# American Institute of Mathematical Sciences

July  2019, 18(4): 1827-1846. doi: 10.3934/cpaa.2019085

## Second order non-autonomous lattice systems and their uniform attractors

 Department of Mathematics, The University of Jordan, Amman 11942 Jordan

Received  July 2018 Revised  November 2018 Published  January 2019

The existence of the uniform global attractor for a second order non-autonomous lattice dynamical system (LDS) with almost periodic symbols has been carefully studied. Considering the nonlinear operators $\left( f_{1i}\left( \overset{.}{u}_{j}\mid j\in I_{iq_{1}}\right) \right) _{i\in \mathbb{Z} ^{n}}$ and $\left( f_{2i}\left( u_{j}\mid j\in I_{iq_{2}}\right) \right) _{i\in \mathbb{Z} ^{n}}$ of this LDS, up to our knowledge it is the first time to investigate the existence of uniform global attractors for such second order LDSs. In fact there are some previous studies for first order autonomous and non-autonomous LDSs with similar nonlinear parts, cf. [3, 24]. Moreover, the LDS under consideration covers a wide range of second order LDSs. In fact, for specific choices of the nonlinear functions $f_{1i}$ and $f_{2i}$ we get the autonomous and non-autonomous second order systems given by [1, 25, 26].

Citation: Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085
##### References:
 [1] A. Y. Abdallah, Upper semicontinuity of the attractor for a second order lattice dynamical system, Discrete. Contin. Dyn. Syst. Ser. B., 5 (2005), 899-916. doi: 10.3934/dcdsb.2005.5.899. [2] A. Y. Abdallah, Long-time behavior for second order lattice dynamical systems, Acta Appl. Math., 106 (2009), 47-59. doi: 10.1007/s10440-008-9281-8. [3] A. Y. Abdallah, Uniform global attractors for first order non-autonomous lattice dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 3219-3228. doi: 10.1090/S0002-9939-10-10440-7. [4] A. Y. Abdallah, Attractors for second order lattice systems with almost periodic symbols in weighted spaces, J. Math. Anal. Appl., 442 (2016), 761-781. doi: 10.1016/j.jmaa.2016.04.071. [5] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153. doi: 10.1142/S0218127401002031. [6] V. Bellrti and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^3$, Disc. Cont. Dyn. Sys., 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719. [7] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Eqs. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010. [8] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Diff. Eqs., 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020. [9] H. Chate and M. Courbage (Eds.), Lattice systems, Phys. D, 103 (1997), 1-612. doi: 10.1016/S0167-2789(96)00256-4. [10] V. V. Chepyzhov and M. I. Vishik, Non-autonomous dynamical systems and their attractors, Appendix in the book Asymptotic Behavior of Solutions of Evolutionary Equaions (M. I. Vishik ed.), Cambridge University Press, 1992. [11] V. V. Chepyzhov and M. I. Vishik, Non-autonomous evolution equations and their attractor, Russ. J. Math. Physics, 1 (1993), 165–190. [12] V. V. Chepyzhov and M. I. Vishik, Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. [13] S. N. Chow, Lattice Dynamical Systems, Dynamical System, Lecture Notes in Mathematics (Springer, Berlin), 2003, pp. 1-102. doi: 10.1007/978-3-540-45204-1_1. [14] J. Huang, X. Han and S. Zhou, Uniform attractors for non-autonomous Klein-Gordon-Schródinger lattice systems, Appl. Math. Mech. Engl. Ed., 30 (2009), 1597-1607. doi: 10.1007/s10483-009-1211-z. [15] X. Jia, C. Zhao and X. Yang, Global attractor and Kolmogorov entropy of three component reversible Gray–Scott model on infinite lattices, Appl. Math. Comp., (2012). doi: 10.1016/j.amc.2012.03.036. [16] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. [17] H. Li and L. Sun, Upper semicontinuity of attractors for small perturbations of Klein-Gordon-Schrödinger lattice system, Adv. Difference Equ., 2014, 2014: 300, 16 pp. doi: 10.1186/1687-1847-2014-300. [18] J. Oliveira, J. Pereira and M. Perla, Attractors for second order periodic lattices with nonlinear damping, J. Diff. Eqs. Appl., 14 (2008), 899-921. doi: 10.1080/10236190701859211. [19] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Appl. Math. Sci. 68. Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [20] 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. [21] X. Yang, C. Zhao and J. Cao, Dynamics of the discrete coupled nonlinear Schrödinger-Boussinesq equations, Appl. Math. Comp., 219 (2013), 8508-8524. doi: 10.1016/j.amc.2013.01.053. [22] C. Zhao, G. Xue and G. Lukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst.-B, 23 (2018), 4021-4044. [23] C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Disc. Cont. Dyn. Sys., 21 (2008), 643-663. doi: 10.3934/dcds.2008.21.643. [24] S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2. [25] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Diff. Eqs., 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005. [26] S. Zhou and M. Zhao, Uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 9 pp. doi: 10.1142/S0218127414500060.

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##### References:
 [1] A. Y. Abdallah, Upper semicontinuity of the attractor for a second order lattice dynamical system, Discrete. Contin. Dyn. Syst. Ser. B., 5 (2005), 899-916. doi: 10.3934/dcdsb.2005.5.899. [2] A. Y. Abdallah, Long-time behavior for second order lattice dynamical systems, Acta Appl. Math., 106 (2009), 47-59. doi: 10.1007/s10440-008-9281-8. [3] A. Y. Abdallah, Uniform global attractors for first order non-autonomous lattice dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 3219-3228. doi: 10.1090/S0002-9939-10-10440-7. [4] A. Y. Abdallah, Attractors for second order lattice systems with almost periodic symbols in weighted spaces, J. Math. Anal. Appl., 442 (2016), 761-781. doi: 10.1016/j.jmaa.2016.04.071. [5] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153. doi: 10.1142/S0218127401002031. [6] V. Bellrti and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^3$, Disc. Cont. Dyn. Sys., 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719. [7] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Eqs. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010. [8] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Diff. Eqs., 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020. [9] H. Chate and M. Courbage (Eds.), Lattice systems, Phys. D, 103 (1997), 1-612. doi: 10.1016/S0167-2789(96)00256-4. [10] V. V. Chepyzhov and M. I. Vishik, Non-autonomous dynamical systems and their attractors, Appendix in the book Asymptotic Behavior of Solutions of Evolutionary Equaions (M. I. Vishik ed.), Cambridge University Press, 1992. [11] V. V. Chepyzhov and M. I. Vishik, Non-autonomous evolution equations and their attractor, Russ. J. Math. Physics, 1 (1993), 165–190. [12] V. V. Chepyzhov and M. I. Vishik, Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. [13] S. N. Chow, Lattice Dynamical Systems, Dynamical System, Lecture Notes in Mathematics (Springer, Berlin), 2003, pp. 1-102. doi: 10.1007/978-3-540-45204-1_1. [14] J. Huang, X. Han and S. Zhou, Uniform attractors for non-autonomous Klein-Gordon-Schródinger lattice systems, Appl. Math. Mech. Engl. Ed., 30 (2009), 1597-1607. doi: 10.1007/s10483-009-1211-z. [15] X. Jia, C. Zhao and X. Yang, Global attractor and Kolmogorov entropy of three component reversible Gray–Scott model on infinite lattices, Appl. Math. Comp., (2012). doi: 10.1016/j.amc.2012.03.036. [16] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. [17] H. Li and L. Sun, Upper semicontinuity of attractors for small perturbations of Klein-Gordon-Schrödinger lattice system, Adv. Difference Equ., 2014, 2014: 300, 16 pp. doi: 10.1186/1687-1847-2014-300. [18] J. Oliveira, J. Pereira and M. Perla, Attractors for second order periodic lattices with nonlinear damping, J. Diff. Eqs. Appl., 14 (2008), 899-921. doi: 10.1080/10236190701859211. [19] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Appl. Math. Sci. 68. Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [20] 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. [21] X. Yang, C. Zhao and J. Cao, Dynamics of the discrete coupled nonlinear Schrödinger-Boussinesq equations, Appl. Math. Comp., 219 (2013), 8508-8524. doi: 10.1016/j.amc.2013.01.053. [22] C. Zhao, G. Xue and G. Lukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst.-B, 23 (2018), 4021-4044. [23] C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Disc. Cont. Dyn. Sys., 21 (2008), 643-663. doi: 10.3934/dcds.2008.21.643. [24] S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2. [25] S. Zhou, Attractors and approximations for lattice dynamical systems, J. Diff. Eqs., 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005. [26] S. Zhou and M. Zhao, Uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 9 pp. doi: 10.1142/S0218127414500060.
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