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November  2016, 15(6): 2221-2245. doi: 10.3934/cpaa.2016035

Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients

1. 

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004

Received  January 2016 Revised  June 2016 Published  September 2016

In this paper, we consider the existence of random attractors in a weighted space $ l_\rho ^2 $ for first-order non-autonomous stochastic lattice system with random coupled coefficients and multiplicative/additive white noise, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.
Citation: Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
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R. A. Adams and J. J. Fournier, Sobolev Spaces,, 2$^{nd}$ edition, (2003). Google Scholar

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L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). Google Scholar

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P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, \emph{Internat. J. Bifur. Chaos.}, 11 (2001), 143. doi: 10.1142/s0218127401002031. Google Scholar

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P. W. Bates, K. Lu and B. Wang, Attractors for non-autonomous stochastic lattice systems in weighted space,, \emph{Physica D}, 289 (2014), 32. doi: 10.1016/j.physd.2014.08.004. Google Scholar

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I. Chueshov, Monotone Random Systems Theory and Applications,, Springer-Verlag, (2002). Google Scholar

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T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems,, \emph{Dynamics of Continuous, 10 (2003), 491. Google Scholar

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T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise,, \emph{Front. Math. China}, 3 (2008), 317. doi: 10.1007/s11464-008-0028-7. Google Scholar

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T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise,, \emph{Nonlinear Anal.}, 130 (2016), 255. doi: 10.1016/j.na.2015.09.025. Google Scholar

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H. Crauel, A. Debussche and F. Flandoli, Random attractors,, \emph{J. Dyn. Diff. Eqns.}, 9 (1997), 307. Google Scholar

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H. Crauel and F. Flandoli, Attractors for random dynamical systems,, \emph{Probab. Th. Re. Fields}, 100 (1994), 365. doi: 10.1007/bf01193705. Google Scholar

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J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, \emph{Ann. Probab.}, 31 (2003), 2109. Google Scholar

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F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, \emph{Stochastics: An Inter. J. Probability and Stoch. Processes.}, 59 (1996), 21. Google Scholar

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J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, \emph{J. Differential Equations}, 73 (1988), 197. doi: 10.1016/0022-0396(88)90104-0. Google Scholar

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X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces,, \emph{J. Differential Equations}, 250 (2011), 1235. doi: 10.1016/j.jde.2010.10.018. Google Scholar

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X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise,, \emph{J. Math. Anal. Appl.}, 376 (2011), 481. doi: 10.1016/j.jmaa.2010.11.032. Google Scholar

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J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises,, \emph{Phys. D.}, 233 (2007), 83. doi: 10.1016/j.physd.2007.06.008. Google Scholar

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Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems,, \emph{Chaos Solitons Fractals}, 27 (2006), 1080. doi: 10.1016/j.chaos.2005.04.089. Google Scholar

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Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations,, \emph{Phys. D.}, 221 (2006), 157. doi: 10.1016/j.physd.2006.07.023. Google Scholar

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G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I: global attractors and global regularity of solutions,, \emph{Journal of American Mathematical Society}, 6 (1993), 503. doi: 10.1090/s0894-0347-1993-1179539-4. Google Scholar

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E. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems,, \emph{Phys. D.}, 212 (2005), 317. doi: 10.1016/j.physd.2005.10.006. Google Scholar

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B. Wang, Dynamics of systems on infinite lattices,, \emph{J. Differential Equations}, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. Google Scholar

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B. Wang, Asymptotic behavior of non-autonomous lattice systems,, \emph{J. Math. Anal. Appl.}, 331 (2007), 121. doi: 10.1016/j.jmaa.2006.08.070. Google Scholar

[27]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, \emph{J. Differential Equations}, 253 (2012), 1544. doi: 10.1016/j.jde.2012.05.015. Google Scholar

[28]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms,, \emph{Stochastics and Dynamics}, 14 (2014). doi: 10.1142/s0219493714500099. Google Scholar

[29]

Y. Wang, Y. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems,, \emph{J. Difference Eqns. Appl.}, 14 (2008), 799. doi: 10.1080/10236190701859542. Google Scholar

[30]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems,, \emph{Nonlinear Anal.}, 72 (2010), 483. doi: 10.1016/j.na.2009.06.094. Google Scholar

[31]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays,, \emph{Discrete Contin. Dyn. Syst.}, 21 (2008), 643. doi: 10.3934/dcds.2008.21.643. Google Scholar

[32]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays,, \emph{Nonlinear Anal.}, 70 (2009), 1330. doi: 10.1016/j.na.2008.02.015. Google Scholar

[33]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems,, \emph{Nonlinearity}, 20 (2007), 1987. doi: 10.1088/0951-7715/20/8/010. Google Scholar

[34]

C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications,, \emph{J. Math. Anal. Appl.}, 354 (2009), 78. doi: 10.1016/j.jmaa.2008.12.036. Google Scholar

[35]

S. Zhou, Attractors for second order lattice dynamical systems,, \emph{J. Differential Equations}, 179 (2002), 605. doi: 10.1006/jdeq.2001.4032. Google Scholar

[36]

S. Zhou, Attractors for first order dissipative lattice dynamical systems,, \emph{Phys. D.}, 178 (2003), 51. doi: 10.1016/s0167-2789(02)00807-2. Google Scholar

[37]

S. Zhou and W. Shi, Attractors and dimension ofdissipative lattice systems,, \emph{J. Differential Equations}, 224 (2006), 172. doi: 10.1016/j.jde.2005.06.024. Google Scholar

[38]

S. Zhou, Attractors and approximations for lattice dynamical systems,, \emph{J. Differential Equations}, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. Google Scholar

[39]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients,, \emph{J. Math. Anal. Appl.}, 395 (2012), 42. doi: 10.1016/j.jmaa.2012.04.080. Google Scholar

[40]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems,, \emph{Discrete Contin. Dyn. Syst. Ser. B.}, 9 (2008), 763. Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces,, 2$^{nd}$ edition, (2003). Google Scholar

[2]

L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). Google Scholar

[3]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems,, \emph{Internat. J. Bifur. Chaos.}, 11 (2001), 143. doi: 10.1142/s0218127401002031. Google Scholar

[4]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, \emph{Stoch. Dyn.}, 6 (2006), 1. doi: 10.1142/S0219493706001621. Google Scholar

[5]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains,, \emph{J. Differential Equations}, 246 (2009), 845. Google Scholar

[6]

P. W. Bates, K. Lu and B. Wang, Attractors for non-autonomous stochastic lattice systems in weighted space,, \emph{Physica D}, 289 (2014), 32. doi: 10.1016/j.physd.2014.08.004. Google Scholar

[7]

I. Chueshov, Monotone Random Systems Theory and Applications,, Springer-Verlag, (2002). Google Scholar

[8]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems,, \emph{Dynamics of Continuous, 10 (2003), 491. Google Scholar

[9]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise,, \emph{Front. Math. China}, 3 (2008), 317. doi: 10.1007/s11464-008-0028-7. Google Scholar

[10]

T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise,, \emph{Nonlinear Anal.}, 130 (2016), 255. doi: 10.1016/j.na.2015.09.025. Google Scholar

[11]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, \emph{J. Dyn. Diff. Eqns.}, 9 (1997), 307. Google Scholar

[12]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, \emph{Probab. Th. Re. Fields}, 100 (1994), 365. doi: 10.1007/bf01193705. Google Scholar

[13]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations,, \emph{Ann. Probab.}, 31 (2003), 2109. Google Scholar

[14]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, \emph{Stochastics: An Inter. J. Probability and Stoch. Processes.}, 59 (1996), 21. Google Scholar

[15]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, \emph{J. Differential Equations}, 73 (1988), 197. doi: 10.1016/0022-0396(88)90104-0. Google Scholar

[16]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces,, \emph{J. Differential Equations}, 250 (2011), 1235. doi: 10.1016/j.jde.2010.10.018. Google Scholar

[17]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise,, \emph{J. Math. Anal. Appl.}, 376 (2011), 481. doi: 10.1016/j.jmaa.2010.11.032. Google Scholar

[18]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises,, \emph{Phys. D.}, 233 (2007), 83. doi: 10.1016/j.physd.2007.06.008. Google Scholar

[19]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems,, \emph{Chaos Solitons Fractals}, 27 (2006), 1080. doi: 10.1016/j.chaos.2005.04.089. Google Scholar

[20]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations,, \emph{Phys. D.}, 221 (2006), 157. doi: 10.1016/j.physd.2006.07.023. Google Scholar

[21]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[22]

D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces,, \emph{Commu. Math. Phys.}, 93 (1984), 285. doi: 10.1142/9789812833709_0019. Google Scholar

[23]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I: global attractors and global regularity of solutions,, \emph{Journal of American Mathematical Society}, 6 (1993), 503. doi: 10.1090/s0894-0347-1993-1179539-4. Google Scholar

[24]

E. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems,, \emph{Phys. D.}, 212 (2005), 317. doi: 10.1016/j.physd.2005.10.006. Google Scholar

[25]

B. Wang, Dynamics of systems on infinite lattices,, \emph{J. Differential Equations}, 221 (2006), 224. doi: 10.1016/j.jde.2005.01.003. Google Scholar

[26]

B. Wang, Asymptotic behavior of non-autonomous lattice systems,, \emph{J. Math. Anal. Appl.}, 331 (2007), 121. doi: 10.1016/j.jmaa.2006.08.070. Google Scholar

[27]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, \emph{J. Differential Equations}, 253 (2012), 1544. doi: 10.1016/j.jde.2012.05.015. Google Scholar

[28]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms,, \emph{Stochastics and Dynamics}, 14 (2014). doi: 10.1142/s0219493714500099. Google Scholar

[29]

Y. Wang, Y. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems,, \emph{J. Difference Eqns. Appl.}, 14 (2008), 799. doi: 10.1080/10236190701859542. Google Scholar

[30]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems,, \emph{Nonlinear Anal.}, 72 (2010), 483. doi: 10.1016/j.na.2009.06.094. Google Scholar

[31]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays,, \emph{Discrete Contin. Dyn. Syst.}, 21 (2008), 643. doi: 10.3934/dcds.2008.21.643. Google Scholar

[32]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays,, \emph{Nonlinear Anal.}, 70 (2009), 1330. doi: 10.1016/j.na.2008.02.015. Google Scholar

[33]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems,, \emph{Nonlinearity}, 20 (2007), 1987. doi: 10.1088/0951-7715/20/8/010. Google Scholar

[34]

C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications,, \emph{J. Math. Anal. Appl.}, 354 (2009), 78. doi: 10.1016/j.jmaa.2008.12.036. Google Scholar

[35]

S. Zhou, Attractors for second order lattice dynamical systems,, \emph{J. Differential Equations}, 179 (2002), 605. doi: 10.1006/jdeq.2001.4032. Google Scholar

[36]

S. Zhou, Attractors for first order dissipative lattice dynamical systems,, \emph{Phys. D.}, 178 (2003), 51. doi: 10.1016/s0167-2789(02)00807-2. Google Scholar

[37]

S. Zhou and W. Shi, Attractors and dimension ofdissipative lattice systems,, \emph{J. Differential Equations}, 224 (2006), 172. doi: 10.1016/j.jde.2005.06.024. Google Scholar

[38]

S. Zhou, Attractors and approximations for lattice dynamical systems,, \emph{J. Differential Equations}, 200 (2004), 342. doi: 10.1016/j.jde.2004.02.005. Google Scholar

[39]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients,, \emph{J. Math. Anal. Appl.}, 395 (2012), 42. doi: 10.1016/j.jmaa.2012.04.080. Google Scholar

[40]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems,, \emph{Discrete Contin. Dyn. Syst. Ser. B.}, 9 (2008), 763. Google Scholar

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