Nonlinear Oscillations

, Volume 8, Issue 2, pp 172–183 | Cite as

Qualitative Behavior of Solutions of a Randomly Perturbed Reaction-Diffusion Equation

  • O. V. Kapustyan
  • O. V. Pereguda
  • J. Valero


On the basis of the developed abstract theory of random attractors of probability dissipative systems, we investigate the qualitative behavior of solutions of a nonuniquely solvable reaction-diffusion equation perturbed by a stochastic “cadlag” process.


Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Equation Qualitative Behavior 


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  1. 1.
    L. Arnold, Random Dynamical Systems, Springer, Berlin (1998).MATHGoogle Scholar
  2. 2.
    H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probab. Theory Related Fields, 100, 365–393 (1994).CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    H. Crauel, “Global random attractors are uniquely determined by attracting deterministic compact sets,” Ann. Mat. Pura Appl., 126, No.4, 57–72 (1999).MathSciNetGoogle Scholar
  4. 4.
    K. R. Schenk-Hoppe, “Random attractors—general properties, existence and applications to stochastic bifurcation theory,” Discr. Contin. Dynam. Syst., 4, No.1, 99–130 (1998).MATHMathSciNetGoogle Scholar
  5. 5.
    T. Caraballo, J. A. Langa, and J. Valero, “Global attractors for multivalued random dynamical systems,” Nonlin. Analysis, 48, 805–829 (2002).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes [in Russian], Nauka, Moscow (1977).MATHGoogle Scholar
  7. 7.
    A. V. Babin and M. I. Vishik, Attractors of Evolution Equations [in Russian], Nauka, Moscow (1989).MATHGoogle Scholar
  8. 8.
    A. V. Kapustyan, “Global attractors of a nonautonomous reaction-diffusion equation,” Differents. Uravn., 38, No.10, 1378–1382 (2002).CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston (1990)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • O. V. Kapustyan
    • 1
  • O. V. Pereguda
    • 1
  • J. Valero
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyivUkraine

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