In this paper, we consider the stochastic higher-order Kirchhoff-type equation with nonlinear strongly dissipation and white noise. We first deal with random term by using Ornstein-Uhlenbeck process and establish the wellness of the solution, then the existence of global random attractor are proved.
References
[1]
Yang, Z.-J. (2007) Long-Time Behavior of the Kirchhoff Type Equation with Strong Damping in RN. Journal of Differential Equations, 242, 269-286.
[2]
Yang, Z.J. and Ding, P.Y. (2016) Longtime Dynamics of the Kirchhoff Equation with Strong Damping and Critical Nonlinearity on RN. Journal of Mathematical Analysis and Applications, 434, 1826-1851.
[3]
Chueshov, I. (2012) Longtime Dynamics of Kirchhoff Wave Models with Strong Nonlinear Damping. Journal of Differential Equations, 252, 1229-1262.
[4]
Ono, K. (1997) On Global Existence, Asymptotic Stability and Blowing up of Solutions for Some Degenerate Non-Linear Wave Equations of Kirchhoff Type with a Strong Dissipation. Mathematical Methods in the Applied Sciences, 20, 151-177. https://doi.org/10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.0.CO;2-0
[5]
Wang, Z.J., Zhou, S.F. and Gu, A.H. (2011) Random Attractor for a Stochastic Damped Wave Equation with Multiplicative Noise on Unbounded Domains. Nonlinear Analysis: Real World Applications, 12, 3468-3482.
[6]
Xu, G.G., Wang, L.B. and Lin, G.G. (2013) Global Random Attractor for the Strongly Damped Stochastic Wave Equation with White Noise and Nonlinear Term. Theoretical Mathematics & Applications, 3, 111-122.
[7]
Zhong, C.K., Yang, M.H. and Sun, C.Y. (2006) The Existence of Global Attractors for the Norm-to-Weak Continuous Semigroup. Journal of Differential Equations, 223, 367-399.
[8]
Wang, Z.J., Zhou, S.F. and Gu, A.H. (2012) Random Attractor of the Stochastic Strongly Damped Wave Equation. Communications in Nonlinear Science and Numerical Simulation, 17, 1649-1658.
[9]
Arnold, L. (1998) Random Dynamical Systems. Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-662-12878-7
[10]
Chueshov, I. (2002) Monotone Random Systems Theory and Applications. Springer-Verlag, Berlin. https://doi.org/10.1007/b83277
[11]
Guo, B. and Pu, X. (2009) Random Infinite Dimensional Dynamical System. Beijing University of Aeronautics and Astronautics Press, Beijing.
[12]
Lu, D. (1986) Stochastic Process and Its Application. Tsinghua University Press, Beijing.
[13]
Fan, X. (2010) Attractors for a Damped Stochastic Wave Equation of Sine-Gordon Type with Sub-Linear Multiplicative Noise. Stochastic Analysis and Applications, 24, 767-793. https://doi.org/10.1080/07362990600751860
[14]
Pazy, A. (1983) Semi-Groups of Linear Operators and Applications to Partial Differential Equations. Springer Verlag, New York.
[15]
Massatt, P. (1983) Limiting Behaviour for a Strong Damped Nonlinear Wave Equation. Journal of Differential Equations, 48, 334-349.
[16]
Crauelh, N.D. (1988) Random Nonlinear Wave Equations: Smoothness of the Solutions. Probability Theory and Related Fields, 79, 469-508. https://doi.org/10.1007/BF00318783
[17]
Prato, G.D. and Zabczyk, J. (1992) Stochastic Equations in Infinite Dimensions. Cambridge University Press, London, 115-236.
[18]
Lin, G. (2011) Nonlinear Evolution Equation. Yunnan University Press, Kunming.
[19]
Qin, C. and Lin, G. (2010) Random Attractor of the Weakly Damped Stochastic Kirchhoff Equations. Journal of Yunnan University (Nature), 32, 101-108.
[20]
Chen, L., Wang, W. and Lin, G. (2016) Exponential Attractor and Inertial Manifolds for the Higher-Order Nonlinear Kirchhoff-Type Equation. International Journal of Modern Communication Technologies and Research, 4, 6-12.
[21]
Zhao, C. and Zhou, S. (2009) Sufficient Conditions for the Existence of Global Random Attractors for Stochatic Lattice Dunamical Systems and Applications. Journal of Mathematical Analysis and Applications, 354, 78-95.
