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
Stability of positive steady-state solutions to a time-delayed system with some applications
DCDS-B Home
This Issue
Next Article
On the family of cubic parabolic polynomials

doi: 10.3934/dcdsb.2021272

Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems - B, 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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. 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. 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. 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]	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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[55]	S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Bates, H. 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. 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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. 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. 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. 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]	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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[55]	S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001
[11]	John M. Ball. Global attractors for damped semilinear wave equations. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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