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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

Zhaojuan Wang 1, and  Shengfan Zhou 2,

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.
Keywords: Non-autonomous stochastic lattice dynamical system, random attractor, random coupled coefficient..
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 37L55, 60H1.
Citation: Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
References:
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R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Elsevier Ltd., 2003.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, New York and Berlin, 1998.

[3]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos., 11 (2001), 143-153. doi: 10.1142/s0218127401002031.

[4]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.

[5]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.

[6]

P. W. Bates, K. Lu and B. Wang, Attractors for non-autonomous stochastic lattice systems in weighted space, Physica D, 289 (2014), 32-50. doi: 10.1016/j.physd.2014.08.004.

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002.

[8]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.

[9]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.

[10]

T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278. doi: 10.1016/j.na.2015.09.025.

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/bf01193705.

[13]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.

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F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics: An Inter. J. Probability and Stoch. Processes., 59 (1996), 21-45.

[15]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214. doi: 10.1016/0022-0396(88)90104-0.

[16]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.

[17]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493. doi: 10.1016/j.jmaa.2010.11.032.

[18]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Phys. D., 233 (2007), 83-94. doi: 10.1016/j.physd.2007.06.008.

[19]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27 (2006), 1080-1090. doi: 10.1016/j.chaos.2005.04.089.

[20]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Phys. D., 221 (2006), 157-169. doi: 10.1016/j.physd.2006.07.023.

[21]

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

[22]

D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Commu. Math. Phys., 93 (1984), 285-300. doi: 10.1142/9789812833709_0019.

[23]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I: global attractors and global regularity of solutions, Journal of American Mathematical Society, 6 (1993), 503-568. doi: 10.1090/s0894-0347-1993-1179539-4.

[24]

E. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D., 212 (2005), 317-336. doi: 10.1016/j.physd.2005.10.006.

[25]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.

[26]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.

[27]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[28]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stochastics and Dynamics, 14 (2014), 31 pages. doi: 10.1142/s0219493714500099.

[29]

Y. Wang, Y. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems, J. Difference Eqns. Appl., 14 (2008), 799-817. doi: 10.1080/10236190701859542.

[30]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494. doi: 10.1016/j.na.2009.06.094.

[31]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Contin. Dyn. Syst., 21 (2008), 643-663. doi: 10.3934/dcds.2008.21.643.

[32]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Anal., 70 (2009), 1330-1348. doi: 10.1016/j.na.2008.02.015.

[33]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006. doi: 10.1088/0951-7715/20/8/010.

[34]

C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.

[35]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.

[36]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D., 178 (2003), 51-61. doi: 10.1016/s0167-2789(02)00807-2.

[37]

S. Zhou and W. Shi, Attractors and dimension ofdissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.

[38]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.

[39]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55. doi: 10.1016/j.jmaa.2012.04.080.

[40]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B., 9 (2008), 763-785.

show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Elsevier Ltd., 2003.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, New York and Berlin, 1998.

[3]

P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos., 11 (2001), 143-153. doi: 10.1142/s0218127401002031.

[4]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.

[5]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.

[6]

P. W. Bates, K. Lu and B. Wang, Attractors for non-autonomous stochastic lattice systems in weighted space, Physica D, 289 (2014), 32-50. doi: 10.1016/j.physd.2014.08.004.

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002.

[8]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.

[9]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.

[10]

T. Caraballo, X. Han, B. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278. doi: 10.1016/j.na.2015.09.025.

[11]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.

[12]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/bf01193705.

[13]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.

[14]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics: An Inter. J. Probability and Stoch. Processes., 59 (1996), 21-45.

[15]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214. doi: 10.1016/0022-0396(88)90104-0.

[16]

X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.

[17]

X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493. doi: 10.1016/j.jmaa.2010.11.032.

[18]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Phys. D., 233 (2007), 83-94. doi: 10.1016/j.physd.2007.06.008.

[19]

Y. Lv and J. Sun, Dynamical behavior for stochastic lattice systems, Chaos Solitons Fractals, 27 (2006), 1080-1090. doi: 10.1016/j.chaos.2005.04.089.

[20]

Y. Lv and J. Sun, Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Phys. D., 221 (2006), 157-169. doi: 10.1016/j.physd.2006.07.023.

[21]

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

[22]

D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Commu. Math. Phys., 93 (1984), 285-300. doi: 10.1142/9789812833709_0019.

[23]

G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I: global attractors and global regularity of solutions, Journal of American Mathematical Society, 6 (1993), 503-568. doi: 10.1090/s0894-0347-1993-1179539-4.

[24]

E. V. Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D., 212 (2005), 317-336. doi: 10.1016/j.physd.2005.10.006.

[25]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.

[26]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.

[27]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015.

[28]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stochastics and Dynamics, 14 (2014), 31 pages. doi: 10.1142/s0219493714500099.

[29]

Y. Wang, Y. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems, J. Difference Eqns. Appl., 14 (2008), 799-817. doi: 10.1080/10236190701859542.

[30]

X. Wang, S. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494. doi: 10.1016/j.na.2009.06.094.

[31]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Contin. Dyn. Syst., 21 (2008), 643-663. doi: 10.3934/dcds.2008.21.643.

[32]

C. Zhao, S. Zhou and W. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Anal., 70 (2009), 1330-1348. doi: 10.1016/j.na.2008.02.015.

[33]

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity, 20 (2007), 1987-2006. doi: 10.1088/0951-7715/20/8/010.

[34]

C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.

[35]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.

[36]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D., 178 (2003), 51-61. doi: 10.1016/s0167-2789(02)00807-2.

[37]

S. Zhou and W. Shi, Attractors and dimension ofdissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.

[38]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.

[39]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55. doi: 10.1016/j.jmaa.2012.04.080.

[40]

X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B., 9 (2008), 763-785.

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