### Electronic Research Archive

2021, Issue 6: 3655-3686. doi: 10.3934/era.2021056

# Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on ${\mathbb{R}}^N$

• Received: 01 January 2021 Revised: 01 June 2021 Published: 13 August 2021
• Primary: 35R60, 35B40, 35B41; Secondary: 35B65

• 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 ${\mathbb{R}}^N$[J]. Electronic Research Archive, 2021, 29(6): 3655-3686. doi: 10.3934/era.2021056

### Related Papers:

• 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.

 [1] L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7 [2] The $p$-Laplacian equation in thin domains: The unfolding approach. J. Differential Equations (2021) 274: 1-34. [3] Wong-Zakai approximations for stochastic Landau-Lifshitz-Gilbert equations. J. Differential Equations (2019) 267: 776-825. [4] O. Calin, An Informal Introduction to Stochastic Calculus with Applications, World Scientific Publishing: Singapore, 2015. doi: 10.1142/9620 [5] Comparison of the long-time behavior of linear Itô and Stratonovich partial differential equations. Stochastic Anal. Appl. (2001) 19: 183-195. [6] Stabilisation of linear PDFs by Stratonovich noise. Systems Control Lett. (2004) 53: 41-50. [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 [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 [9] Random attractors. J. Dynam. Differential Equations (1997) 9: 307-341. [10] Attracors for random dynamical systems. Probab. Theory Related Fields (1994) 100: 365-393. [11] On the existence of regular global attractor for $p$-Laplacian evolution equations. Appl. Math. Optim. (2015) 71: 517-532. [12] Long-time dynamics of the parabolic $p$-Laplacian equation. Commun. Pure Appl. Anal. (2013) 12: 735-754. [13] Wong-Zakai approximations and periodic solutions in distribution of dissipative stochastic differential equations. J. Differential Equations (2021) 274: 652-765. [14] Global attractors for one dimensional $p$-Laplacian equation. Nonliear Anal. (2009) 71: 155-171. [15] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, 2011. doi: 10.1090/surv/176 [16] Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise. Appl. Math. Comput. (2014) 246: 365-376. [17] Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains. J. Math. Anal. Appl. (2014) 417: 1018-1038. [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. [19] Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations. J. Differential Equations (2015) 258: 504-534. [20] Existence, regularity and approximation of global attractors for weaky dissipative $p$-Laplacian equations. Discrete Contin. Dyn. Syst. Ser. S (2016) 9: 1939-1957. [21] Wong-Zakai approximations and long term behavior of stochastic partial differential equations. J. Dyn. Diff. Equat. (2019) 31: 1341-1371. [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 [23] A comparison of the Itô and Stratonovich formulations of problems in finance. J. Economic Dynamics and Control (1981) 3: 343-356. [24] The Wong-Zakai approximations of invariant manifolds and foliations for stochastic evolution equations. J. Differential Equations (2019) 266: 4568-4623. [25] Existence and upper semicontinuity of global attractors for $p(x)$-Laplacian systems. J. Math. Anal. Appl. (2012) 388: 23-38. [26] Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equation on the unbounded domains. J. Appl. Anal. Comput. (2020) 10: 2338-2361. [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 [28] Suffcient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differential Equations (2012) 253: 1544-1583. [29] Wong-Zakai approximations and attractors forstochastic reaction-diffusion equations onunbounded domains. J. Differential Equations (2018) 264: 378-424. [30] On the relation between ordinary and stochastic differentialequations. Internat. J. Engrg. Sci. (1965) 3: 213-229. [31] On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. (1965) 36: 1560-1564. [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 [33] Global attractor for $p$-Laplacian equation. J. Math. Anal. Appl. (2007) 327: 1130-1142. [34] 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. (2017) 40: 4863-4879. [35] Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise. J. Math. Anal. App. (2017) 455: 1178-1203. [36] Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on ${\mathbb{R}}^N$. Comput. Math. Appl. (2018) 75: 3801-3824. [37] Long-time random dynamics of stochastic parabolic $p$-Laplacian equations on $\mathbb{R}^N$. Nonliner Anal. (2017) 152: 196-219. [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. [39] High-order Wong-Zakai approximations for non-autonomous stochastic $p$-Laplacian equations on ${\mathbb{R}}^N$. Commun. Pure Appl. Anal. (2021) 20: 243-280. [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
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.604 0.8