doi: 10.3934/dcdsb.2021272
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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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021272
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show all references

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.  Google Scholar

[2]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[3]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[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.  Google Scholar

[5]

H. CuiJ. 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.  Google Scholar

[6]

S. N. ChowJ. M. Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differ. Equ., 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.  Google Scholar

[7]

T. CaraballoA. N. CarvalhoJ. 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.  Google Scholar

[8]

T. CaraballoI. 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.  Google Scholar

[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.  Google Scholar

[10]

T. CaraballoG. 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.  Google Scholar

[11]

T. CaraballoA. 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.  Google Scholar

[12]

T. CaraballoP. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. HuangX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

X. HanP. 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.  Google Scholar

[24]

X. HanW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. E. KloedenP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

P. E. KloedenJ. 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.  Google Scholar

[31]

J. C. Robinson, Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511933912.  Google Scholar

[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.  Google Scholar
[33]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[34]

J. C. Robinson, Global attractors: Topology and finite-dimensional dynamics, J. Dynam. Differential Equations, 11 (1999), 557-581.  doi: 10.1023/A:1021918004832.  Google Scholar

[35]

L. ShiR. WangK. 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.  Google Scholar

[36]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.  Google Scholar

[44]

B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.  doi: 10.1016/j.jde.2019.08.007.  Google Scholar

[45]

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