American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions
  • CPAA Home
  • This Issue
  • Next Article
    Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge
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 & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
References:
[1]

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

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[1]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[2]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022
[3]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
[4]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025
[5]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523
[6]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269
[7]

Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119
[8]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115
[9]

Hong Lu, Jiangang Qi, Bixiang Wang, Mingji Zhang. Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 683-706. doi: 10.3934/dcds.2019028
[10]

Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643
[11]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[12]

Dingshi Li, Xuemin Wang. Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains. Electronic Research Archive, 2021, 29 (2) : 1969-1990. doi: 10.3934/era.2020100
[13]

Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1
[14]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
[15]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757
[16]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[17]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[18]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[19]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088
[20]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences