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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
References:
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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

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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