Pullback attractors for a class of nonlinear lattices with delays

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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

Yejuan Wang ^1, and Kuang Bai ^1,

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

Abstract
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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.

Keywords: Nonlinear lattices, variable delay, pullback attractor..

Mathematics Subject Classification: 34K05, 34K31, 35B40, 35B41, 35L0.

Citation: Yejuan Wang, Kuang Bai. Pullback attractors for a class of nonlinear lattices with delays. Discrete and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

show all references

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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