# American Institute of Mathematical Sciences

doi: 10.3934/era.2021056
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## Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on ${\mathbb{R}}^N$

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

* Corresponding author: Wenqiang Zhao

Received  January 2021 Revised  June 2021 Early access August 2021

Fund Project: The second author was supported by NSF grant 11871122, Chongqing NSF Grant cstc2019jcyj-msxmX0115 and CTBU Grant (KFJJ2018101, ZDPTTD201909)

In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of $L^2( {\mathbb{R}}^N)$ space. We establish the Wong-Zakai approximations of solutions in $L^l( {\mathbb{R}}^N)$ for arbitrary $l\geq q$ in the sense of upper semi-continuity of their random attractors, where $q$ is the growth exponent of the nonlinearity. The $L^l$-pre-compactness of attractors is proved by using the truncation estimate in $L^q$ and the higher-order bound of solutions.

Citation: Guifen Liu, Wenqiang Zhao. Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on ${\mathbb{R}}^N$. Electronic Research Archive, doi: 10.3934/era.2021056
##### References:
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Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [29] X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors forstochastic reaction-diffusion equations onunbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar [30] E. Wong and M. Zakai, On the relation between ordinary and stochastic differentialequations, Internat. J. Engrg. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.  Google Scholar [31] 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 [32] 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 [33] 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 [34] J. Yin and Y. Li, 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.   Google Scholar [35] 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 [36] W. Zhao, Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on ${\mathbb{R}}^N$, Comput. Math. Appl., 75 (2018), 3801-3824.  doi: 10.1016/j.camwa.2018.02.031.  Google Scholar [37] 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 [38] W. Zhao, Existences and upper semi-continuity of pullback attractors in $H^1({\mathbb{R}}^N)$ for non-autonomous reaction-diffusion equations perturbed by multiplicative noise, Electronic J. Differential Equations, 2016 (2016), Paper No. 294, 28 pp.  Google Scholar [39] W. Zhao and Y. Zhang, High-order Wong-Zakai approximations for non-autonomous stochastic $p$-Laplacian equations on ${\mathbb{R}}^N$, Commun. Pure Appl. Anal., 20 (2021), 243-280.  doi: 10.3934/cpaa.2020265.  Google Scholar [40] W. Zhao, Y. Zhang and S. Chen, Higher-orderWong-Zakai approximations of stochastic reaction-diffusion equations on $\mathbb{R}^N$, Phys. D, 401 (2020), 132147, 15 pp. doi: 10.1016/j.physd.2019.132147.  Google Scholar

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##### References:
 [1] L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] J. M. Arrieta, J. C. Nakasato and M. C. Pereira, The $p$-Laplacian equation in thin domains: The unfolding approach, J. Differential Equations, 274 (2021), 1-34.  doi: 10.1016/j.jde.2020.12.004.  Google Scholar [3] Z. Brzeźniak, U. Manna and D. Mukherjee, Wong-Zakai approximations for stochastic Landau-Lifshitz-Gilbert equations, J. Differential Equations, 267 (2019), 776-825.  doi: 10.1016/j.jde.2019.01.025.  Google Scholar [4] O. Calin, An Informal Introduction to Stochastic Calculus with Applications, World Scientific Publishing: Singapore, 2015. doi: 10.1142/9620.  Google Scholar [5] T. Caraballo and J. A. Langa, Comparison of the long-time behavior of linear Itô and Stratonovich partial differential equations, Stochastic Anal. Appl., 19 (2001), 183-195.  doi: 10.1081/SAP-100000758.  Google Scholar [6] T. Caraballo and J. C. Robinson, Stabilisation of linear PDFs by Stratonovich noise, Systems Control Lett., 53 (2004), 41-50.  doi: 10.1016/j.sysconle.2004.02.020.  Google Scholar [7] Y.-H. Cheng, Reconstruction and stability of inverse nodal problems for energy-dependent $p$-Laplacian equations, J. Math. Anal. Appl., 491 (2020), 124388, 16 pp. doi: 10.1016/j.jmaa.2020.124388.  Google Scholar [8] S.-Y. Chung and J. Hwang, A complete characterization of Fujita's blow-up solutions for discrete $p$-Laplacian parabolic equations under the mixed boundary conditions on networks, J. Math. Anal. Appl., 497 (2021), 124859, 21 pp doi: 10.1016/j.jmaa.2020.124859.  Google Scholar [9] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar [10] H. Crauel and F. Flandoli, Attracors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar [11] 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 [12] P. G. Geredeli and A. 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 [13] X. Jiang and Y. Li, Wong-Zakai approximations and periodic solutions in distribution of dissipative stochastic differential equations, J. Differential Equations, 274 (2021), 652-765.  doi: 10.1016/j.jde.2020.10.022.  Google Scholar [14] A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation, Nonliear Anal., 71 (2009), 155-171.  doi: 10.1016/j.na.2008.10.037.  Google Scholar [15] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, 2011. doi: 10.1090/surv/176.  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] 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 [18] J. Li, Y. Li and H. Cui, Existence and upper semicontinuity of random attractors of stochastic $p$-Laplacian equations on unbounded domains, Electron. J. Differential Equations, 2014 (2014), No. 87, 27 pp.  Google Scholar [19] Y. Li, A. Gu and J. Li, Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar [20] 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 [21] K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dyn. Diff. Equat., 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.  Google Scholar [22] U. Manna, D. Mukherjee and A. A. Panda, Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations with anisotropy energy, J. Math. Anal. Appl., 480 (2019), 123384, 13 pp. doi: 10.1016/j.jmaa.2019.123384.  Google Scholar [23] S. P. Sethi and J. P. Lehoczky, A comparison of the Itô and Stratonovich formulations of problems in finance, J. Economic Dynamics and Control, 3 (1981), 343-356.  doi: 10.1016/0165-1889(81)90026-9.  Google Scholar [24] J. Shen, J. Zhao, K. Lu and B. Wang, The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations, J. Differential Equations, 266 (2019), 4568-4623.  doi: 10.1016/j.jde.2018.10.008.  Google Scholar [25] J. Simsen and M. S. Simsen, Existence and upper semicontinuity of global attractors for $p(x)$-Laplacian systems, J. Math. Anal. Appl., 388 (2012), 23-38.  doi: 10.1016/j.jmaa.2011.10.003.  Google Scholar [26] Y. Sun and H. Gao, Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equation on the unbounded domains, J. Appl. Anal. Comput., 10 (2020), 2338-2361.  doi: 10.11948/20190215.  Google Scholar [27] B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.  Google Scholar [28] B. Wang, Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar [29] X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors forstochastic reaction-diffusion equations onunbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar [30] E. Wong and M. Zakai, On the relation between ordinary and stochastic differentialequations, Internat. J. Engrg. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.  Google Scholar [31] 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 [32] 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 [33] 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 [34] J. Yin and Y. Li, 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.   Google Scholar [35] 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 [36] W. Zhao, Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on ${\mathbb{R}}^N$, Comput. Math. Appl., 75 (2018), 3801-3824.  doi: 10.1016/j.camwa.2018.02.031.  Google Scholar [37] 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 [38] W. Zhao, Existences and upper semi-continuity of pullback attractors in $H^1({\mathbb{R}}^N)$ for non-autonomous reaction-diffusion equations perturbed by multiplicative noise, Electronic J. Differential Equations, 2016 (2016), Paper No. 294, 28 pp.  Google Scholar [39] W. Zhao and Y. Zhang, High-order Wong-Zakai approximations for non-autonomous stochastic $p$-Laplacian equations on ${\mathbb{R}}^N$, Commun. Pure Appl. Anal., 20 (2021), 243-280.  doi: 10.3934/cpaa.2020265.  Google Scholar [40] W. Zhao, Y. Zhang and S. Chen, Higher-orderWong-Zakai approximations of stochastic reaction-diffusion equations on $\mathbb{R}^N$, Phys. D, 401 (2020), 132147, 15 pp. doi: 10.1016/j.physd.2019.132147.  Google Scholar
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