Asymptotic H^2 regularity of a stochastic reaction-diffusion equation

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October 2022, 27(10): 5653-5671. doi: 10.3934/dcdsb.2021290

Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation

Hongyong Cui ^1, and Yangrong Li ^2,,

1.	School of Mathematics and Statistics, and Hubei Key Laboratory of Engineering Modeling and Science Computing, Huazhong University of Science & Technology, Wuhan 430074, China
2.	School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^* Corresponding author

Received March 2021 Revised September 2021 Published October 2022 Early access December 2021

In this paper we study the asymptotic dynamics for the weak solutions of the following stochastic reaction-diffusion equation defined on a bounded smooth domain

$ {\mathcal{O}} \subset {\mathbb{R}}^N $

,

$ N \leqslant 3 $

, with Dirichlet boundary condition:

$ \begin{equation} \nonumber\begin{aligned} { {{\rm{d}}} u } +(-\Delta u + u ^3- \beta u ) {{\rm{d}}} t = g(x) {{\rm{d}}} t+h(x) {{\rm{d}}} W , \quad u|_{t = 0} = u_0\in H: = L^2( {\mathcal{O}}), \end{aligned} \end{equation} $

where

$ \beta>0 $

,

$ g\in H $

, and

$ W $

a scalar and two-sided Wiener process with a regular perturbation intensity

$ h $

. We first construct an

$ H^2 $

tempered random absorbing set of the system, and then prove an

$ (H,H^2) $

-smoothing property and conclude that the random attractor of the system is in fact a finite-dimensional tempered random set in

$ H^2 $

and pullback attracts tempered random sets in

$ H $

under the topology of

$ H^2 $

. The main technique we shall employ is comparing the regularity of the stochastic equation to that of the corresponding deterministic equation for which the asymptotic

$ H^2 $

regularity is already known.

Keywords: Smoothing property, random attractor, regularity, fractal dimension.

Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.

Citation: Hongyong Cui, Yangrong Li. Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 5653-5671. doi: 10.3934/dcdsb.2021290

References:

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction–diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.
[3]	T. Caraballo, H. Crauel, J. A. Langa and J. C. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382. doi: 10.1090/S0002-9939-06-08593-5.
[4]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[5]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[6]	H. Cui, A. C. Cunha and J. A. Langa, Finite-dimensionality of tempered random uniform attractors, Journal of Nonlinear Science, in press.
[7]	H. Cui, J. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235. doi: 10.1016/j.na.2016.03.012.
[8]	H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898. doi: 10.1007/s10884-017-9617-z.
[9]	H. Cui, Y. Li and J. Yin, Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104. doi: 10.11948/2016071.
[10]	H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324. doi: 10.1016/j.na.2015.08.009.
[11]	A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491. doi: 10.1080/07362999708809490.
[12]	J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294. doi: 10.1016/j.matpur.2005.08.001.
[13]	Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.
[14]	J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22 (1976), 331-348. doi: 10.1016/0022-0396(76)90032-2.
[15]	R. Mañé, On the Dimension of the Compact Invariant Sets of Certain Non-Linear Maps, in Dynamical Systems and Turbulence, Warwick 1980, D. Rand and L. -S. Young, eds., Berlin, Heidelberg, 1981, Springer Berlin Heidelberg, pp. 230–242.
[16]	J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, vol. 28, Cambridge University Press, 2001.
[17]	J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, 2011.
[18]	A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281. doi: 10.1007/s40072-013-0007-1.
[19]	C. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007.
[20]	W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203. doi: 10.1016/j.jmaa.2017.06.025.
[21]	C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.
[22]	S. Zhou, Y. Tian and Z. Wang, Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl. Math. Comput., 276 (2016), 80-95. doi: 10.1016/j.amc.2015.12.009.

show all references

References:

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[2]	P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction–diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.
[3]	T. Caraballo, H. Crauel, J. A. Langa and J. C. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382. doi: 10.1090/S0002-9939-06-08593-5.
[4]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[5]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[6]	H. Cui, A. C. Cunha and J. A. Langa, Finite-dimensionality of tempered random uniform attractors, Journal of Nonlinear Science, in press.
[7]	H. Cui, J. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235. doi: 10.1016/j.na.2016.03.012.
[8]	H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898. doi: 10.1007/s10884-017-9617-z.
[9]	H. Cui, Y. Li and J. Yin, Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104. doi: 10.11948/2016071.
[10]	H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Anal., 128 (2015), 303-324. doi: 10.1016/j.na.2015.08.009.
[11]	A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491. doi: 10.1080/07362999708809490.
[12]	J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294. doi: 10.1016/j.matpur.2005.08.001.
[13]	Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.
[14]	J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations, 22 (1976), 331-348. doi: 10.1016/0022-0396(76)90032-2.
[15]	R. Mañé, On the Dimension of the Compact Invariant Sets of Certain Non-Linear Maps, in Dynamical Systems and Turbulence, Warwick 1980, D. Rand and L. -S. Young, eds., Berlin, Heidelberg, 1981, Springer Berlin Heidelberg, pp. 230–242.
[16]	J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, vol. 28, Cambridge University Press, 2001.
[17]	J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, 2011.
[18]	A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281. doi: 10.1007/s40072-013-0007-1.
[19]	C. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007.
[20]	W. Zhao, Random dynamics of stochastic $p$-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203. doi: 10.1016/j.jmaa.2017.06.025.
[21]	C.-K. Zhong, M.-H. Yang and C.-Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.
[22]	S. Zhou, Y. Tian and Z. Wang, Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Appl. Math. Comput., 276 (2016), 80-95. doi: 10.1016/j.amc.2015.12.009.

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