Asymptotic behavior of non-autonomous stochastic Boussinesq lattice system

Ailing Ban; Ailing Ban

doi:10.3934/math.2025040

AIMS Mathematics

2025, Volume 10, Issue 1: 839-857. doi: 10.3934/math.2025040

Previous Article Next Article

Research article

Asymptotic behavior of non-autonomous stochastic Boussinesq lattice system

Ailing Ban ^,

School of big data and artificial intelligence, Chizhou University, Chizhou, Anhui 247000, China

Received: 24 October 2024 Revised: 25 December 2024 Accepted: 30 December 2024 Published: 15 January 2025
MSC : 37L55, 35B41, 35B40

In this paper, we investigate the existence of a random uniform exponential attractor for the non-autonomous stochastic Boussinesq lattice equation with multiplicative white noise and quasi-periodic forces. We first show the existence and uniqueness of the solution of the considered Boussinesq system. Then, we consider the existence of a uniform absorbing random set for a jointly continuous non-autonomous random dynamical system (NRDS) generated by the system, and make an estimate on the tail of solutions. Third, we verify the Lipschitz continuity of the skew-product cocycle defined on the phase space and the symbol space. Finally, we prove the boundedness of the expectation of some random variables and obtain the existence of a random uniform exponential attractor for the considered system.
- random uniform exponential attractor,
- Boussinesq lattice equations,
- multiplicative white noise,
- quasi-periodic forces
Citation: Ailing Ban. Asymptotic behavior of non-autonomous stochastic Boussinesq lattice system[J]. AIMS Mathematics, 2025, 10(1): 839-857. doi: 10.3934/math.2025040

Related Papers:

Abstract

In this paper, we investigate the existence of a random uniform exponential attractor for the non-autonomous stochastic Boussinesq lattice equation with multiplicative white noise and quasi-periodic forces. We first show the existence and uniqueness of the solution of the considered Boussinesq system. Then, we consider the existence of a uniform absorbing random set for a jointly continuous non-autonomous random dynamical system (NRDS) generated by the system, and make an estimate on the tail of solutions. Third, we verify the Lipschitz continuity of the skew-product cocycle defined on the phase space and the symbol space. Finally, we prove the boundedness of the expectation of some random variables and obtain the existence of a random uniform exponential attractor for the considered system.

References

[1]	A. Y. Abdallah, Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation, Abstr. Appl. Anal., 6 (2005), 655–671. http://dx.doi.org/10.1155/aaa.2005.655 doi: 10.1155/aaa.2005.655
[2]	P. W. Bates, K. Lu, B. Wang, Attractors for lattice dynamical systems, Int. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143–153. https://dx.doi.org/10.1142/S0218127401002031 doi: 10.1142/S0218127401002031
[3]	V. V. Chepyzhov, M. I. Vishik, Attractors for equations of mathematical physics, Providence: American Mathematical Society, 2002.
[4]	B. Wang, Dynamics of systems on infinite lattices, J. Diff. Equ., 221 (2006), 224–245. https://dx.doi.org/10.1016/j.jde.2005.01.003 doi: 10.1016/j.jde.2005.01.003
[5]	B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121–136. https://dx.doi.org/10.1016/j.jmaa.2006.08.070 doi: 10.1016/j.jmaa.2006.08.070
[6]	S. Zhou, W. Shi, Attractors and dimension of dissipative lattice system, J. Diff. Equ., 224 (2006), 172–204. https://dx.doi.org/10.1016/j.jde.2005.06.024 doi: 10.1016/j.jde.2005.06.024
[7]	A. Adili, B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser., 18 (2013), 643–666. http://dx.doi.org/10.3934/dcdsb.2013.18.643 doi: 10.3934/dcdsb.2013.18.643
[8]	L. Arnold, Random dynamical systems, Berlin: Springer, 1998. https://dx.doi.org/10.1007/978-3-662-12878-7
[9]	P. W. Bates, K. Lu, B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32–50. http://dx.doi.org/10.1016/j.physd.2014.08.004 doi: 10.1016/j.physd.2014.08.004
[10]	T. Caraballo, F. Morillas, J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Diff. Equ., 253 (2012), 667–693. http://dx.doi.org/10.1016/j.jde.2012.03.020 doi: 10.1016/j.jde.2012.03.020
[11]	H. Cui, M. Freitas, J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 3379–3407. http://dx.doi.org/10.3934/dcdsb.2017142 doi: 10.3934/dcdsb.2017142
[12]	H. Cui, J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Diff. Equ., 263 (2017), 1225–1268. http://dx.doi.org/10.1016/j.jde.2017.03.018 doi: 10.1016/j.jde.2017.03.018
[13]	X. Fan, Attractors for a damped stochastic wave equation of Sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767–793. http://dx.doi.org/10.1080/07362990600751860 doi: 10.1080/07362990600751860
[14]	X. Han, W. Shen, S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Diff. Equ., 250 (2011), 1235–1266. http://dx.doi.org/10.1016/j.jde.2010.10.018 doi: 10.1016/j.jde.2010.10.018
[15]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Diff. Equ., 253 (2012), 1544–1583. http://dx.doi.org/10.1016/j.jde.2012.05.015 doi: 10.1016/j.jde.2012.05.015
[16]	Z. Wang, S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwan J. Math., 20 (2016), 589–616. http://dx.doi.org/10.11650/tjm.20.2016.6699 doi: 10.11650/tjm.20.2016.6699
[17]	M. Zhao, S. Zhou, Random attractor of non-autonomous stochastic Boussinesq lattice system, J. Math. Phys., 56 (2015), 092702. http://dx.doi.org/10.1063/1.4930195 doi: 10.1063/1.4930195
[18]	Z. Li, W. Zhao, Stability of stochastic reaction-diffusion equation under random influences in high regular spaces, J. Math. Phys., 64 (2023), 081508. http://dx.doi.org/10.1063/5.0148290 doi: 10.1063/5.0148290
[19]	Q. Zhang, Well-posedness and dynamics of stochastic retarded FitzHugh-Nagumo Lattice systems, J. Math. Phys., 64 (2023), 121506. http://dx.doi.org/10.1063/5.0173334 doi: 10.1063/5.0173334
[20]	A. Eden, C. Foias, B. Nicolaenko, R. Temam, Exponential attractors for dissipative evolution equations, Amer. Math. Monthly, 37 (1995), 825–825. https://dx.doi.org/10.1137/1038025 doi: 10.1137/1038025
[21]	A. Y. Abdallah, Uniform exponential attractors for first-order non-autonomous lattice dynamical systems, J. Diff. Equ., 251 (2011), 1489–1504. http://dx.doi.org/10.1016/j.jde.2011.05.030 doi: 10.1016/j.jde.2011.05.030
[22]	A. N. Carvalho, S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141–1165. http://dx.doi.org/10.3934/cpaa.2014.13.1141 doi: 10.3934/cpaa.2014.13.1141
[23]	M. Efendiev, S. Zelik, A. Miranville, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A., 135 (2005), 703–730. http://dx.doi.org/10.1017/s030821050000408x doi: 10.1017/s030821050000408x
[24]	J. A. Langa, A. Miranville, J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329–1357. http://dx.doi.org/10.3934/dcds.2010.26.1329 doi: 10.3934/dcds.2010.26.1329
[25]	M. Zhao, S. Zhou, Exponential attractor for lattice system of nonlinear Boussinesq equation, Discrete. Dyn. Nat. Soc., 2013 (2013), 1–6. http://dx.doi.org/10.1155/2013/869621 doi: 10.1155/2013/869621
[26]	M. Zhao, S. Zhou, Pullback and uniform exponential attractors for nonautonomous Boussinesq lattice system, Int. J. Bifurc. Chaos, 25 (2015), 1–10. http://dx.doi.org/10.1142/S021812741550100X doi: 10.1142/S021812741550100X
[27]	S. Zhou, X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasi-periodic external forces, Nonlinear Anal., 78 (2013), 141–155. http://dx.doi.org/10.1016/j.na.2012.10.001 doi: 10.1016/j.na.2012.10.001
[28]	T. Caraballo, S. Sonner, Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383–6403. http://dx.doi.org/10.3934/dcds.2017277 doi: 10.3934/dcds.2017277
[29]	A. Shirikyan, S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Diff. Equ. Anal. Comput., 1 (2013), 241–281. http://dx.doi.org/10.1007/s40072-013-0007-1 doi: 10.1007/s40072-013-0007-1
[30]	H. Su, S. Zhou, L.Wu, Random exponential attractor for second order non-autonomous stochastic lattice dynamical systems with multiplicative white noise in weighted spaces, Adv. Diff. Equ., 2019 (2019), 1–21. http://dx.doi.org/10.1186/s13662-019-1983-x doi: 10.1186/s13662-019-1983-x
[31]	S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Diff. Equ., 263 (2017), 2247–2279. http://dx.doi.org/10.1016/j.jde.2017.03.044 doi: 10.1016/j.jde.2017.03.044
[32]	S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in R$^{3}$, J. Diff. Equ., 263 (2017), 6347–6383. http://dx.doi.org/10.1016/j.jde.2017.07.013 doi: 10.1016/j.jde.2017.07.013
[33]	Z. Han, S. Zhou, Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh-Nagumo lattice systems with quasi-periodic forces and multiplicative noise, Stoch. Dyn., 20 (2020), 2050036. http://dx.doi.org/10.1142/s0219493720500367 doi: 10.1142/s0219493720500367
[34]	R. Lin, M. Zhao, Random uniform exponential attractors for non-autonomous stochastic discrete long wave-short wave resonance equations, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 5775–5799. http://dx.doi.org/10.3934/dcdsb.2023077 doi: 10.3934/dcdsb.2023077
[35]	S. Zhang, S. Zhou, Random uniform exponential attractors for Schr$\ddot{o}$dinger lattice systems with quasi-periodic forces and multiplicative white noise, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 753–772. http://dx.doi.org/10.3934/dcdss.2022056 doi: 10.3934/dcdss.2022056
[36]	B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Diff. Equ., 268 (2019), 1–59. http://dx.doi.org/10.1016/j.jde.2019.08.007 doi: 10.1016/j.jde.2019.08.007
[37]	B. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104–132. http://dx.doi.org/10.1016/j.jmaa.2019.04.015 doi: 10.1016/j.jmaa.2019.04.015

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)