American Institute of Mathematical Sciences

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

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

Linfang Liu 1, , Xianlong Fu 1, and  Yuncheng You 2,,

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.

Keywords: Pullback random attractor, stochastic reaction-diffusion equation, pullback asymptotic compactness, pullback flattening property.
Mathematics Subject Classification: 35B40, 35B41, 35R60, 37L30.
Citation: Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

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

T. Caraballo, G. 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. Google Scholar

[8]

Caraballo, G. 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. Google Scholar

[9]

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

[10]

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

[11]

H. Li, Y. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

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

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[25]

K. Wiesenfeld, D. Pierson, E. Pantazelou, C. Dames and F. Moss, Stochastic resonance on a circle, Phys. Rev. Lett., 72 (1994), 2125-2129. doi: 10.1103/PhysRevLett.72.2125. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

C. Zhong, M. 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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

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

T. Caraballo, G. 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. Google Scholar

[8]

Caraballo, G. 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. Google Scholar

[9]

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

[10]

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

[11]

H. Li, Y. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

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

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

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[25]

K. Wiesenfeld, D. Pierson, E. Pantazelou, C. Dames and F. Moss, Stochastic resonance on a circle, Phys. Rev. Lett., 72 (1994), 2125-2129. doi: 10.1103/PhysRevLett.72.2125. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

C. Zhong, M. 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. Google Scholar

[1]

Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114
[2]

Saugata Bandyopadhyay, Bernard Dacorogna, Olivier Kneuss. The Pullback equation for degenerate forms. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 657-691. doi: 10.3934/dcds.2010.27.657
[3]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[4]

María Anguiano, Tomás Caraballo, José Real, José Valero. Pullback attractors for reaction-diffusion equations in some unbounded domains with an $H^{-1}$-valued non-autonomous forcing term and without uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307
[5]

Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326
[6]

Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525
[7]

Yuncheng You. Pullback uniform dissipativity of stochastic reversible Schnackenberg equations. Conference Publications, 2015, 2015 (special) : 1134-1142. doi: 10.3934/proc.2015.1134
[8]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[9]

Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
[10]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[11]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[12]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060
[13]

Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023
[14]

Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855
[15]

José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329
[16]

Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301
[17]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875
[18]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[19]

Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749
[20]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (25)
  • HTML views (7)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences