December  2017, 22(10): 3629-3651. doi: 10.3934/dcdsb.2017143

Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$

1. 

Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

2. 

Department of Mathematics and Statistics, Uniersity of South Florida, Tampa, FL 33620, USA

* Corresponding author: Yuncheng You

Received  October 2016 Revised  January 2017 Published  April 2017

Fund Project: The second author is supported by NSF grant of China (Nos. 11671142 and 11371087), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407).

In this paper we study the asymptotic dynamics of the weak solutions of nonautonomous stochastic reaction-diffusion equations driven by a time-dependent forcing term and the multiplicative noise. By conducting the uniform estimates we show that the cocycle generated by this SRDE has a pullback $(L^2, H^1)$ absorbing set and it is pullback asymptotically compact through the pullback flattening approach. The existence of a pullback $(L^2, H^1)$ random attractor for this random dynamical system in space $H^{1}(\mathbb{R}^{n})$ is proved.

Citation: Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143
References:
[1]

L. Arnold, Random Dynamical Systems Spring-Verlag, New York and Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

J. Ball, Continuity properties and global attractors of generalized semiflows and the Naiver-Stokes equations, J. Nonlinear Science, 7 (1997), 475-502.  doi: 10.1007/s003329900037.

[3]

J. Ball, Global attractors for damped semilinear wave equation, Discrete and Continuous Dynamical Systems, Ser. A, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[4]

T. Bao, Existence and upper semi-continuity of uniform attractors for non-autonomous reaction-diffusion equations on $\mathbb{R}^{n}$, Electronic Journal of Differential Equations, 2012 (2012), 1-18. 

[5]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[6] A. N. CarvalhoJ. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013. 
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T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[8]

CaraballoG. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C.R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

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V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimensions, J. Math. Pures Appl., 73 (1994), 279-333. 

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P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[11]

H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, Journal of Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.

[12]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, Journal of Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.

[13]

G. Lukaszewicz and A. Tarasinska, On $H^{1}$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Analysis, 71 (2009), 782-788.  doi: 10.1016/j.na.2008.10.124.

[14]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Computation, 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.

[15]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (1992), 185-192.

[16] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002.  doi: 10.1007/978-1-4757-5037-9.
[17]

B. Q. Tang, Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains Stochastics and Dynamics 16 (2016), 1650006, 29pp. doi: 10.1142/S0219493716500064.

[18] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1998.  doi: 10.1007/978-1-4684-0313-8.
[19] H. Tuckwell, INTRODUCTION to Theoretical Neurobiology: Nonlinear and Stochastic Theories, Cambridge University Press, Cambridge, 1998. 
[20]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical system, Journal of Differential Equations, 253 (2012), 1544-1563.  doi: 10.1016/j.jde.2012.05.015.

[21]

B. Wang, Random attractors for non-autonomous stochastic wave equation with multiplicative noise, Discrete and Continuous Dynamical Systems, Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[22]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms Stochastics and Dynamics 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099.

[23]

G. Wang and Y. Tang, $(L^2, H^1)$-random attractors for stochastic reaction-diffusion equations on unbounded domains, Abstract and Applied Analysis 2013 (2013), 279509, 23pp.

[24]

Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equation, Dynamical Systems, 23 (2008), 1-16.  doi: 10.1080/14689360701611821.

[25]

K. WiesenfeldD. PiersonE. PantazelouC. Dames and F. Moss, Stochastic resonance on a circle, Phys. Rev. Lett., 72 (1994), 2125-2129.  doi: 10.1103/PhysRevLett.72.2125.

[26]

K. Wiesenfeld and F. Moss, Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs, Nature, 373 (1995), 33-35.  doi: 10.1038/373033a0.

[27]

Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete and Continuous Dynamical Systems, Ser. A, 34 (2014), 301-333.  doi: 10.3934/dcds.2014.34.301.

[28]

Y. You, Random dynamics of stochastic reaction-diffusion systems with additive noise, J. Dynamics and Differential Equations, 29 (2017), 83-112.  doi: 10.1007/s10884-015-9431-4.

[29]

W. Xhao, $H^{1}$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noise, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.

[30]

C. ZhongM. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, Journal of Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems Spring-Verlag, New York and Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

J. Ball, Continuity properties and global attractors of generalized semiflows and the Naiver-Stokes equations, J. Nonlinear Science, 7 (1997), 475-502.  doi: 10.1007/s003329900037.

[3]

J. Ball, Global attractors for damped semilinear wave equation, Discrete and Continuous Dynamical Systems, Ser. A, 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[4]

T. Bao, Existence and upper semi-continuity of uniform attractors for non-autonomous reaction-diffusion equations on $\mathbb{R}^{n}$, Electronic Journal of Differential Equations, 2012 (2012), 1-18. 

[5]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[6] A. N. CarvalhoJ. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013. 
[7]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[8]

CaraballoG. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C.R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015.

[9]

V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimensions, J. Math. Pures Appl., 73 (1994), 279-333. 

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[11]

H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, Journal of Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.

[12]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, Journal of Differential Equations, 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.

[13]

G. Lukaszewicz and A. Tarasinska, On $H^{1}$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Analysis, 71 (2009), 782-788.  doi: 10.1016/j.na.2008.10.124.

[14]

Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Computation, 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.

[15]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (1992), 185-192.

[16] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002.  doi: 10.1007/978-1-4757-5037-9.
[17]

B. Q. Tang, Regularity of random attractors for stochastic reaction-diffusion equations on unbounded domains Stochastics and Dynamics 16 (2016), 1650006, 29pp. doi: 10.1142/S0219493716500064.

[18] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1998.  doi: 10.1007/978-1-4684-0313-8.
[19] H. Tuckwell, INTRODUCTION to Theoretical Neurobiology: Nonlinear and Stochastic Theories, Cambridge University Press, Cambridge, 1998. 
[20]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical system, Journal of Differential Equations, 253 (2012), 1544-1563.  doi: 10.1016/j.jde.2012.05.015.

[21]

B. Wang, Random attractors for non-autonomous stochastic wave equation with multiplicative noise, Discrete and Continuous Dynamical Systems, Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[22]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms Stochastics and Dynamics 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099.

[23]

G. Wang and Y. Tang, $(L^2, H^1)$-random attractors for stochastic reaction-diffusion equations on unbounded domains, Abstract and Applied Analysis 2013 (2013), 279509, 23pp.

[24]

Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equation, Dynamical Systems, 23 (2008), 1-16.  doi: 10.1080/14689360701611821.

[25]

K. WiesenfeldD. PiersonE. PantazelouC. Dames and F. Moss, Stochastic resonance on a circle, Phys. Rev. Lett., 72 (1994), 2125-2129.  doi: 10.1103/PhysRevLett.72.2125.

[26]

K. Wiesenfeld and F. Moss, Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs, Nature, 373 (1995), 33-35.  doi: 10.1038/373033a0.

[27]

Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete and Continuous Dynamical Systems, Ser. A, 34 (2014), 301-333.  doi: 10.3934/dcds.2014.34.301.

[28]

Y. You, Random dynamics of stochastic reaction-diffusion systems with additive noise, J. Dynamics and Differential Equations, 29 (2017), 83-112.  doi: 10.1007/s10884-015-9431-4.

[29]

W. Xhao, $H^{1}$-random attractors and random equilibria for stochastic reaction-diffusion equations with multiplicative noise, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 2707-2721.  doi: 10.1016/j.cnsns.2013.03.012.

[30]

C. ZhongM. Yang and C. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, Journal of Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.

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