Advanced Search
Article Contents
Article Contents

High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $

  • * Corresponding author

    * Corresponding author 

The first author is supported by CTBU Grant (KFJJ2018101, ZDPTTD201909), Chongqing NSF Grant cstc2019jcyj-msxmX0115 and China NSF Grant 11871122

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we investigate the approximations of stochastic $ p $-Laplacian equation with additive white noise by a family of piecewise deterministic partial differential equations driven by a stationary stochastic process. We firstly obtain the tempered pullback attractors for the random $ p $-Laplacian equation with a general diffusion. We secondly prove the convergence of solutions and the upper semi-continuity of pullback attractors of the Wong-Zakai approximation equations in a Hilbert space for the additive case. Thirdly, by a truncation technique, the uniform compactness of pullback attractor with respect to the quantity of approximations is derived in the space of $ q $-times integrable functions, where the upper semi-continuity of the attractors of the approximation equations is well established.

    Mathematics Subject Classification: Primary:35R60, 35B40, 35B41, 35B65;Secondary:60H15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] S. Al-azzawiJ. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.
    [2] L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
    [3] I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.
    [4] H. Crauel and F. Flandoli, Attracors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.
    [5] H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.
    [6] P. G. Geredeli, On the existence of regular global attractor for $p$-Laplacian evolution equations, Appl. Math. Optim, 71 (2015), 517-532.  doi: 10.1007/s00245-014-9268-y.
    [7] P. G. Geredeli and A. Kh. Khanmamedov, Long-time dynamics of the parabolic $p$-Laplacian equation, Commun. Pure Appl. Anal., 12 (2013), 735-754.  doi: 10.3934/cpaa.2013.12.735.
    [8] A. GuK. Lu and B. Wang, Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 185-218.  doi: 10.3934/dcds.2019008.
    [9] A. Gu and B. Wang, Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720.  doi: 10.3934/dcdsb.2018072.
    [10] I. Gyongy, On the approximation of stochastic partial differential equations Ⅰ, Stochastics, 25 (1988), 59-85.  doi: 10.1080/17442508808833533.
    [11] I. Gyongy, On the approximation of stochastic partial differential equations Ⅱ, Stochastics, 26 (1989), 129-164.  doi: 10.1080/17442508908833554.
    [12] I. Gyongy and A. Shmatkov, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Appl. Math. Optim., 54 (2006), 315-341.  doi: 10.1007/s00245-006-1001-z.
    [13] H. Hu, X. Liu and et al, On smooth approximation for random attractor of stochastic partial differential equations with multiplicative noise, Stoch. Dyn., 18 (2018), 22pp. doi: 10.1142/S0219493718500405.
    [14] A. Kh. Khanmamedov, Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615.  doi: 10.1016/j.jmaa.2005.05.003.
    [15] A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.  doi: 10.1016/j.jmaa.2014.03.037.
    [16] A. KrauseM. Lewis and B. Wang, Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.  doi: 10.1016/j.amc.2014.08.033.
    [17] J. LiH. Cui and Y. Li, Existence and upper semicontinuity of random attractors of stochastic $p$-Laplacian equations on unbounded domains, Electron J. Differ. Equ., 2014 (2014), 1-27. 
    [18] Y. LiA. Gu and J. Li, Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.
    [19] Y. Li and J. Yin, Existence, regularity and approximation of global attractors for weaky dissipative $p$-Laplacian equations, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1939-1957.  doi: 10.3934/dcdss.2016079.
    [20] J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.
    [21] K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Differ. Equ., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.
    [22] K. Lu and Q. Wang, Chaotic behavior in differential equations driven by a Brownian motion, J. Differ. Equ., 251 (2011), 2853-2895.  doi: 10.1016/j.jde.2011.05.032.
    [23] T. Nakayama and S. Tappe, Wong-Zakai approximations with convergence rate for stochastic partial differential equations, Stoch. Anal. Appl., 36 (2018), 832-857.  doi: 10.1080/07362994.2018.1471402.
    [24] J. C. RobinsonInfnite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0.
    [25] B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universität, (1992), 185-192.
    [26] J. ShenK. Lu and W. Zhang, Heteroclinic chaotic behavior driven by a Brownian motion, J. Differ. Equ., 255 (2013), 4185-4225.  doi: 10.1016/j.jde.2013.08.003.
    [27] J. Shen and K. Lu, Wong-Zakai approximations and center manifolds of stochastic differential equations, J. Differ. Equ., 263 (2017), 4929-4977.  doi: 10.1016/j.jde.2017.06.005.
    [28] J. ShenJ. ZhaoK. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differ. Equ., 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008.
    [29] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
    [30] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009. doi: 10.1142/S0219493714500099.
    [31] B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.
    [32] X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors forstochastic reaction-diffusion equations onunbounded domains, J. Differ. Equ., 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.
    [33] E. Wong and M. Zakai, On the relation between ordinary and stochastic differentialequations, Int. J. Eng. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.
    [34] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Stat., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.
    [35] M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $\mathbb{R}^N$, Nonlinear Anal., 66 (2007), 1-13. doi: 10.1016/j.na.2005.11.004.
    [36] M. YangC. Sun and C. Zhong, Global attractor for $p$-Laplacian equation, J. Math. Anal. Appl., 327 (2007), 1130-1142.  doi: 10.1016/j.jmaa.2006.04.085.
    [37] J. YinY. Li and H. Cui, Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic $p$-Laplacian equations on $\mathbb{R}^N$, Math. Meth. Appl. Sci., 40 (2017), 4863-4879.  doi: 10.1002/mma.4353.
    [38] W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_\rho^p$, Appl. Math. Comput., 291 (2016), 226-243.  doi: 10.1016/j.amc.2016.06.045.
    [39] W. Zhao and Y. Zhang, Higher-order Wong-Zakai approximations of stochastic reaction-diffusion equations on $\mathbb{R}^N$, Physica D, 401 (2020), 132147. doi: 10.1016/j.physd.2019.132147.
    [40] W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. App., 455 (2017), 1178-1203.  doi: 10.1016/j.jmaa.2017.06.025.
    [41] W. Zhao, Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$, Nonliner Anal., 152 (2017), 196-219.  doi: 10.1016/j.na.2017.01.004.
  • 加载中

Article Metrics

HTML views(138) PDF downloads(272) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint