Advertisement

Journal of Mathematical Sciences

, Volume 136, Issue 2, pp 3655–3671 | Cite as

Convergence of discretized attractors for parabolic equations on the line

  • W. -J. Beyn
  • V. S. Kolezhuk
  • S. Yu. Pilyugin
Article
  • 27 Downloads

Abstract

We show that, for a semilinear parabolic equation on the real line satisfying a dissipativity condition, global attractors of time-space discretizations converge (with respect to the Hausdorff semi-distance) to the attractor of the continuous system as the discretization steps tend to zero. The attractors considered correspond to pairs of function spaces (in the sense of Babin-Vishik) with weighted and locally uniform norms (taken from Mielke-Schneider) used both for the continuous and discrete systems. Bibliography: 13 titles.

Keywords

Function Space Parabolic Equation Real Line Discrete System Global Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. V. Babin and M. I. Vishik, Attractors of Evolutionary Equations [in Russian], Nauka, Moscow (1989).Google Scholar
  2. 2.
    A. V. Babin and M. I. Vishik, “Attractors of partial differential equations in an unbounded domain,” Proc. R. Soc. Edinburgh, 116A, 221–243 (1990).MathSciNetGoogle Scholar
  3. 3.
    W.-J. Beyn and S. Yu. Pilyugin, “Attractors of reaction diffusion systems on infinite lattices,” J. Dynam. Differ. Equat., 15, 485–515 (2003).MathSciNetCrossRefGoogle Scholar
  4. 4.
    W.-J. Beyn, V. S. Koleshuk, and S. Yu. Pilyugin, “Convergence of discretized attractors for parabolic equations on the line,” Preprint 04-13, DFG Research Group “Spectral Analysis, Asymptotic Expansions, and Stochastic Dynamics,” Univ. of Bielefeld (2004).Google Scholar
  5. 5.
    D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press (2001).Google Scholar
  6. 6.
    J. K. Hale, X.-B. Lin, and G. Raugel, “Upper semicontinuity of attractors and partial differential equations,” Math. Comp., 50, 80–123 (1988).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. S. Kolezhuk, “Dynamical systems generated by parabolic equations on the line,” Preprint 04-44, DFG Research Group “Spectral Analysis, Asymptotic Expansions, and Stochastic Dynamics,” Univ. Bielefeld (2004).Google Scholar
  8. 8.
    V. S. Kolezhuk, “Interpolation and projection operators in weighted function spaces on the real line,” Preprint 04-45, DFG Research Group “Spectral Analysis, Asymptotic Expansions, and Stochastic Dynamics,” Univ. Bielefeld (2004).Google Scholar
  9. 9.
    O. A. Ladyzhenskaya, “Globally stable difference schemes and their attractors,” Preprint POMI P-5-91, St. Petersburg (1991).Google Scholar
  10. 10.
    S. Larsson, “Nonsmooth data error estimates with applications to the study of the long-time behavior of finite element solutions of semilinear parabolic problems,” Preprint 1992-36, Dept. of Math., Chalmers Univ. of Technology (1992).Google Scholar
  11. 11.
    A. Mielke and G. Schneider, “Attractors for modulation equations on unbounded domains: existence and comparison,” Nonlinearity, 8, 743–768 (1995).MathSciNetCrossRefGoogle Scholar
  12. 12.
    G. R. Sell and Y. You, Dynamics of Evolutionary Equations. Applied Mathematical Sciences, 143, Springer, Berlin, Heidelberg, New York (2002).Google Scholar
  13. 13.
    A. Stuart, “Perturbation theory for infinite dimensional dynamical systems,” in: Theory and Numerics of Ordinary and Partial Differential Equations (Leicester, 1994), Adv. Num. Anal. IV, Oxford University Press (1995), pp. 181–290.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • W. -J. Beyn
    • 1
  • V. S. Kolezhuk
    • 2
  • S. Yu. Pilyugin
    • 2
  1. 1.Bielefeld UniversityGermany
  2. 2.St.Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations