Weak Mean Random Attractor of Reversible Selkov Lattice Systems Driven By Locally Lipschitz Lévy Noises

Authors

Guofu Li School of Mathematical Sciences Guizhou Normal University Guiyang 550025, China https://orcid.org/0009-0002-5845-2934
Xiulan Yang School of Mathematical Sciences Guizhou Normal University Guiyang 550025, China https://orcid.org/0009-0007-8759-7709
Liguang Zhou School of Mathematical Sciences Guizhou Normal University Guiyang 550025, China https://orcid.org/0009-0006-9343-5530
Yan Wang School of Mathematical Sciences Guizhou Normal University Guiyang 550025, China https://orcid.org/0009-0005-8518-9395

https://doi.org/10.61383/ejam.20242165

Keywords:

Selkov systems, Weak pullback mean random attractor, Lévy noises

Abstract

This paper is concerned with weak pullback mean random attractor of reversible Selkov lattice systems defined on the entire integer set \(\mathbb{Z}\) driven by locally Lipschitz Lévy noises. Firstly, we formulate the stochastic lattice equations to an abstract system defined in the non-concrete space \(\ell^2\times\ell^2\) of square-summable sequences. Secondly, we establish the global well-posedness of the systems with locally Lipschitz diffusion terms. Under certain conditions, we show that the long-time dynamics can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space \(L^2(\Omega,\ell^2\times\ell^2)\). To overcome the difficulty caused by the drift and diffusion terms, we adopt a stopping time technique to prove the convergence of solutions in probability. The mean random dynamical systems theory proposed by Wang (J. Differ. Equ., 31:2177-2204, 2019) is used to deal with the difficulty caused by the nonlinear noise.

References

D. Applebaum, Lévy processes and stochastic calculus, Cambridge university press (2009), pp. 1-460 DOI: 10.1017/CBO9780511809781. DOI: https://doi.org/10.1017/CBO9780511809781

S. Peszat, J. Zabczyk, Stochastic partial differential equations with Lévy noise: An evolution equation approach, Cambridge University Press (2007), pp.1-415, DOI: 10.1017/cbo9780511721373. DOI: https://doi.org/10.1017/CBO9780511721373

D. Applebaum, M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, Journal of Applied Probability, 46 (2009), no.4, pp. 1116-1129, DOI: 10.1239/jap/1261670692. DOI: https://doi.org/10.1239/jap/1261670692

Z. Chen, D. Yang, S. Zhong, Weak mean attractor and periodic measure for stochastic lattice systems driven by Lévy noises, Stochastic Analysis and Applications, 41 (2023), no.3, pp. 509-544, DOI: 10.1080/07362994.2022.2038624. DOI: https://doi.org/10.1080/07362994.2022.2038624

X. Liu, J. Duan, J. Liu, P. E. Kloeden, Synchronization of systems of Marcus canonical equations driven by α-stable noises, Nonlinear Analysis: Real World Applications, 11 (2010), no.5, pp. 3437-3445, DOI: 10.1016/j.nonrwa.2009.12.004. DOI: https://doi.org/10.1016/j.nonrwa.2009.12.004

E.E. SelKov, Self-Oscillations in Glycolysis, A Simple Kinetic Model, European Journal of Biochemistry, 4 (1968), no.1, pp. 79-86, DOI: 10.1111/j.1432-1033.1968.tb00175.x. DOI: https://doi.org/10.1111/j.1432-1033.1968.tb00175.x

P. Richter, P. Rehmus, J. Ross, Control and dissipation in oscillatory chemical engines, Progress of Theoretical Physics, 66 (1981), no.2, pp. 385-405, DOI: 10.1143/PTP.66.385. DOI: https://doi.org/10.1143/PTP.66.385

J.C. Artés, J. Llibre, C. Valls, Dynamics of the Higgins-Selkov and Selkov systems, Chaos, Solitons & Fractals, 114 (2018), pp. 145-150, DOI: 10.1016/j.chaos.2018.07.007. DOI: https://doi.org/10.1016/j.chaos.2018.07.007

P. Gray, S. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chemical Engineering Science, 38(1983), no.1, pp. 29-43, DOI: 10.1016/0009-2509(83)80132-8. DOI: https://doi.org/10.1016/0009-2509(83)80132-8

P. Gray, S. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C, Chemical Engineering Science 39 (1984), no.6, pp. 1087-1097, DOI: 10.1016/0009-2509(84)87017-7. DOI: https://doi.org/10.1016/0009-2509(84)87017-7

Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), no.6, pp. 3049-3071, DOI: 10.1016/j.na.2011.12.002. DOI: https://doi.org/10.1016/j.na.2011.12.002

Y. You, Upper-Semicontinuity of Global Attractors for Reversible Schnackenberg Equations, Studies in Applied Mathematics, 130 (2013), no.3, pp. 232-263, DOI: 10.1111/j.1467-9590.2012.00565.x. DOI: https://doi.org/10.1111/j.1467-9590.2012.00565.x

Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete and continuous dynamical systems, 34 (2013), no.1, pp. 301-333, DOI: 10.3934/dcds.2014.34.301. DOI: https://doi.org/10.3934/dcds.2014.34.301

C. Guo, Y. Guo, X. Li, Upper semicontinuity of random attractors for nonautonomous stochastic reversible Selkov system with multiplicative noise, Advances in Mathematical Physics 2019, DOI: 10.1155/2019/2763245. DOI: https://doi.org/10.1155/2019/2763245

A. Gu, S. Zhou, Z. Wang, Uniform attractor of non-autonomous three-component reversible GrayScott system, Applied Mathematics and Computation, 219 (2013), no.16, pp. 8718-8729, DOI: 10.1016/j.amc.2013.02.056. DOI: https://doi.org/10.1016/j.amc.2013.02.056

A. Gu, Pullback-Attractor of Nonautonomous Three-Component Reversible Gray-Scott System on Unbounded Domains, Abstract and Applied Analysis, (2013), DOI: 10.1155/2013/719063. DOI: https://doi.org/10.1155/2013/719063

H. Li, A random dynamical system of the stochastic lattice reversible Selkov equations, Differential Equations and Applications (2019), DOI: 10.1109/ICEBE.2016.038. DOI: https://doi.org/10.1109/ICEBE.2016.038

L.O. Chua, T. Roska, The CNN paradigm, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40 (1993), no.3, pp. 147-156, DOI: 10.1109/81.222795. DOI: https://doi.org/10.1109/81.222795

J.P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM Journal on Applied Mathematics, 47 (1987), no.3, pp. 556-572, DOI: 10.1137/0147038. DOI: https://doi.org/10.1137/0147038

J.P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, Journal of theoretical biology, 148 (1991), no.1, pp. 49-82, DOI: 10.1016/S0022-5193(05)80465-5. DOI: https://doi.org/10.1016/S0022-5193(05)80465-5

Y. Wang, C. Guo, Y. Wu, Existence and stability of invariant/periodic measures of lattice reversible Selkov systems driven by locally Lipschitz noise, Revista de la Real Academia de Ciencias Exactas,

Físicas y Naturales. Serie A. Matemáticas, 118 (2024), no.1, pp. 42, DOI: 10.1007/s13398-023-01543-2. DOI: https://doi.org/10.1007/s13398-023-01543-2

V.S. Afraimovich, V.I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, International Journal of bifurcation and chaos, 4 (1994), no.03, pp. 631-637, DOI: 10.1142/S0218127494000459. DOI: https://doi.org/10.1142/S0218127494000459

P.W. Bates, X. Chen, A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM Journal on Mathematical Analysis, 35 (2003), no.2, pp. 520-546, DOI: 10.1137/S0036141000374002. DOI: https://doi.org/10.1137/S0036141000374002

S.N. Chow, J. Mallet-Paret, W. Shen, Traveling waves in lattice dynamical systems, Journal of differential equations, 149 (1998), no.2, pp. 248-291, DOI: 10.1006/jdeq.1998.3478. DOI: https://doi.org/10.1006/jdeq.1998.3478

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, Journal of differential equations, 96 (1992), no.1, pp. 1-27, DOI: 10.1016/0022-0396(92)90142-A. DOI: https://doi.org/10.1016/0022-0396(92)90142-A

S.N. Chow, W. Shen, Dynamics in a discrete Nagumo equation: spatial topological chaos, SIAM Journal on Applied Mathematics, 55 (1995), no.6, pp. 1764-1781, DOI: 10.1137/S0036139994261757. DOI: https://doi.org/10.1137/S0036139994261757

S.N. Chow, J. Mallet-Paret, E.S.V. Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random and Computational Dynamics, 4 (1996), no.2, pp. 109-178, Corpus ID: 18876239

W.J. Beyn, S.Y. Pilyugin, Attractors of reaction-diffusion systems on infinite lattices, Journal of Dynamics and Differential Equations, 15 (2003), no.2, pp. 485-515, DOI: 10.1023/B:JODY.0000009745.41889.30. DOI: https://doi.org/10.1023/B:JODY.0000009745.41889.30

N.I. Karachalios, N. Athanasios, Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrodinger equation, Journal of Differential Equations, 217 (2005), no.1, pp. 88-123, DOI: 10.1016/j.jde.2005.06.002. DOI: https://doi.org/10.1016/j.jde.2005.06.002

P.W. Bates, K. Lu, B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D: Nonlinear Phenomena, 289 (2014), pp. 32-50, DOI: 10.1016/j.physd.2014.08.004. DOI: https://doi.org/10.1016/j.physd.2014.08.004

T. Caraballo, K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Frontiers of Mathematics in China, 3 (2008), pp. 317-335, DOI: 10.1007/s11464-008-0028-7. DOI: https://doi.org/10.1007/s11464-008-0028-7

P.W. Bates, K. Lu, B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded do mains, Journal of Differential Equations, 246 (2009), no.2, pp. 845-869, DOI: 10.1016/j.jde.2008.05.017. DOI: https://doi.org/10.1016/j.jde.2008.05.017

I. Chueshov, M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), no.2, pp. 127-144, DOI: 10.1080/1468936042000207792. DOI: https://doi.org/10.1080/1468936042000207792

J, Huang, W. Shen, Pullback attractors for nonautonomous and random parabolicequations on non smooth domains, Discrete and Continuous Dynamical Systems 24 (2009), no.3, pp. 855-882, DOI: 10.3934/dcds.2009.24.855. DOI: https://doi.org/10.3934/dcds.2009.24.855

F. Flandoli, B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multi plicative white noise, Stochastics: An International Journal of Probability and Stochastic Processes 59 (1996), no. 1-2, pp. 21-45, DOI: 10.1080/17442509608834083. DOI: https://doi.org/10.1080/17442509608834083

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact ran dom dynamical systems, Journal of Differential Equations, 253 (2012), no.5, pp. 1544-1583, DOI: 10.1016/j.jde.2012.05.015. DOI: https://doi.org/10.1016/j.jde.2012.05.015

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and continuous dynamical systems, 34 (2013), no.1, pp.1 269-300, DOI: 10.3934/dcds.2014.34.269. DOI: https://doi.org/10.3934/dcds.2014.34.269

P.W. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics and Dynamics, 6 (2006), no.01, pp. 1-21, DOI: 10.1142/S0219493706001621. DOI: https://doi.org/10.1142/S0219493706001621

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, Journal of Dynamics and Differential Equations, 31 (2019), no.4, pp. 2177-2204, DOI: 10.1007/s10884-018-9696 5. DOI: https://doi.org/10.1007/s10884-018-9696-5

R. Wang, B. Wang, Global well-posedness and long-term behavior of discrete reaction-diffusion equations driven by superlinear noise, Stochastic Analysis and Applications, 39 (2021), no.4, pp. 667-696, DOI: 10.1080/07362994.2020.1828917. DOI: https://doi.org/10.1080/07362994.2020.1828917

Z. Brzeniak, T. Caraballo, J.A. Langa, Y. Li, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, Journal of Differential Equations, 255 (2013), no.11, pp. 3897-3919, DOI: 10.1016/j.jde.2013.07.043. DOI: https://doi.org/10.1016/j.jde.2013.07.043

D. Li, B. wang, X. wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on unbounded thin domains, Journal of Mathematical Physics 60 (2019), no.3, pp.1575-1602, DOI: 10.1016/j.jde.2016.10.024. DOI: https://doi.org/10.1016/j.jde.2016.10.024

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in R3, Journal of Differential Equations, 263 (2017), no.10, pp. 6347-6383, DOI: 10.1016/j.jde.2017.07.013. DOI: https://doi.org/10.1016/j.jde.2017.07.013

J. Xu, T. Caraballo, Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and WongZakai Approximations, SIAM Journal on Mathematical Analysis, 54 (2022), no.3, pp. 2792-2844, DOI: 10.1137/21M1412645. DOI: https://doi.org/10.1137/21M1412645

X. Wang, K. Lu, B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM Journal on Applied Dynamical Systems, 14 (2015), no.2, pp. 1018-1047, DOI: 10.1137/140991819. DOI: https://doi.org/10.1137/140991819

X. Wang, K. Lu, B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, Journal of Differential Equations, 264 (2018), no.1, pp. 378-424, DOI: 10.1016/j.jde.2017.09.006. DOI: https://doi.org/10.1016/j.jde.2017.09.006

B. Wang, Well-posedness and long-term behavior of supercritical wave equations driven by nonlinear colored noise on Rn, Journal of Functional Analysis, 283 (2022), no.2, pp. 109498, DOI: 10.1016/j.jfa.2022.109498. DOI: https://doi.org/10.1016/j.jfa.2022.109498

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on R3, Transactions of the American Mathematical Society, 363 (2011), no.7, pp. 3639-3663, DOI: 10.1090/S0002-9947-2011 05247-5. DOI: https://doi.org/10.1090/S0002-9947-2011-05247-5

J.C. Robinson, Stability of random attractors under perturbation and approximation, Journal of Differential Equations, 186 (2002), no.2, pp. 652-669, DOI: 10.1016/S0022-0396(02)00038-4. DOI: https://doi.org/10.1016/S0022-0396(02)00038-4

B. Wang, Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise, Journal of Mathematical Analysis and Applications, 477 (2019), no.1, pp. 104-132, DOI: 10.1016/j.jmaa.2019.04.015. DOI: https://doi.org/10.1016/j.jmaa.2019.04.015

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Published

2024 Mar 25

How to Cite

[1]

G. Li, X. Yang, L. Zhou, and Y. Wang, “Weak Mean Random Attractor of Reversible Selkov Lattice Systems Driven By Locally Lipschitz Lévy Noises”, Electron. J. Appl. Math., vol. 2, no. 1, pp. 40–63, Mar. 2024.