格点时滞反应扩散方程的拉回吸引子与不变测度

Abstract

Abstract:

In this article, the authors study the pullback attractor and invariant measures for retarded lattice reaction-diffusion equations. They first prove the global well-posedness of the addressed problem, and then show that the solution mappings generates a continuous process possessing a pullback attractor. Afterwards, they construct a family of invariant Borel probability measures for the process via the pullback attractor and the notion of generalized Banach limit.

Key words: retarded reaction-diffusion equations, pullback attractors, invariant measures

CLC Number:

O.175.8

Ni Siyan, Zou Tianfang, Zhao Caidi. Pullback Attractors and Invariant Measures for Retarded Lattice Reaction-Diffusion Equations[J].Acta mathematica scientia,Series A, 2025, 45(4): 1128-1143.

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References 34

[1]	Evans L C. Partial Differential Equations Providence, RI: American Mathematical Society, 2022
[2]	叶其孝. 反应扩散方程引论. 北京: 科学出版社, 2019
	Ye Q X. Introduction to Reaction-Diffusion Equations. Beijing: Science Press, 2019
[3]	Wu J. Theory and Applications of Partial Functional Differential Equations. New York: Springer Science and Business Media, 1996
[4]	Chate H, Courbage M. Lattice systems. Physica D: Nonlinear Phenomena, 1997, 1.3: 1-612
[5]	Keener J P. Propagation and its failure in coupled systems of discrete excitable cells. Journal of Mathematical Chemistry, 1987, 47: 556-572
[6]	Winslow R L, Kimball A L, Varghese A, et al. Simulating cardiac sinus and atrial network dynamics on the connection machine. Physica D: Nonlinear Phenomena, 1993, 64: 281-298
[7]	Erneux T, Nicolis G. Propagating waves in discrete bistable reaction-diffusion systems. Physica D: Nonlinear Phenomena, 1993, 67: 237-244
[8]	Kapral R. Discrete models for chemically reacting systems. Journal of Mathematical Chemistry, 1991, 6: 113-163
[9]	Pecora L M, Carroll T L. Synchronization in chaotic systems. Physical Review Letters, 1990, 64: 821
[10]	Fabiny L, Colet P, Roy R, et al. Coherence and phase dynamics of spatially coupled solid-state lasers. Physical Review A, 1993, 47: 4287-4298 pmid: 9909435
[11]	Hillert M. A solid-solution model for inhomogeneous systems. Acta Metallurgica, 1961, 9: 525-535
[12]	Chow S N, Paret J M, Van Vleck E S. Pattern formation and spatial chaos in spatially discrete evolution equations Random Comput. Random and Computational Dynamics, 1996, 4: 109-178
[13]	Abdallah A. Uniform exponential attractor for first order non-autonomous lattice dynamical systems. Journal of Differential Equations, 2011, 2.1: 1489-1504
[14]	Bates P, Lu K, Wang B. Attractors of non-autonomous stochastic lattice systems in weighted spaces. Physica D: Nonlinear Phenomena, 2014, 2.9: 32-50
[15]	Caraballo T, Morillas F, Valero J. Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearity. Journal of Differential Equations, 2012, 2.3: 667-693
[16]	Caraballo T, Morillas F, Valero J. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34: 51-77
[17]	Wang X, Lu K, Wang B. Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise. Journal of Dynamics and Differential Equations, 2016, 28: 1309-1355
[18]	Zhou S. Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise. Journal of Differential Equations, 2017, 2.3: 2247-2279
[19]	Chekroun M, Glatt-Holtz N. Invariant measures for dissipative dynamical systems: abstract results and applications. Communications in Mathematical Physics, 2012, 3.6(3): 723-761
[20]	Łukaszewicz G, Robinson J C. Invariant measures for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34: 4211-4222
[21]	Li X, Shen W, Sun C. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete and Continuous Dynamical Systems-B, 2017, 22(6): 2427-2446
[22]	Moussa G, Cláudia B. Invariant measures for multivalued semigroups. Journal of Mathematical Analysis and Applications, 2017, 4.5(2): 1234-1248
[23]	李永军, 桑燕苗, 赵才地. 一阶格点系统的不变测度与 Liouville 型方程. 数学物理学报, 2020, 40A(2): 328-339
	Li Y J, Sang Y M, Zhao C D. Invariant measures and Liouville type theorem for first-order lattice systems. Acta Math Sci, 2020, 40A(2): 328-339
[24]	Zhao C, Wang J, Caraballo T. Invariant sample measures and random Liouville type theorem for the two-dimensional stochastic Navier-Stokes equations. Journal of Differential Equations, 2022, 3.7: 474-494
[25]	邹天芳, 赵才地. 加权空间中一阶格点系统的统计解及其 Kolmogorov 熵. 数学物理学报, 2023, 43A(5): 1559-1574
	Zhou T F, Zhao C D. Statistical solutions and Kolmogorov entropy for first-order lattice systems in weighted spaces. Acta Math Sci, 2023, 43A(5): 1559-1574
[26]	Zhao C, Zhuang R. Statistical solutions and Liouville theorem for the second order lattice systems with varying coefficients. Journal of Differential Equations, 2023, 3.2: 194-234
[27]	赵才地, 李艳娇, 阳玲, 等. Ladyzhenskaya 流体力学方程组的拉回吸引子与不变测度. 数学学报, 2018, 61: 1-12
	Zhao C D, Li Y J, Yang L, et al. Pullback attractors and invariant measures for Ladyzhenskaya model. Acta Mathematica Sinica, 2018, 61: 823-834
[28]	杨虎军, 韩晓玲, 赵才地. 三维热带气候模型轨道统计解的存在性, 退化正则性与极限行为. 数学学报, 2025, 68(2): 325-349 doi: 10.12386/A20230094
	Yang H J, Han X L, Zhao C D. Existence, degenerate regularity and limit behavior of trajectory statistical solution for the 3D tropical climate model. Acta Mathematica Sinica, 2025, 68(2): 325-349 doi: 10.12386/A20230094
[29]	Boyer F, Fabrie P. Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. New York: Springer, 2013
[30]	Zhao C. Absorbing estimate implies trajectory statistical solutions for nonlinear elliptic equations in half-cylindrical domains. Mathematische Annalen, 2025, 3.1: 1711-1730
[31]	Carvalho A, Langa J A, Robinson J C. Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. New York: Springer, 2013
[32]	Caraballo T, Morillas F, Valero J. Attractors for non-autonomous retarded lattice dynamical systems. Nonautonomous Dynamical Systems, 2015, 2: 31-51
[33]	Foias C, Manley O, Rosa R, Temam R. Navier-Stokes Equations and Turbulence. Cambridge: Cambridge University Press, 2001
[34]	Yang H, Han X, Zhao C. Pullback dynamics and statistical solutions for dissipative non-autonomous Zakharov equations. Journal of Differential Equations, 2024, 3.0: 1-57

Pullback Attractors and Invariant Measures for Retarded Lattice Reaction-Diffusion Equations

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