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January  2021, 20(1): 243-280. doi: 10.3934/cpaa.2020265

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

Wenqiang Zhao 1,, and  Yijin Zhang 2,

1. 

Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
2. 

Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

* Corresponding author

Received  May 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

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

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.

Keywords: Wong-Zakai approximation, stochastic $ p $-Laplacian, additive white noise, random dynamical systems, tempered random attractor, upper semi-continuity.
Mathematics Subject Classification: Primary:35R60, 35B40, 35B41, 35B65;Secondary:60H15.
Citation: Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265
References:
[1]

S. Al-azzawi, J. 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.  Google Scholar

[2]

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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A. Gu, K. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[16]

A. Krause, M. 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.  Google Scholar

[17]

J. Li, H. 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.   Google Scholar

[18]

Y. Li, A. 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.  Google Scholar

[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.  Google Scholar

[20]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] J. C. Robinson, Infnite-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.  Google Scholar
[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. Google Scholar

[26]

J. Shen, K. 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.  Google Scholar

[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.  Google Scholar

[28]

J. Shen, J. Zhao, K. 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.  Google Scholar

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

X. Wang, K. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

M. Yang, C. 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.  Google Scholar

[37]

J. Yin, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. Al-azzawi, J. 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.  Google Scholar

[2]

L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[3]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[4]

H. Crauel and F. Flandoli, Attracors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[5]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Gu, K. 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.  Google Scholar

[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.  Google Scholar

[10]

I. Gyongy, On the approximation of stochastic partial differential equations Ⅰ, Stochastics, 25 (1988), 59-85.  doi: 10.1080/17442508808833533.  Google Scholar

[11]

I. Gyongy, On the approximation of stochastic partial differential equations Ⅱ, Stochastics, 26 (1989), 129-164.  doi: 10.1080/17442508908833554.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Krause, M. 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.  Google Scholar

[17]

J. Li, H. 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.   Google Scholar

[18]

Y. Li, A. 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.  Google Scholar

[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.  Google Scholar

[20]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] J. C. Robinson, Infnite-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.  Google Scholar
[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. Google Scholar

[26]

J. Shen, K. 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.  Google Scholar

[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.  Google Scholar

[28]

J. Shen, J. Zhao, K. 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.  Google Scholar

[29]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

X. Wang, K. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

M. Yang, C. 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.  Google Scholar

[37]

J. Yin, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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