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Ukrainian Mathematical Journal

, Volume 48, Issue 8, pp 1153–1170 | Cite as

Theorem on the central manifold of a nonlinear parabolic equation

  • E. P. Belan
  • O. B. Lykova
Article

Abstract

Under certain assumptions, we prove the existence of anm-parameter family of solutions that form the central invariant manifold of a nonlinear parabolic equation. For this purpose, we use an abstract scheme that corresponds to energy methods for strongly parabolic equations of arbitrary order.

Keywords

Parabolic Equation Real Axis Invariant Manifold Unique Fixed Point Zero Solution 
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.

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References

  1. 1.
    A. M. Lyapunov,General Problem of Stability of Motion [in Russian] Nauka, Moscow 1950.Google Scholar
  2. 2.
    N. N. Bogolyubov,On Some Statistical Methods in Mathematical Physics [in Russian] Academy of Sciences of the USSR, Lvov 1945.Google Scholar
  3. 3.
    Yu. A. Mitroporsl’ii and O. B. Lykova,Integral Manifolds in Nonlinear Mechanics [in Russian] Nauka, Moscow 1973.Google Scholar
  4. 4.
    O. B. Lykova and Ya. S. Baris,Approximate Integral Manifolds [in Russian] Naukova Dumka, Kiev 1993.Google Scholar
  5. 5.
    O. B. Lykova, “On the behavior of solutions of a system of nonlinear differential equations in a neighborhood of an isolated statistical solution,”Dokl. Akad. Nauk SSSR,115, No. 3, 447–449 (1957).MATHMathSciNetGoogle Scholar
  6. 6.
    O. B. Lykova, “Investigation of solutions of nonlinear systems close to integrable systems by the method of integral manifolds,” in:Proceedings of the International Symposium on Nonlinear Oscillations [in Russian], Ukrainian Academy of Sciences, Kiev (1963), pp. 315–327.Google Scholar
  7. 7.
    V. A. Pliss,Nonlocal Problems in the Theory of Oscillations [in Russian] Nauka, Moscow 1964.Google Scholar
  8. 8.
    A. Kelly, “The stable, center-stable, unstable manifolds,”J. Math. Anal Appl.,18, No. 2, 330–344 (1967).Google Scholar
  9. 9.
    Yu. L. Daletskii and M. G. Krein,Stability of Solutions of Differential Equations in Banach Spaces [in Russian] Nauka, Moscow 1970.Google Scholar
  10. 10.
    O. B. Lykova, “Reduction principle in Banach spaces,”Ukr. Mat. Zh.,23, No. 4, 464–471 (1971).MATHMathSciNetGoogle Scholar
  11. 11.
    O. B. Lykova, “On the reduction principle for differential equations with unbounded operator coefficients,”Ukr. Mat. Zh.,17, No. 2, 240–243 (1975).Google Scholar
  12. 12.
    D. Henry,Geometric Theory of Semilinear Parabolic Equations [Russian translation] Mir, Moscow 1985.MATHGoogle Scholar
  13. 13.
    J. E. Marsden and M. McCracken,The Hopf Bifurcation and Its ApplicAtions, Springer-Verlag, New York 1976.MATHGoogle Scholar
  14. 14.
    B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan,Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge 1981.MATHGoogle Scholar
  15. 15.
    W. A. Coppel and K. J. Palmer, “Averaging and integral manifolds,”Bull. Math. Soc., 197–222 (1970).Google Scholar
  16. 16.
    V. V. Zhikov, “Some problems of admissibility and dichotomy. Averaging Principle,”Izv. Akad. Nauk SSSR, Ser. Mat.,40, No. 6, 1380–1408 (1976).MATHMathSciNetGoogle Scholar
  17. 17.
    E. P. Belan and O. B. Lykova, “Integral manifolds and exponential splitting of linear parabolic equations with rapidly varying coefficients,”Ukr. Mat. Zh.,47, No. 12, 1593–1608 (1995).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    J. L. Lions,Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod Gauthier-Villars, Paris 1969.MATHGoogle Scholar
  19. 19.
    J. Carr,Application of Center Manifolds Theory, Springer, New York 1981.Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • E. P. Belan
    • 1
  • O. B. Lykova
    • 2
  1. 1.Simferopol UniversitySimferopol
  2. 2.Institute of MathematicsUkrainian Academy of SciencesKiev

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