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

Linfang Liu; Xianlong Fu; Yuncheng You

doi:10.3934/dcdsb.2017143

Previous Article
A PDE model of intraguild predation with cross-diffusion
DCDS-B Home
This Issue
Next Article

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 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. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013.
[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.
[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.
[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. 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.
[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. 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.
[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. 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.

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. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013.
[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.
[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.
[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. 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.
[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. 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.
[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. 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.

[1]	Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114
[2]	Julia García-Luengo, Pedro Marín-Rubio. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2127-2146. doi: 10.3934/cpaa.2020094
[3]	Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2699-2722. doi: 10.3934/dcdsb.2020028
[4]	Xiaolei Dong, Yuming Qin. Strong pullback attractors for a nonclassical diffusion equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (11) : 6217-6231. doi: 10.3934/dcdsb.2021313
[5]	Saugata Bandyopadhyay, Bernard Dacorogna, Olivier Kneuss. The Pullback equation for degenerate forms. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 657-691. doi: 10.3934/dcds.2010.27.657
[6]	Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058
[7]	Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1695-1724. doi: 10.3934/dcdsb.2021107
[8]	Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[9]	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 and Continuous Dynamical Systems - B, 2010, 14 (2) : 307-326. doi: 10.3934/dcdsb.2010.14.307
[10]	Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326
[11]	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
[12]	Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[13]	Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525
[14]	Yuncheng You. Pullback uniform dissipativity of stochastic reversible Schnackenberg equations. Conference Publications, 2015, 2015 (special) : 1134-1142. doi: 10.3934/proc.2015.1134
[15]	Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
[16]	Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[17]	T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[18]	Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060
[19]	Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 167-198. doi: 10.3934/dcdsb.2021036
[20]	Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L², L^q) pullback attractors for a stochastic p-laplacian equation. Communications on Pure and Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors