带可加噪声的非自治随机Boussinesq格点方程的随机吸引子

Abstract

Abstract: In this paper, we study the asymptomatic behavior of non-autonomous stochastic Boussinesq lattice equations with time-dependent coupled coefficients, time-dependent deterministic forces and additive white noises. Firstly, we prove the existence of a random attractor for the continuous cocycle generated by the solutions of the non-autonomous stochastic Boussinesq lattice equations. Lastly we establish the upper semi-continuity of random attractors for the random systems as the intensity of the noises tends to zero.

Key words: Stochastic Boussinesq lattice equations, Continuous cocycle, Random attractor, Upper semi-continuity, Additive white noise

CLC Number:

O193

Zhao Min, Zhou Shengfan. Random Attractor for Non-Autonomous Stochastic Boussinesq Lattice Equations with Additive White Noises[J].Acta mathematica scientia,Series A, 2018, 38(5): 924-940.

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References

[1] Jia Y, Zhang X, Dong B. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Commun Pure Appl Anal, 2013, 12(2):923-937
[2] Wu G, Zheng X. Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. J Differential Equations, 2013, 255(9):2891-2926
[3] Yang X, Fan Z, Li K. Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity. Math Method Appl Sci, 2016, 39(11):3075-3087
[4] Yang Z. Longtime dynamics of the damped Boussinesq equation. J Math Anal Appl, 2013, 399(1):180-190
[5] Zhao L, Yang M, Zhang C. Dynamics for the damped Boussinesq equation with delay. Stoch Dynam, 2016, 16(6):1650021
[6] 耿范, 李瑞斋, 葛翔宇. 具阻尼项的Boussinesq型方程的长时间行为. 数学物理学报, 2016, 36A(6):1196-1210 Geng F, Li R, Ge X. Long time behavior of Boussinesq type equation with damping term. Acta Math Sci, 2016, 36A(6):1196-1210
[7] Abdallah A. Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation. Abstr Appl Anal, 2005, 6:655-671
[8] Zhao M, Zhou S. Exponential attractor for lattice system of nonlinear Boussinesq equation. Discrete Dyn Nat Soc, 2013, 2013, Art ID:869621
[9] Zhao M, Zhou S. Pullback and uniform exponential attractors for nonautonomous Boussinesq lattice system. Internat J Bifur Chaos, 2015, 25(8):1550100
[10] Zhao M, Zhou S. Random attractor of non-autonomous stochastic Boussinesq lattice system. J Math Phys, 2015, 56(9):092702
[11] Crauel H, Debussche A, Flandoli F. Random attractors. J Dynam Differential Equations, 1997, 9(2):307-341
[12] Crauel H, Flandoli F. Attractors for random dynamical systems. Probab Theory Related Fields, 1994, 100(3):365-393.
[13] Flandoli F, Schmalfuss B. Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stoch Stoch Rep, 1996, 59(1/2):21-45
[14] Imkeller P, Schmalfuss B. The conjugacy of stochastic and random differential equations and the existence of global attractors. J Dynam Differential Equations, 2001, 13(2):215-249
[15] Arnold L. Random Dynamical Systems. Berlin:Springer-Verlag, 1998
[16] Zhao M, Zhou S. Random attractor for stochastic boissonade system with time-dependent deterministic forces and white noises. Discrete Cont Dyn-B, 2017, 22(4):1683-1717
[17] Caraballo T, Lu K. Attractors for stochastic lattice dynamical systems with a multiplication noise. Front Math China, 2008, 3(3):317-335
[18] Chueshov I. Monotone Random Systems Theory and Applications. Berlin:Springer-Verlag, 2002
[19] Zhou S, Zhao M. Random attractors for damped non-autonomous wave equations with memory and white noise. Nonlinear Anal, 2015, 120:202-226
[20] Shen Z, Zhou S, Shen W. One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation. J Differential Equations, 2010, 248(6):1432-1457
[21] 黎定仕, 范英飞. 随机非局部扩散方程的随机吸引子的存在性. 数学物理学报,2017, 37(1):158-172 Li D, Fan Y. Random attractors for the stochastic nonclassical diffusion equation on unbounded domains. Acta Math Sci, 2017, 37(1):158-172
[22] Wang B. Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J Differential Equations, 2012, 253(5):1544-1583
[23] Zhou S, Wei L. A random attractor for a stochastic second order lattice system with random coupled coefficients. J Math Anal Appl, 2012, 395(1):42-55
[24] Caraballo T, Langa J, Robinson J. Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commu Partial Differential Equations, 1998, 23(9/10):1557-1581
[25] Caraballo T, Langa J. On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dynam Cont Dis Ser A, 2003, 10(4), 491-513
[26] Robinson J. Infinite-Dimensional Dynamical Systems. Cambridge:Cambridge University Press, 2001
[27] Zhou S. Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises. Nonlinear Anal, 2012, 75(5):2793-2805
[28] Zhou S, Zhao C, Wang Y. Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete Contin Dyn Syst, 2008, 21(4):1259-1277
[29] Wang B. Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch Dyn, 14(4):1450009
[30] Bates P, Lu K, Wang B. Attractors of non-autonomous stochastic lattice systems in weighted spaces. Physica D Nonlinear Phenomena, 2014, 289:32-50
[31] Xiang X, Zhou S. Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term. J Difference Equ Appl, 2016, 22(2):235-252
[32] Pazy A. Semigroups of Linear Operator and Applications to Partial Differential Equations. New York:Springer-Verlag, 2007

Random Attractor for Non-Autonomous Stochastic Boussinesq Lattice Equations with Additive White Noises

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