Pullback attractor for a non-autonomous reaction-diffusion equation in some unbounded domains

Abstract

The existence of a pullback attractor in L2 (Ω) for the following non-autonomous reaction-diffusion equation #### (1) is proved in this paper, when the domain Ω is not necessarily bounded but satisfying the Poincaré inequality, and h ∈ L 2loc (ℝ; H−1 (Ω)). The main concept used in the proof is the asymptotic compactness of the process generated by the problem.

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References

  1. [1]

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

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    T. Caraballo, G. Lukaszewicz, and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains. C. R. Math. Acad. Sci. Paris, 342 (2006), 263–268.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    T. Caraballo, J. A. Langa, and J. Valero, Dimension of attractors of nonautonomous reaction-diffusion equations. ANZIAM J., 45 (2003), 207–222.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    H. Crauel, A. Debussche, and F. Flandoli, Random attractors. J. Dyn. Diff. Eq., 9 (1995), no. 2, 307–341.

    MathSciNet  Article  Google Scholar 

  5. [5]

    M.J. Garrido-Atienza and P. Marín-Rubio, Navier-Stokes equations with delays on unbounded domains. Nonlinear Anal., 64 (2006), no. 5, 1100–1118.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    P.E. Kloeden and J.A. Langa, Flatenning, squeezing and the existence of random attractors. Proc. Roy. Soc. Lond. Series A, 463 (2007), 163–181.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    Y. Li and C.K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations. Applied Mathematics and Computation, 190 (2007), 1020–1029.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris 1969.

    Google Scholar 

  9. [9]

    F. Morillas and J. Valero, Attractors for reaction-diffusion equations inN with continuous nonlinearity. Asymptotic Analysis, 44 (2005), 111–130.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    M. Prizzi, A remark on reaction-diffusion equations in unbounded domains. Discrete Contin. Dyn. Syst., 9 (2003), 281–286.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    R. Rosa, The Global Attractor for the 2D Navier-Stokes flow on some unbounded domains. Nonlinear Anal., 32 (1998), no. 1, 71–85.

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations. J. Math. Anal. Appl., 325 (2007), 1200–1215.

    MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    R. Temam, Navier-Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 66. SIAM, Philadelphia, 1983 (2nd edition, 1995).

    Google Scholar 

  14. [14]

    Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-difusion equations. Dynamical Systems, 23 (2008), 1–16.

    MathSciNet  MATH  Google Scholar 

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Correspondence to María Anguiano.

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Anguiano, M. Pullback attractor for a non-autonomous reaction-diffusion equation in some unbounded domains. SeMA 51, 9–16 (2010). https://doi.org/10.1007/BF03322548

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Key words

  • pullback attractor
  • asymptotic compactness
  • evolution process
  • non-autonomous reaction-diffusion equation

AMS subject classifications

  • 35B41
  • 35Q35
  • 35Q30
  • 35K90
  • 37L30