# American Institute of Mathematical Sciences

September  2022, 27(9): 5225-5253. doi: 10.3934/dcdsb.2021272

## Existence and approximation of attractors for nonlinear coupled lattice wave equations

 1 School of Mathematics and Compute Science, Liupanshui Normal University, Liupanshui, Guizhou 553004, China 2 Faculty of Mathematics, Federal University of Pará, Raimundo Santana Cruz Street, S/N, 68721-000, Salinópolis, Pará, Brazil 3 Ph.D Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, 01, 66075-110, Belém, Pará, Brazil 4 Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China

* Corresponding author: Renhai Wang (rwang-math@outlook.com)

Received  June 2021 Revised  August 2021 Published  September 2022 Early access  November 2021

Fund Project: Lianbing She was supported by the Science and Technology Foundation of Guizhou Province ([2020]1Y007), School level Foundation of Liupanshui Normal University(LPSSYKYJJ201801, LPSSYKJTD201907). Renhai Wang was supported by China Postdoctoral Science Foundation under grant numbers 2020TQ0053 and 2020M680456

This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in $E: = \ell^2\times\ell^2\times\ell^2\times\ell^2$. We then prove that the solution semigroup has a unique global attractor in $E$. We finally prove that this attractor can be approximated in terms of upper semicontinuity of $E$ by a finite-dimensional global attractor of a $2(2n+1)$-dimensional truncation system as $n$ goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.

Citation: Lianbing She, Mirelson M. Freitas, Mauricio S. Vinhote, Renhai Wang. Existence and approximation of attractors for nonlinear coupled lattice wave equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5225-5253. doi: 10.3934/dcdsb.2021272
##### References:
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Differ. Equ., 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478. [7] T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037. [8] T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2006/07), 1489-1507.  doi: 10.1137/050647281. [9] T. Caraballo, B. Guo, N. H. Tuan and R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, (2020), 1–31. doi: 10.1017/prm.2020.77. [10] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015. [11] T. Caraballo, A. M. Mérquez-Durén and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin Dyn Syst., 15 (2006), 559-578.  doi: 10.3934/dcds.2006.15.559. [12] 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. [13] T. L. Carrol and L. M. Pecora, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821. [14] T. Caraballo and J. Real, Navier-Stokes equations with delays, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807. [15] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [16] J. Huang, X. Han and S. Zhou, Uniform attractors for non-autonomous Klein-Gordon Schrödinger lattice systems, Appl. Math. Mech., 30 (2009), 1597-1607.  doi: 10.1007/s10483-009-1211-z. [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. [18] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467.  doi: 10.3934/dcds.2011.31.445. [19] X. Han, Asymptotic dynamics of stochastic lattice differential equations: A review, Continuous and Distributed Systems II. Stud. Syst. Decis. Control, 30 (2015), 121-136.  doi: 10.1007/978-3-319-19075-4_7. [20] X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024.  doi: 10.1142/S0219493711500249. [21] X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on Zk in weighted spaces, J. Math. Anal. 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Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753. [27] P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031. [28] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735. [29] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, vol. 176 of Mathematical Surveys and Monographs, Americal Mathematical Society, 2011. doi: 10.1090/surv/176. [30] P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017. [31] J. C. Robinson, Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511933912. [32] J. C. 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D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2. [38] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012. [39] B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006. [40] B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356. [41] 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. [42] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. 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##### References:
 [1] A. Y. Abdallah, Uniform exponential attractors for first order non-autonomous lattice dynamical systems, J. Differ. Equ., 251 (2011), 1489-1504.  doi: 10.1016/j.jde.2011.05.030. [2] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621. [3] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031. [4] H. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186.  doi: 10.1016/j.jde.2018.07.028. [5] H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z. [6] S. N. Chow, J. M. Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differ. Equ., 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478. [7] T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037. [8] T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2006/07), 1489-1507.  doi: 10.1137/050647281. [9] T. Caraballo, B. Guo, N. H. Tuan and R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, (2020), 1–31. doi: 10.1017/prm.2020.77. [10] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.  doi: 10.1016/j.crma.2005.12.015. [11] T. Caraballo, A. M. Mérquez-Durén and J. Real, Pullback and forward attractors for a 3D LANS-$\alpha$ model with delay, Discrete Contin Dyn Syst., 15 (2006), 559-578.  doi: 10.3934/dcds.2006.15.559. [12] 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. [13] T. L. Carrol and L. M. Pecora, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821. [14] T. Caraballo and J. Real, Navier-Stokes equations with delays, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807. [15] T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I. [16] J. Huang, X. Han and S. Zhou, Uniform attractors for non-autonomous Klein-Gordon Schrödinger lattice systems, Appl. Math. Mech., 30 (2009), 1597-1607.  doi: 10.1007/s10483-009-1211-z. [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. [18] X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Contin. Dyn. Syst., 31 (2011), 445-467.  doi: 10.3934/dcds.2011.31.445. [19] X. Han, Asymptotic dynamics of stochastic lattice differential equations: A review, Continuous and Distributed Systems II. Stud. Syst. Decis. Control, 30 (2015), 121-136.  doi: 10.1007/978-3-319-19075-4_7. [20] X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024.  doi: 10.1142/S0219493711500249. [21] X. Han, Asymptotic behaviors for second order stochastic lattice dynamical systems on Zk in weighted spaces, J. Math. Anal. Appl., 397 (2013), 242-254.  doi: 10.1016/j.jmaa.2012.07.015. [22] X. Han and P. E. Kloeden, Attractors Under Discretisation, SpringerBriefs in Mathematics. BCAM SpringerBriefs. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2017. doi: 10.1007/978-3-319-61934-7. [23] X. Han, P. E. Kloeden and S. Sonner, Discretisation of global attractors for lattice dynamical systems, J. Dynam. Differential Equations, 32 (2020), 1457-1474.  doi: 10.1007/s10884-019-09770-1. [24] X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equ., 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018. [25] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038. [26] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753. [27] P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.  doi: 10.1016/j.jde.2007.10.031. [28] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735. [29] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, vol. 176 of Mathematical Surveys and Monographs, Americal Mathematical Society, 2011. doi: 10.1090/surv/176. [30] P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017. [31] J. C. Robinson, Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511933912. [32] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0. [33] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, 2001. doi: 10.1007/978-94-010-0732-0. [34] J. C. Robinson, Global attractors: Topology and finite-dimensional dynamics, J. Dynam. Differential Equations, 11 (1999), 557-581.  doi: 10.1023/A:1021918004832. [35] L. Shi, R. Wang, K. Lu and B. Wang, Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains, J. Differential Equations, 267 (2019), 4373-4409.  doi: 10.1016/j.jde.2019.05.002. [36] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003. [37] B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2. [38] B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012. [39] B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006. [40] B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc., 147 (2019), 1627-1638.  doi: 10.1090/proc/14356. [41] 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. [42] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. 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