Weak Pullback Mean Attractor for \(p\)-Laplacian Selkov Lattice Systems with Locally Lipschitz Delay Diffusion Terms

Authors

Yan Wang School of Mathematics Science Guizhou Normal University Guiyang 550025, China
Xiaolan Qin School of Mathematics Science Guizhou Normal University Guiyang 550025, China
Hailang Bai School of Mathematics Science Guizhou Normal University Guiyang 550025, China
Yu Wang School of Mathematics Science Guizhou Normal University Guiyang 550025, China

https://doi.org/10.61383/ejam.20231233

Keywords:

Nonlinear p-Laplacian, Weak pullback mean attractor, Selkov systems, Dalay time, Lipschitze noise

Abstract

This paper focuses on the dynamics of a class of nonlinear, reversible, random \(p\)-Laplace Selkov delay lattice systems defined by local lipschitz noise-driven \(\mathbb{Z}^d\). We first establish the global fitness of the system using the local Lipschitz delayed diffusion term. Under certain conditions, we demonstrate the existence and uniqueness of the mean stochastic dynamical system in relation to the stochastic equation in the product Hilbert space \(L^2(\Omega, \mathcal{F}_\tau; \ell^2\times\ell^2） \times L^2 （\Omega， \mathcal{F}_\tau; L^2((-\rho, 0), \ell^2\times\ell^2) \). The average stochastic dynamical system theory proposed by Wang (J.Equ., 31:2177-2204, 2019) is used to deal with the difficulties caused by nonlinear noise. Even if the discrete \(p\)-Laplace is replaced by the usual discrete Laplace, the results of this paper are new.

References

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

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

T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst. 34 (2014), no. 1, pp. 51-77, DOI: 10.3934/dcds.2014.34.51. DOI: https://doi.org/10.3934/dcds.2014.34.4019

E. Van Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D: Nonlinear Phenom. 212 (2005), no. 3-4, pp. 317-336, DOI: 10.1016/j.physd.2005.10.006. DOI: https://doi.org/10.1016/j.physd.2005.10.006

T. Caraballo, K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China 3 (2008), no. 3, pp. 317-335, DOI: 10.1007/s11464-008-0028-7. DOI: https://doi.org/10.1007/s11464-008-0028-7

T. Caraballo, F. Morillas and J. Valero, Pullback attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity, J. Differ. Equ. 253 (2012), no. 2, pp. 667-693, DOI: 10.1016/j.jde.2012.03.020. DOI: https://doi.org/10.1016/j.jde.2012.03.020

B. Wang, Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise, J. Math. Anal. Appl. 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

Z. Chen and B. Wang,Weak mean attractors and invariant measures for stochastic Schr¨odinger delay lattice systems, J. Dyn. Differ. Equ. (2021), pp. 1-40, DOI: 10.1007/s10884-021-10085-3. DOI: https://doi.org/10.1007/s10884-021-10085-3

Z. Chen, D. Yang, S. Zhong, Limiting Dynamics for Stochastic FitzHugh-Nagumo Lattice Systems in weighted spaces, J. Dyn. Differ. Equ. (2022), DOI: 10.1007/s10884-022-10145-2. DOI: https://doi.org/10.1007/s10884-022-10145-2

R. Wang and B. Wang, Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stoch. Process. Their Appl 130 (2020), no. 12, pp. 7431-7462, DOI:10.1016/j.spa.2020.08.002. DOI: https://doi.org/10.1016/j.spa.2020.08.002

J.C. Artes, J. Llibre and C. Valls, Dynamics of the Higgins-Selkov and Selkov systems, Chaos Solit. 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

S. Dhatt and P. Chaudhury, Study of oscillatory dynamics in a Selkov glycolytic model using sensitivity analysis, Indian J. Phys. 96 (2022), no. 6, pp. 1649-1654, DOI: 10.1007/s12648-021-02102-4. DOI: https://doi.org/10.1007/s12648-021-02102-4

A. Goldbeter and J. Keizer, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Phys. Today 1 March 51 (1998), no. 3, pp. 86-86, DOI: 10.1063/1.882194. DOI: https://doi.org/10.1063/1.882194

E.E. Selkov, Self-oscillations in glycolysis, Eur. J. Biochem 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

J. Llibre, A. Nabavi, Phase portraits of the Selkov model in the Poincare disc, Discrete Contin. Dyn. Syst. -B 27 (2022), no. 12, pp. 7607-7623, DOI: 10.3934/dcdsb.2022056. DOI: https://doi.org/10.3934/dcdsb.2022056

Y. Wang, C. Gun and R. Wang, Limiting behavior of invariant or periodic measures of lattice Peversible selkov systems driven by locally lipschitz noise, (2023), submitted.

H. Li, Attractors for the Stochastic Lattice Selkov Equations with Additive Noises, J. Appl. Math. and Physics 7 (2019), no. 6, pp. 1329-1339, DOI: 10.4236/jamp.2019.76090. DOI: https://doi.org/10.4236/jamp.2019.76090

H. Li, Random Attractor of the Stochastic Lattice Reversible Selkov Equations with Additive Noises, IEEE International Conference on E-business Engineering, (2016), pp. 176-181, DOI: 10.1109/ICEBE.2016.038. DOI: https://doi.org/10.1109/ICEBE.2016.038

Y. You, Asymptotical dynamics of Selkov equations, Discrete Contin. Dyn. Syst. -S 2 (2009), no. 1, pp. 193-219, DOI: 10.3934/dcdss.2009.2.193. DOI: https://doi.org/10.3934/dcdss.2009.2.193

C. Guo, Y. Guo, X. Li, Upper Semicontinuity of Random Attractors for Nonautonomous Stochastic Reversible Selkov System with Multiplicative Noise, Adv. Math. Phys. 4 (2019), pp. 1-15, DOI: 10.1155/2019/2763245. DOI: https://doi.org/10.1155/2019/2763245

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields 100 (1994), no. 3, pp. 365-393, DOI: 10.1007/BF01193705. DOI: https://doi.org/10.1007/BF01193705

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. stoch. rep. 59 (1996), no. 1, pp. 21-45, DOI: 10.1080/17442509608834083. DOI: https://doi.org/10.1080/17442509608834083

Z. Brzezniak, T. Caraballo, J.A. Langa, Y. Li, G. Lukaszewicz and J. Real, Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains, J. Differ. Equ. 255 (2013), no. 11, pp. 3897-3919, DOI: DOI:10.1016/j.jde.2013.07.043. DOI: https://doi.org/10.1016/j.jde.2013.07.043

T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuss and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst. - S 21 (2008), no. 2, pp. 415-443. DOI: https://doi.org/10.3934/dcds.2008.21.415

T. Caraballo, J. Real and I.D. Chueshov, Pullback attractors for stochastic heat equationsin materials with memory, Discrete Contin. Dyn. Syst. - B 9 (2008), no. 3, pp. 525-539, DOI: 10.3934/dcdsb.2008.9.525. DOI: https://doi.org/10.3934/dcdsb.2008.9.525

T. Caraballo and J.A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 10 (2003), no. 4, pp. 491-513, DOI: 10.1016/S0166-218X(03)00183-5. DOI: https://doi.org/10.1016/S0166-218X(03)00183-5

T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. - B 14 (2010), no. 2, pp. 439-455, DOI: 10.3934/dcdsb.2010.14.439. DOI: https://doi.org/10.3934/dcdsb.2010.14.439

T. Caraballo, M.J. Garrido-Atienza and T.Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations witha fractional Brownian motion, Adv. Nonlinear Anal. 74 (2011), no. 11, pp. 3671-3684, DOI: 10.1016/j.na.2011.02.047. DOI: https://doi.org/10.1016/j.na.2011.02.047

T. Caraballo, J.A. Langa, V.S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic mtltivalued dynamical systems, Set-Valued Analysis 11 (2003), no. 2, pp. 153-201, DOI: 10.1023/A:1022902802385. DOI: https://doi.org/10.1023/A:1022902802385

T. Caraballo, P.E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evo-lution equations and their perturbation, Appl. Math. Optim. 50 (2004), no. 3, pp. 183-207, DOI:10.1007/s00245-004-0802-1. DOI: https://doi.org/10.1007/s00245-004-0802-1

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. R. Soc. Edinb. A: Math. 151 (2021), no. 6, pp. 1-31, DOI: 10.1017/prm.2020.77. DOI: https://doi.org/10.1017/prm.2020.77

A. Gu, D. Li, B. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on RN , J. Differ. Equ. 264 (2018), no. 12, pp. 7094-7137, DOI: 10.1016/j.jde.2018.02.011. DOI: https://doi.org/10.1016/j.jde.2018.02.011

Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochasticsemilinear Laplacian equations, J. Differ. Equ. 258 (2015), no. 2, pp. 504-534, DOI: 10.1016/j.jde.2014.09.021. DOI: https://doi.org/10.1016/j.jde.2014.09.021

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dyn. Differ. Equ. 31 (2019), no. 4, pp. 2177-2204, DOI:10.1007/s10884-018-9696-5. DOI: https://doi.org/10.1007/s10884-018-9696-5

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on R3, Discrete Contin. Dyn. Syst. -B 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

B. Wang, Weak pullback attractors for stochastic Navier-Stokes equations with nonlinear diffusion terms, Proc. Amer. Math. Soc. 147 (2019), no. 4, pp. 1627-1638, DOI: 10.1090/proc/14356. DOI: https://doi.org/10.1090/proc/14356

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equationon unbounded domains, J. Differ. Equ. 246 (2009), no. 6, pp. 2506-2537, DOI: 10.1016/j.jde.2008.10.012 DOI: https://doi.org/10.1016/j.jde.2008.10.012

B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differ. Equ. 268 (2019), no. 1, pp. 1-59, DOI: 10.1016/j.jde.2019.08.007. DOI: https://doi.org/10.1016/j.jde.2019.08.007

R. Wang, B. Guo and B. Wang, Well-posedness and dynamics of fractional Fitz Hugh-Nagumo systems on RN driven by nonlinear noise, Sci. China Math. 64 (2020), no. 11, pp. 1-42, DOI: 10.1007/s11425-019-1714-2. DOI: https://doi.org/10.1007/s11425-019-1714-2

R. Wang, L. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on RN , Nonlinearity 32 (2019),no. 11, pp. 4524-4556, DOI: 10.1088/1361-6544/ab32d7. DOI: https://doi.org/10.1088/1361-6544/ab32d7

R. Wang, B. Guo, W. Liu and D.T. Nguyen, Fractal dimension of random invariant sets and regular random attractors for stochastic hydrodynamical equations, Math. Ann. (2023), DOI: https://doi.org/10.1007/s00208-023-02661-3. DOI: https://doi.org/10.1007/s00208-023-02661-3

R. Wang, K. Kinra and M.T. Mohan, Asymptotically autonomous robustness in probability of random attractors for stochastic Navier-Stokes equations on unbounded Poincar’e domains, SIAM J. Math. Anal. 55 (2023), no. 4, pp. 2644-2676, DOI: 10.48550/arXiv.2208.06808. DOI: https://doi.org/10.1137/22M1517111

J. Xu, T. Caraballo and J. Valero, Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion, J. Differ. Equ. 327 (2022), pp. 418-447, DOI: 10.1016/j.jde.2022.04.033. DOI: https://doi.org/10.1016/j.jde.2022.04.033

J. Xu, Z. Zhang and T. Caraballo, Mild Solutions to Time Fractional Stochastic 2D-Stokes Equations with Bounded and Unbounded Delay, J. Dyn. Differ. Equ. 34 (2022), no. 1, pp. 583-603, DOI: 10.1007/s10884-019-09809-3. DOI: https://doi.org/10.1007/s10884-019-09809-3

C. Zhao, Y. Li and T. Caraballo, Trajectory statistical solutions and Liouville type equations for evolution equations: Abstract results and applications, J. Differ. Equ. 269 (2020), no. 1, pp. 467-494, DOI: 10.1016/j.jde.2019.12.011. DOI: https://doi.org/10.1016/j.jde.2019.12.011

C. Zhao and S. Zhou. Pullback trajectory attractors for evolution equations and application to 3D incompressible non-Newtonian fluid, Nonlinearity 21 (2008), no. 8, pp. 1691-1718, DOI: 10.1088/0951-7715/21/8/002. DOI: https://doi.org/10.1088/0951-7715/21/8/002

C. Zhao and W. Sun, Global well-posedness and pullback attractors for a two dimensional non-autonomous micropolar fluid flows with ininite delays, Commun. Math. Sci. 15 (2017), no. 1, pp. 97-121, DOI: 10.4310/CMS.2017.v15.n1.a5. DOI: https://doi.org/10.4310/CMS.2017.v15.n1.a5

C. Zhao, G. Liu and R. An, Global well-posedness and pullback attractors for an incompressible non-Newtonian fluid flows with infinite delays, Differ. Equations Dyn. Syst. 25 (2017), no. 1, 39-64, DOI: 10.1007/s12591-014-0231-9. DOI: https://doi.org/10.1007/s12591-014-0231-9

C. Zhao and L. Yang, Pullback attractor and invariant measures for three dimensional globally modified Navier-Stokes equations, Commun. Math. Sci. 15 (2017), no. 6, pp. 1565-1580, DOI: 10.4310/CMS.2017.v15.n6.a4. DOI: https://doi.org/10.4310/CMS.2017.v15.n6.a4

Z. Zhu and C. Zhao, Pullback attractors and invariant measures for three dimensional regularized MHD equations, Discrete Contin. Dyn. Syst. 38 (2018), no. 3, pp. 1461-1477, DOI: 10.3934/dcds.2018060. DOI: https://doi.org/10.3934/dcds.2018060

Z. Zhu, Y. Sang and C. Zhao, Pullback attractor and invariant measures for the discrete Zakharov equations, J. Appl. Anal. Comput. 9(2019), no. 6, pp. 2333-2357, DOI:10.11948/20190091. DOI: https://doi.org/10.11948/20190091

X. Wang, K. Lu and B. Wang, Exponential Stability of Non-Autonomous Stochastic Delay Lattice Systems with Multiplicative Noise, J. Dyn. Differ. Equ. 28 (2016), no. 3-4, pp. 1309-1335, DOI: 10.1007/s10884-015-9448-8. DOI: https://doi.org/10.1007/s10884-015-9448-8

W.P. Yan, Y. Li and S.G. Ji, Random attractors for first order stochastic retarded lattice dynamical systems, J. Math. Phys. 51 (2010), no. 3, 032702-032702-17, DOI: 10.1063/1.3319566. DOI: https://doi.org/10.1063/1.3319566

C. Zhao, G. Xue, and G. Lukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrodinger equations, Discrete Contin. Dyn. Syst. -B 23 (2018), no. 9, pp. 4021-4044, DOI: 10.3934/dcdsb.2018122. DOI: https://doi.org/10.3934/dcdsb.2018122

J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Q. Appl. Math. 42 (1984), no. 1, pp. 1-14, DOI: 10.1090/qam/73650421. DOI: https://doi.org/10.1090/qam/736501

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

T.L. Carrol and L.M. Pecora, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) , no. 8, pp. 821-824, DOI: 10.1103/PhysRevLett.64.821. DOI: https://doi.org/10.1103/PhysRevLett.64.821

B. Wang, Dynamics of systems on infinite lattices, J. Differ. Equ. 221 (2006), no. 1, a pp. 224-245, DOI: 10.1016/j.jde.2005.01.003. DOI: https://doi.org/10.1016/j.jde.2005.01.003

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equations 9 (1997) pp. 307-341, DOI: 10.1007/BF02219225. DOI: https://doi.org/10.1007/BF02219225

P.E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differ. Equ. 253 (2012), no. 5, pp. 1422-1438, DOI:10.1016/j.jde.2012.05.016. DOI: https://doi.org/10.1016/j.jde.2012.05.016

A. Gu and Y. Li, Sufficient Criteria for Existence of Pullback Attractors for Stochastic Lattice Dynamical Systems with Deterministic Non-autonomous Terms, Eprint Arxiv (2014), DOI:10.48550/arXiv.1404.0488.

R. Wang, Y. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst. 39 (2019), no. 7, pp. 4091-4126, DOI: 10.3934/dcds.2019165. DOI: https://doi.org/10.3934/dcds.2019165

C. Zhao and S. Zhou, Attractors of retarded first order lattice systems, Nonlinearity 20 (2007)no. 8, pp. 1987-2006, DOI: 10.1088/0951-7715/20/8/010. DOI: https://doi.org/10.1088/0951-7715/20/8/010

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differ. Equ. 263 (2017), no. 4, pp. 2247-2279, DOI: 10.1016/j.jde.2017.03.044. DOI: https://doi.org/10.1016/j.jde.2017.03.044

B. Wang, R. Wang, Asymptotic behavior of stochastic Schrodinger lattice systems driven by nonlinear noise, Stochastic Anal. Appl. 38 (2019), no. 5, pp. 1-25. DOI: https://doi.org/10.1080/07362994.2019.1679646

R. Wang, Long-Time Dynamics of Stochastic Lattice Plate Equations with Nonlinear Noise and Damping, J. Dyn. Differ. Equ. 33 (2021), pp. 767-803, DOI:10.1007/s10884-020-09830-x. DOI: https://doi.org/10.1007/s10884-020-09830-x

R. Wang and B. Wang, Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise, Discrete Contin. Dyn. Syst.-B 25 (2020), no. 7, pp. 2461-2493, DOI: 10.3934/dcdsb.2020019. DOI: https://doi.org/10.3934/dcdsb.2020019

X. Wang and S. Zhou, Random attractors for non-autonous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficient, Taiwan. J. Math. 20 (2016), no. 3, pp. 589-616, DOI:10.11650/tjm.20.2016.6699. DOI: https://doi.org/10.11650/tjm.20.2016.6699

R. Wang and N.H. Tuan, Multi-parameter stability of evolution systems of probability measures for highly nonlinear p-Laplacian Ito Schr ¨odinger equations on Zd with delay, 2023, submitted.

R. Wang and B. Wang, Random dynamics of p -Laplacian lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Processes Appl. 130 (2020), no. 12, pp. 7431-7462, DOI: 10.1016/j.spa.2020.08.002. DOI: https://doi.org/10.1016/j.spa.2020.08.002

Downloads

Published

2023 Sep 09

How to Cite

[1]

Y. Wang, X. QIN, H. Bai, and Y. Wang, “Weak Pullback Mean Attractor for \(p\)-Laplacian Selkov Lattice Systems with Locally Lipschitz Delay Diffusion Terms”, Electron. J. Appl. Math., vol. 1, no. 2, pp. 1–17, Sep. 2023.