nonlinear Theory and Application·Volume 08 (2019)·Pages 41–52·Published 22 April 2019·DOI10.4236/ijmnta.2019.82003
Copy link · social · email
Abstract
The existence of random attractor family for a class of nonlinear nonlocal higher-order Kirchhoff partial differential equations with additive white noise is studied. The weak solution of the equation is established by the Ornstein-Uhlenbeck process to deal with the random term, and a bounded random absorption set is obtained. And then, the existence of the random attractor family is proved by the isomorphism mapping method.
KeywordsStochastic Dynamical SystemsRandom Attractor FamilyWhite NoiseOrnstein-Uhlenbeck ProcessTempered Sets
- Li, X.T. and Xu, L. (2014) Existence of Random Attractors for Stochastic Delay Discrete Wave Equations. Journal of Jilin University (Science Edition), 52, 261-262.
- Ban, A.L. (2018) Asymptotic Behavior of a Class of Stochastic Wave Equations. Journal of Anhui Normal University (Natural Science Edition), 41, 329-334.
- Zhao, C.D. and Zhou, S.F. (2009) Sufficient Conditions for the Existence of Global Random Attractors for Stochastic Lattice Dynamical Systems and Applications. Journal of Mathematical Analysis and Applications, 354, 78-95. https://doi.org/10.1016/j.jmaa.2008.12.036
- Lin, G.G., Chen, L. and Wang, W. (2017) Random Attractors of the Stochastic Strongly Damped for the Higher-Order Nonlinear Kirchhoff-Type Equation. International Journal of Modern Nonlinear Theory and Application, 6, 59-69. https://doi.org/10.4236/ijmnta.2017.62005
- Cai, D.H., Fan, X.M. and Ye, J.J. (2014) The Random Attractor of Dissipative KDV Type Equation with Multiplicative Noise Is Considered. Journal of Southwest University for Nationalities (Natural Science Edition), 40, 900-904.
- Yin, F.Q., Zhou, S.F. and Li, H.Y. (2006) Existence of Stochastic Attractors for Stochastic Sine-Gordon Equation with Strong Damping. Journal of Shanghai University (Natural Science Edition), 12, 260-265.
- Hao, H.J. and Zhou, S.F. (2010) Existence of Random Attractors for Strongly Damped Stochastic Sine-Gordon Equations, Journal of Shanghai Normal University (Natural Science Edition), 2, 121-127.
- Yin, J.Y., Li, Y.R. and Zhao, H.J. (2013) Random Attractor of Dissipative Hamiltonian Amplitude Modulated Waves with Multiplicative White Noise. Journal of Southwest Normal University (Natural Science Edition), 38, 44-48.
- Wang, R. and Li, Y.R. (2012) Random Attractor of Generalized Ginzburg-Landau Equation with Multiplicable White Noise. Journal of Southwest University (Natural Science Edition), 34, 92-95.
- Cheng, Y.Y. and Li, Y.R. (2012) Random Attractor of Generalized Kuramoto-Sivashinsky Equation with Multiplicative White Noise. Journal of Southwest Normal University (Natural Science Edition), 37, 27-30.
- Xu, G.G., Wang, L.B. and Lin, G.G. (2017) Random Attractors for Nonautonomous Stochastic Wave Equations with Dispersive and Dissipative Terms. Journal of Applied Functional Analysis, 19, 131-149.
- Qin, C.L., Du, J.J. and Lin, G.G. (2017) Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises. International Journal of Modern Nonlinear Theory and Application, 6, 134-147. https://doi.org/10.4236/ijmnta.2017.64012