A sharp condition for existence of an inertial manifold

  • Milan Miklavčič


It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semi-linear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel's theorem for Banach valued functions are used.

Key words

Inertial invariant exponentially attracting manifolds non-linear evolution equations 


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  1. 1.
    Babin, A. V., and Vishik, M. I. (1986). Unstable invariant sets of semigroups of nonlinear operators and their perturbations.Russ. Math. Surv. 41, 1–41.Google Scholar
  2. 2.
    Brunovský, P., and Tereščák, I. (1991). Regularity of invariant manifolds.J. Diff. Eq. 3, 313–337.Google Scholar
  3. 3.
    Chow, S.-N., and Lu, K. (1988). Invariant manifolds for flows in Banach spaces.J. Diff. Eq. 74, 285–317.Google Scholar
  4. 4.
    Foias, C., Sell, G. R., and Titi, E. S. (1989). Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations.J. Dynam. Diff. Eq. 1, 199–244.Google Scholar
  5. 5.
    Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Am. Math. Soc., Providence.Google Scholar
  6. 6.
    Henry, D. (1981).Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, New York.Google Scholar
  7. 7.
    Mallet-Paret, J., and Sell, G. R. (1988). Inertial manifolds for reaction diffusion equations in higher space dimensions.J. Am. Math. Soc. 1, 805–866.Google Scholar
  8. 8.
    Matano, H. (1989). Personal communication.Google Scholar
  9. 9.
    Temam, R. (1988).Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.Google Scholar
  10. 10.
    Vagi, S. (1969). A remark on Plancherel's theorem for Banach space valued functions.Ann. Scuola Norm. Sup. Pisa 23, 305–315.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Milan Miklavčič
    • 1
  1. 1.Department of MathematicsMichigan State UniversityEast Lansing

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