Ukrainian Mathematical Journal

, Volume 50, Issue 3, pp 361–376 | Cite as

Central manifolds of quasilinear parabolic equations

  • E. P. Belan
  • O. B. Lykova


We investigate central manifolds of quasilinear parabolic equations of arbitrary order in an unbounded domain. We suggest an algorithm for the construction of an approximate central manifold in the form of asymptotically convergent power series. We describe the application of the results obtained in the theory of stability.


Parabolic Equation Invariant Manifold Parabolic System Exponential Dichotomy Nonlinear Parabolic Equation 
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  1. 1.
    N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1963).Google Scholar
  2. 2.
    Yu. A. Mitropol’skii and O. B. Lykova, Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1973).Google Scholar
  3. 3.
    D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).zbMATHGoogle Scholar
  4. 4.
    M. P. Vishnevskii, Invariant Sets of Nonlinear Parabolic Systems. Some Applications of Functional Analysis to Problems of Mathematical Physics [in Russian], Institute of Mathematics, Siberian Division of the Academy of Sciences of the USSR, Novosibirsk (1986), pp. 32–56.Google Scholar
  5. 5.
    V. N. Levitan and V. V. Zhikov, Almost-Periodic Functions and Differential Equations [in Russian], Moscow University, Moscow (1978).zbMATHGoogle Scholar
  6. 6.
    S. D. Éidel’man, Parabolic Systems [in Russian], Nauka, Moscow (1964).Google Scholar
  7. 7.
    V. B. Levenshtam, “Averaging of quasilinear parabolic equations with rapidly oscillating principal part. Exponential dichotomy,” Izv. Akad. Nauk SSSR, Ser. Mat., 56, No. 44, 813–851 (1992).Google Scholar
  8. 8.
    W. A. Coppel and K. J. Palmer, “Averaging and integral manifolds,” Bull. Austral. Math. Soc., 2, 197–222 (1970).zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    E. P. Belan and O. B. Lykova, “A theorem on a central manifold of a nonlinear parabolic equation,” Ukr. Mat. Zh., 48, No. 8, 1021–1036 (1996).CrossRefMathSciNetGoogle Scholar
  10. 10.
    V. V. Zhikov, “Some problems of admissibility and dichotomy. Averaging principle,” Izv. Akad. Nauk SSSR, Ser. Mat., 40, No. 6, 1380–1408 (1976).zbMATHMathSciNetGoogle Scholar
  11. 11.
    A. B. Hausrath, “Stability in the critical case of purely imaginary roots for neutral functional differential equations,” Differents. Equat., 13, 329–357 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Carr, Application of Center Manifold Theory, Springer, New York (1981).Google Scholar
  13. 13.
    Chen Yushu and Xu Jian, “Universal classification of bifurcating solutions to a primary parametric resonance in der Pol-Duffing Matchicu’s systems,” Sci. China, 39, No. 4, 405–417 (1996).zbMATHMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1999

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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