Skip to main content
SpringerLink
Log in

Pullback Exponential Attractors for Non-autonomous Lattice Systems

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We first present some sufficient conditions for the existence and the construction of a pullback exponential attractor for the continuous process (non-autonomous dynamical system) on Banach spaces and weighted spaces of infinite sequences. Then we apply our results to study the existence of pullback exponential attractors for first order non-autonomous differential equations and partly dissipative differential equations on infinite lattices with time-dependent coupled coefficients and time-dependent external terms in weighted spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chepyzhov V.V., Vishik M.I.: Attractors of nonautonomous dynamical systems and their dimension. J. Math. Pures Appl. 73, 279–333 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. Colloquium Publications, American Math. Soc., 49 (2002)

  3. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland (1992)

  4. Chepyzhov V.V., Efendiev M.: Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example. Comm. Pure Appl. Math. 53, 647–665 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheban D.N., Fakeeh D.S.: Global attractos of infinite-dimensional dynamical systems, iii. Bull. Acad. Sci. Rep. Moldova Mat. 18(19(2-3), 3–13 (1995)

    Google Scholar 

  6. Caraballo T., Lukaszewicz G., Real J.: Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal. 64, 484–498 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hale J.K.: Asymptotic Behavior of Dissipative Systems. AMS, Providence (1988)

    MATH  Google Scholar 

  8. Robinson J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  9. Sell G.R.: Nonautonomous differential equations and dynamical systems. Trans. Am. Math. Soc. 127, 241–283 (1967)

    MathSciNet  MATH  Google Scholar 

  10. Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York (1988)

    Book  MATH  Google Scholar 

  11. Chow, S.N.: Lattice dynamical systems. In: “ Dynamical Systems”, 1-102, Lecture Notes in Math., 1822. Springer, Berlin (2003)

  12. Bates P.W., Lu K., Wang B.: Attractors for lattice dynamical systems. Intern. J. Bifurcation Choas 11(1), 143–153 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang B.: Asymptotic behavior of non-autonomous lattice systems. J. Math. Anal. Appl. 331, 121–136 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang B.: Dynamics of systems on infinite lattices. J. Diff. Eqn. 221, 224–245 (2006)

    Article  MATH  Google Scholar 

  15. Zhao X., Zhou S.: Kernel sections for processes and nonautonomous lattice systems. Disc. Cont. Dyn. Syst. B 9, 763–785 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhou S., Zhao C., Wang Y.: Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Disc. Conti. Dyna. Syst. B 21, 1259–1277 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Abdallah A.Y.: Exponential attractors for first-order lattice dynamical systems. J. Math. Anal. Appl. 339, 217–224 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Abdallah A.Y.: Exponential attractors for second order lattice dynamical systems. Commun. Pure Appl. Anal. 8, 803–813 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fan, X.: Exponential attractors for a first-order dissipative lattice dynamical systems. J. Appl. Math. pp. 1–8 (2008)

  20. Fan X., Yang H.: Exponential attractor and its fractal dimension for a second order lattice dynamical system. J. Math. Anal. Appl. 367, 350–359 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li X., Wei K., Zhang H.: Exponential attractors for lattice dynamical systems in weighted spaces. Acta Appl. Math. 114, 157–172 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li X., Wang D.: Attractors for partly dissipative lattice dynamic systems in weighted spaces. J. Math. Anal. Appl. 325, 141–156 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li X., Zhong C.: Attractors for partly dissipative lattice dynamic systems in l 2 ×  l 2. J. Comp. Appl. Math. 177, 159–174 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhao C., Zhou S.: Sufficient conditions for the existence of exponential attractors for lattice systems and applications. Acta Math. Sinica (in Chinese) 53, 233–242 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Han X.: Exponential attractors for lattice dynamical systems in weighted spaces. Discret Contin. Dynam. Syst. A 1, 445–467 (2011)

    Article  Google Scholar 

  26. Abdallah A.Y.: Uniform exponential attractors for first order non-autonomous lattice dynamical systems. J. Diff. Eqns. 251, 1489–1504 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Babin A.V., Nicolaenko B.: Exponential attractors for reaction diffusion equations in unbounded domains. J. Dyna. Diff. Eqns. 7, 567–590 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dung L., Nicolaenko B.: Exponential attractors in banach spaces. J. Dyn. Diff. Eqns. 13, 791–806 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Eden A., Foias C., Kalantarov V.: A remark on two constructions of exponential attractors for α-contractions. J. Dyn. Diff. Eqns. 10, 37–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Eden A., Foias C., Nicolaenko B., Temam R.: Exponential Attractors for Dissipative Evolution Equations,RAM: Research in Applied Mathematics. Masson, Wiley, Ltd., Paris (1994)

    Google Scholar 

  31. Efendiev M., Miranville A., Zelik S.: Infinite dimensional exponential attractors for a non-autonomous reaction-diffusion system. Math. Nachr. 248(249), 72–96 (2003)

    Article  MathSciNet  Google Scholar 

  32. Efendiev M., Miranville A., Zelik S.: Exponential attractors and finite-dimensional re- duction for non-autonomous dynamical systems. Proc. Royal Soc. Edinburg A 135, 703–730 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Efendiev M., Miranville A., Zelik S.: Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains. Proc. Royal Soc. Edinburg A 134, 271–315 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fabrie P., Miranville A.: Exponential attractors for nonautonomous first-order evolution equations. Discrete Contin. Dynam. Syst. 4, 225–240 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Efendiev M., Yamamoto Y., Yagi A.: Exponential attractors for non-autonomous dissipative system. J. Math. Soc. Jpn. 63, 647–673 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Langa J.A., Miranville A., Real J.: Pullback exponential attractors. Discrete Contin. Dynam. Syst. 26, 1329–1357 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Miranville A.: Exponential attractors for nonautonomous evolution equations. Appl. Math. Lett. 11, 19–22 (1998)

    Article  MathSciNet  Google Scholar 

  38. Miranville A., Pata V., Zelik S.: Exponential attractors for singularly perturbed damped wave equations: A simple construction. Asymp. Anal. 53, 1–12 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Han X., Shen W., Zhou S.: Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Diff. Eqn. 250, 1235–1266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, X., Lv, H.: Uniform attractor for the partly dissipative nonautonomous lattice systems. Adv. Differ. Eqns., Article ID 916316, 21 pages (doi:10.1155/2009/916316) (2009)

  41. Pazy A.: Semigroups of Linear Operator and Applications to Partial Differential Equations. Springer-Verlag, (2007)

  42. Adams R.A., Fournier J.J.: Sobolev Spaces, 2nd ed. Elsevier Ltd., Amsterdam (2003)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, People’s Republic of China

    Shengfan Zhou

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Xiaoying Han

Authors
  1. Shengfan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoying Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shengfan Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, S., Han, X. Pullback Exponential Attractors for Non-autonomous Lattice Systems. J Dyn Diff Equat 24, 601–631 (2012). https://doi.org/10.1007/s10884-012-9260-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-012-9260-7

Keywords

Search

Navigation