Existence and approximation of attractors for nonlinear coupled lattice wave equations

Lianbing She; Mirelson M. Freitas; Mauricio S. Vinhote; Renhai Wang

doi:10.3934/dcdsb.2021272

Previous Article
Global generalized solutions to the forager-exploiter model with logistic growth
DCDS-B Home
This Issue
Next Article
Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

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

Existence and approximation of attractors for nonlinear coupled lattice wave equations

Lianbing She ^1, , Mirelson M. Freitas ^2, , Mauricio S. Vinhote ^3, and Renhai Wang ^4,,

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.

Keywords: Coupled discrete wave equations, global attractors, upper semicontinuity, asymptotic compactness.

Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 37L30.

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:

[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.
[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.
[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.
[45]	B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[46]	C. Wang, G. Xue and C. Zhao, Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comp., 339 (2018), 853-865. doi: 10.1016/j.amc.2018.06.059.
[47]	R. Wang, Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping, J. Dynam. Differential Equations, 33 (2021), 767-803. doi: 10.1007/s10884-020-09830-x.
[48]	R. Wang, B. Guo and B. Wang, Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on $\mathbb{R}^N$ driven by nonlinear noise, Sci. China Math., 64 (2021), 2395-2436. doi: 10.1007/s11425-019-1714-2.
[49]	R. Wang, Y. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126. doi: 10.3934/dcds.2019165.
[50]	R. Wang, Y. Li and B. Wang, Bi-spatial pullback attractors of fractional non-classical diffusion equations on unbounded domains with $(p, q)$-growth nonlinearities, Appl. Math. Optim., 84 (2021), 425-461. doi: 10.1007/s00245-019-09650-6.
[51]	R. Wang, L. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N$, Nonlinearity, 32 (2019), 4524-4556. doi: 10.1088/1361-6544/ab32d7.
[52]	R. Wang and B. Wang, Random dynamics of $p$-laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Process. Appl., 130 (2020), 7431-7462. doi: 10.1016/j.spa.2020.08.002.
[53]	C. Zhao, H. Jiang and T. Caraballo, Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices, Appl. Math. Comput., 404 (2021), 126103. doi: 10.1016/j.amc.2021.126103.
[54]	C. Zhao, G. Xue and G. Lukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Contin. Dyn. Syst.B, 23 (2018), 4021-4044. doi: 10.3934/dcdsb.2018122.
[55]	S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.
[56]	S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.
[57]	S. Zhou and X. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dynam. Differential Equations, 24 (2012), 601-631. doi: 10.1007/s10884-012-9260-7.
[58]	S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst., 21 (2008), 1259-1277. doi: 10.3934/dcds.2008.21.1259.
[59]	V. Z. Zhu, Y. Sang and C. Zhao, Pullback attractor and invariant measures for the discrete Zakharov equations, J. Appl. Anal. Comput., 9 (2019), 2333-2357. doi: 10.11948/20190091.

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.
[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.
[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.
[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.
[45]	B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$, Tran. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[46]	C. Wang, G. Xue and C. Zhao, Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comp., 339 (2018), 853-865. doi: 10.1016/j.amc.2018.06.059.
[47]	R. Wang, Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping, J. Dynam. Differential Equations, 33 (2021), 767-803. doi: 10.1007/s10884-020-09830-x.
[48]	R. Wang, B. Guo and B. Wang, Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on $\mathbb{R}^N$ driven by nonlinear noise, Sci. China Math., 64 (2021), 2395-2436. doi: 10.1007/s11425-019-1714-2.
[49]	R. Wang, Y. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126. doi: 10.3934/dcds.2019165.
[50]	R. Wang, Y. Li and B. Wang, Bi-spatial pullback attractors of fractional non-classical diffusion equations on unbounded domains with $(p, q)$-growth nonlinearities, Appl. Math. Optim., 84 (2021), 425-461. doi: 10.1007/s00245-019-09650-6.
[51]	R. Wang, L. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N$, Nonlinearity, 32 (2019), 4524-4556. doi: 10.1088/1361-6544/ab32d7.
[52]	R. Wang and B. Wang, Random dynamics of $p$-laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Process. Appl., 130 (2020), 7431-7462. doi: 10.1016/j.spa.2020.08.002.
[53]	C. Zhao, H. Jiang and T. Caraballo, Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices, Appl. Math. Comput., 404 (2021), 126103. doi: 10.1016/j.amc.2021.126103.
[54]	C. Zhao, G. Xue and G. Lukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Contin. Dyn. Syst.B, 23 (2018), 4021-4044. doi: 10.3934/dcdsb.2018122.
[55]	S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.
[56]	S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.
[57]	S. Zhou and X. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dynam. Differential Equations, 24 (2012), 601-631. doi: 10.1007/s10884-012-9260-7.
[58]	S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst., 21 (2008), 1259-1277. doi: 10.3934/dcds.2008.21.1259.
[59]	V. Z. Zhu, Y. Sang and C. Zhao, Pullback attractor and invariant measures for the discrete Zakharov equations, J. Appl. Anal. Comput., 9 (2019), 2333-2357. doi: 10.11948/20190091.

[1]	Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036
[2]	Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653
[3]	Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189
[4]	Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
[5]	Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
[6]	Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054
[7]	Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[8]	Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252
[9]	Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control and Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010
[10]	María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations and Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001
[11]	John M. Ball. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[12]	Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L², L^q) pullback attractors for a stochastic p-laplacian equation. Communications on Pure and Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023
[13]	Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115
[14]	Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116
[15]	Wenlong Sun. The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28 (3) : 1343-1356. doi: 10.3934/era.2020071
[16]	Na Lei, Shengfan Zhou. Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 73-108. doi: 10.3934/dcds.2021108
[17]	Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592
[18]	Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015
[19]	Sergey Dashkovskiy, Oleksiy Kapustyan. Robustness of global attractors: Abstract framework and application to dissipative wave equations. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021054
[20]	Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

American Institute of Mathematical Sciences

Existence and approximation of attractors for nonlinear coupled lattice wave equations

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Existence and approximation of attractors for nonlinear coupled lattice wave equations

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors