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June  2015, 20(4): 1213-1230. doi: 10.3934/dcdsb.2015.20.1213

## Pullback attractors for a class of nonlinear lattices with delays

 1 School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China, China

Received  January 2014 Revised  December 2014 Published  February 2015

We consider a class of nonlinear delay lattices $$\ddot{u}_i(t)+(-1)^p\triangle^pu_i(t)+\lambda u_i(t)+\dot{u}_i(t)=h_i(u_i(t-\rho(t)))+f_i(t),~~~i \in \mathbb{Z},$$ where $\lambda$ is a real positive constant, $p$ is any positive integer and $\triangle$ is the discrete one-dimensional Laplace operator. Under suitable conditions on $h$ and $f$ we prove the existence of pullback attractors for the multi-valued process associated with the system for which the uniqueness of solutions need not hold.
Citation: Yejuan Wang, Kuang Bai. Pullback attractors for a class of nonlinear lattices with delays. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1213-1230. doi: 10.3934/dcdsb.2015.20.1213
##### References:
 [1] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008.  Google Scholar [2] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.  Google Scholar [3] T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst., 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17.  Google Scholar [4] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.  Google Scholar [5] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar [6] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [7] J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739. doi: 10.1016/j.jmaa.2010.04.074.  Google Scholar [8] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delay in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. doi: 10.3934/dcds.2010.26.989.  Google Scholar [9] B. X. 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 [10] Y. J. Wang and S. F. Zhou, Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Applied Math., 67 (2009), 343-378.  Google Scholar [11] Y. J. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370. doi: 10.3934/dcds.2014.34.4343.  Google Scholar [12] C. D. Zhao, S. F. Zhou and W. M. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Analysis TMA, 70 (2009), 1330-1348. doi: 10.1016/j.na.2008.02.015.  Google Scholar [13] S. F. Zhou, Attractors for second-order lattice dynamical systems with damping, J. Math. Phys., 43 (2002), 452-465. doi: 10.1063/1.1418719.  Google Scholar [14] S. F. 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

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##### References:
 [1] T. Caraballo, P. Marín-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008.  Google Scholar [2] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.  Google Scholar [3] T. Caraballo and P. E. Kloeden, Non-autonomous attractors for integro-differential evolution equations, Discrete Contin. Dyn. Syst., 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17.  Google Scholar [4] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.  Google Scholar [5] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar [6] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [7] J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739. doi: 10.1016/j.jmaa.2010.04.074.  Google Scholar [8] P. Marín-Rubio and J. Real, Pullback attractors for 2D-Navier-Stokes equations with delay in continuous and sub-linear operators, Discrete Contin. Dyn. Syst., 26 (2010), 989-1006. doi: 10.3934/dcds.2010.26.989.  Google Scholar [9] B. X. 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 [10] Y. J. Wang and S. F. Zhou, Kernel sections of multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains, Quart. Applied Math., 67 (2009), 343-378.  Google Scholar [11] Y. J. Wang and P. E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst., 34 (2014), 4343-4370. doi: 10.3934/dcds.2014.34.4343.  Google Scholar [12] C. D. Zhao, S. F. Zhou and W. M. Wang, Compact kernel sections for lattice systems with delays, Nonlinear Analysis TMA, 70 (2009), 1330-1348. doi: 10.1016/j.na.2008.02.015.  Google Scholar [13] S. F. Zhou, Attractors for second-order lattice dynamical systems with damping, J. Math. Phys., 43 (2002), 452-465. doi: 10.1063/1.1418719.  Google Scholar [14] S. F. 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
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