SeMA Journal

, Volume 51, Issue 1, pp 9–16 | Cite as

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

  • María AnguianoEmail author
Actas del NSDS09


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 loc 2 (ℝ; H−1 (Ω)). The main concept used in the proof is the asymptotic compactness of the process generated by the problem.

Key words

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

AMS subject classifications

35B41 35Q35 35Q30 35K90 37L30 


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  1. [1]
    T. Caraballo, G. Lukaszewicz, and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Analysis, 64 (2006), 484–498.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    H. Crauel, A. Debussche, and F. Flandoli, Random attractors. J. Dyn. Diff. Eq., 9 (1995), no. 2, 307–341.MathSciNetCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris 1969.zbMATHGoogle Scholar
  9. [9]
    F. Morillas and J. Valero, Attractors for reaction-diffusion equations inN with continuous nonlinearity. Asymptotic Analysis, 44 (2005), 111–130.MathSciNetzbMATHGoogle Scholar
  10. [10]
    M. Prizzi, A remark on reaction-diffusion equations in unbounded domains. Discrete Contin. Dyn. Syst., 9 (2003), 281–286.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations. J. Math. Anal. Appl., 325 (2007), 1200–1215.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2010

Authors and Affiliations

  1. 1.Dpto. Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain

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