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Acta Applicandae Mathematicae

, Volume 114, Issue 3, pp 157–172 | Cite as

Exponential Attractors for Lattice Dynamical Systems in Weighted Spaces

  • Xiaojun LiEmail author
  • Kaijin Wei
  • Haiyun Zhang
Article

Abstract

The aim of this paper is to investigate the existence of exponential attractors for lattice reaction-diffusion systems in weighted spaces \(l_{\sigma}^{2}\) and for partly dissipative lattice reaction-diffusion systems in weighted spaces \(l_{\mu}^{2}\times l_{\mu}^{2}\), respectively. In contrast to the previous work by Abdallah in J. Math. Anal. Appl. 339, 217–224 (2008) and Commun. Pure Appl. Anal. 8, 803–818 (2009), we get the existence of exponential attractors for lattice dynamical systems in the weak topology spaces.

Keywords

Lattice dynamical system Squeezing property Exponential attractor Weighted space 

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References

  1. 1.
    Abdallah, A.Y.: Exponential attractors for first-order lattice dynamical systems. J. Math. Anal. Appl. 339, 217–224 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abdallah, A.Y.: Exponential attractors for second order lattice dynamical systems. Commun. Pure Appl. Anal. 8, 803–818 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Babin, A., Nicolaenko, B.: Exponential attractors of reaction-diffusion systems in unbounded domains. J. Dyn. Differ. Equ. 7, 567–590 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Babin, A.V., Vishik, M.I.: Attractors of partial differential evolution equations in an unbounded domain. Proc. R. Soc. Edinb. A 116, 221–243 (1990) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bates, P.W., Lu, K., Wang, B.: Attractors for lattice dynamical systems. Int. J. Bifurc. Chaos 11(1), 143–153 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bell, J., Cosner, C.: Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Q. Appl. Math. 42, 1–14 (1984) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Beyn, W.J., Pilyugin, S.Y.: Attractors of reaction diffusion systems on infinite lattices. J. Dyn. Differ. Equ. 15, 485–515 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cahn, J.W.: Theory of crystal growth and interface motion in crystalline materials. Acta Metall. 8, 554–562 (1960) CrossRefGoogle Scholar
  9. 9.
    Carrol, T.L., Pecora, L.M.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chow, S.N., Mallet-Paret, J.: Pattern formation and spatial chaos in lattice dynamical systems, I, II. IEEE Trans. Circuits Syst. 42, 746–751 (1995) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chow, S.N., Mallet-Paret, J., Van Vleck, E.S.: Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comput. Dyn. 4, 109–178 (1996) zbMATHGoogle Scholar
  12. 12.
    Chow, S.N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 49, 248–291 (1998) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chua, L.O., Yang, Y.: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35, 1257–1272 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dung, L., Nicolaenko, B.: Exponential attractors in Banach spaces. J. Dyn. Differ. Equ. 13, 791–806 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations, Res. Appl. Math., vol. 37. Masson/Wiley Co-publication, Paris (1994) zbMATHGoogle Scholar
  16. 16.
    Eden, A., Foias, C., Kalantarov, V.: A remark on two constructions of exponential attractors for α-contractions. J. Dyn. Differ. Equ. 10, 37–45 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Efendiev, M.A., Zelik, S.V.: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Commun. Pure Appl. Math. 54(6), 625–688 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors and finite-dimensional reduction for nonautonomous dynamical systems. Proc. R. Soc. Edinb. A 13, 703–730 (2005) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction diffusion systems. Physica D 67, 237–244 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fabiny, L., Colet, P., Roy, R.: Coherence and phase dynamics of spatially coupled solid-state lasers. Phys. Rev. A 47, 4287–4296 (1993) CrossRefGoogle Scholar
  21. 21.
    Fabrie, P., Galusinski, C., Miranville, A., Zelik, S.: Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete Contin. Dyn. Syst. 10, 211–238 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Keener, J.P.: The effects of discrete gap junction coupling on propagation in myocardium. J. Theor. Biol. 148, 49–82 (1991) CrossRefGoogle Scholar
  23. 23.
    Li, X., Wang, B.: Attractors for partly dissipative lattice dynamic systems in weighted spaces. J. Math. Anal. Appl. 325, 141–156 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Li, X., Zhong, C.: Attractors for partly dissipative lattice dynamic systems in l 2×l 2. J. Comput. Appl. Math. 177, 159–174 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Rodríguez-Bernal, A., Wang, B.: Attractors for partly dissipative reaction-diffusion system in ℝn. J. Math. Anal. Appl. 252, 790–803 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997) zbMATHGoogle Scholar
  27. 27.
    Vleck, E.V., Wang, B.: Attractors for lattice FitzHugh-Nagumo systems. Physica D 212, 317–336 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Wang, B.: Attractors for reaction-diffusion equations in unbounded domain. Physica D 128, 41–52 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Wang, B.: Dynamics of systems on infinite lattices. J. Differ. Equ. 221, 224–245 (2006) zbMATHCrossRefGoogle Scholar
  30. 30.
    Zhou, S.: Attractors for second order lattice dynamical systems. J. Differ. Equ. 179, 605–624 (2002) zbMATHCrossRefGoogle Scholar
  31. 31.
    Zhou, S.: Attractors for first order dissipative lattice dynamical systems. Physica D 178, 51–61 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Zinner, B.: Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Differ. Equ. 96, 1–27 (1992) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of ScienceHohai UniversityNanjingChina

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