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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 |
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.
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L. Arnold,
Random Dynamical Systems Spring-Verlag, New York and Berlin, 1998.
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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. |
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J. Ball,
Global attractors for damped semilinear wave equation, Discrete and Continuous Dynamical Systems, Ser. A, 10 (2004), 31-52.
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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.
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H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.
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A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Nonautonomous Dynamical Systems, Springer, New York, 2013.
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T. Caraballo, G. Lukaszewicz and J. Real,
Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.
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[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. |
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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. 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.
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[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. |
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