# American Institute of Mathematical Sciences

• Previous Article
Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity
• DCDS-B Home
• This Issue
• Next Article
A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations
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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  doi: 10.1063/1.1418719.  Google Scholar [14] S. F. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342.  doi: 10.1016/j.jde.2004.02.005.  Google Scholar

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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  doi: 10.1063/1.1418719.  Google Scholar [14] S. F. Zhou, Attractors and approximations for lattice dynamical systems,, J. Differential Equations, 200 (2004), 342.  doi: 10.1016/j.jde.2004.02.005.  Google Scholar
 [1] Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195 [2] Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183 [3] T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037 [4] Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060 [5] Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543 [6] Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215 [7] Hai Huyen Dam, Kok Lay Teo. Variable fractional delay filter design with discrete coefficients. Journal of Industrial & Management Optimization, 2016, 12 (3) : 819-831. doi: 10.3934/jimo.2016.12.819 [8] Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559 [9] Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114 [10] Christopher Chong, P.G. Kevrekidis, Guido Schneider. Justification of leading order quasicontinuum approximations of strongly nonlinear lattices. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3403-3418. doi: 10.3934/dcds.2014.34.3403 [11] Cheng Hou Tsang, Boris A. Malomed, Kwok Wing Chow. Exact solutions for periodic and solitary matter waves in nonlinear lattices. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1299-1325. doi: 10.3934/dcdss.2011.4.1299 [12] Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143 [13] Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058 [14] Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 [15] Jong Yeoul Park, Jae Ug Jeong. Pullback attractors for a $2D$-non-autonomous incompressible non-Newtonian fluid with variable delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2687-2702. doi: 10.3934/dcdsb.2016068 [16] Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717 [17] S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593 [18] Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165 [19] Goro Akagi. Doubly nonlinear parabolic equations involving variable exponents. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 1-16. doi: 10.3934/dcdss.2014.7.1 [20] Theodore Tachim Medjo. Pullback $\mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (15)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]