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doi: 10.3934/dcdsb.2021290
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Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation

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.
Citation: Hongyong Cui, Yangrong Li. Asymptotic $ H^2$ regularity of a stochastic reaction-diffusion equation. Discrete and 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.
[2]

P. W. BatesK. 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. CaraballoH. CrauelJ. 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. CrauelA. 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. CuiJ. 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. CuiJ. 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. CuiY. 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. CuiY. 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. LiA. 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. ZhongM.-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. ZhouY. 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. BatesK. 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. CaraballoH. CrauelJ. 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. CrauelA. 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. CuiJ. 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. CuiJ. 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. CuiY. 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. CuiY. 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. LiA. 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. ZhongM.-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. ZhouY. 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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