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

Hongyong Cui; Yangrong Li

doi:10.3934/dcdsb.2021290

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doi: 10.3934/dcdsb.2021290

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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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021290

References:

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[5]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[6]	H. Cui, A. C. Cunha and J. A. Langa, Finite-dimensionality of tempered random uniform attractors, Journal of Nonlinear Science, in press. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491. doi: 10.1080/07362999708809490. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, 2011. Google Scholar
[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. Google Scholar
[19]	C. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[5]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[6]	H. Cui, A. C. Cunha and J. A. Langa, Finite-dimensionality of tempered random uniform attractors, Journal of Nonlinear Science, in press. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491. doi: 10.1080/07362999708809490. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[17]	J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge University Press, 2011. Google Scholar
[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. Google Scholar
[19]	C. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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American Institute of Mathematical Sciences

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

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

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