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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 $

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.

Citation: Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure and Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265
References:
[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. 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.
[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.

show all references

References:
[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. 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.
[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.

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