Abstract
An example of a dissipative semilinear parabolic equation in a Hilbert space without smooth inertial manifolds is constructed. Moreover, the attractor of this equation can be embedded in no finite-dimensionalC 1 invariant submanifold of the phase space. The class of scalar reaction-diffusion equations in bounded domains Ω ⊂ ℝm without inertial manifolds\(\mathcal{M} \subset L^{\text{2}} (\Omega )\) with the property of absolute normal hyperbolicity on the setE of stationary points of the phase semiflow is described. Such equations may have inertial manifolds with the weaker property of normal hyperbolicity onE. Three-dimensional reaction-diffusion systems without inertial manifolds normally hyperbolic at stationary points are found.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. D. Chueshov, “Global attractors in nonlinear problems of mathematical physics,”Uspekhi Mat. Nauk [Russian Math. Surveys],48, No. 3, 135–162 (1993).
X. Mora and J. Sola-Morales, “Existence and non-existence of finite-dimensional globally attracting invariant manifolds in semilinear damped wave equations,” in:Dynamics of Infinite-Dimensional Systems, Springer, New York (1987), pp. 187–210.
M. W. Hirsch, C. C. Pugh, and M. Shub, “Invariant manifolds,” in:Lecture Notes in Math, Vol. 583, Springer, New York (1977).
J. Mallet-Paret, G. R. Sell, and Z. Shao, “Obstructions to the existence of normally hyperbolic inertial manifolds,”Indiana Univ. Math. J.,42, No. 3, 1027–1055 (1993).
J. Mallet-Paret and G. R. Sell, “Inertial manifolds for reactions-diffusion equations in higher space dimensions,”J. Amer. Math. Soc.,1, No. 4, 805–866 (1988).
A. V. Romanov, “Sharp estimates of the dimensionality of inertial manifolds for nonlinear parabolic equations,”Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.],57, No. 4, 36–54 (1993).
D. Henry,Geometric Theory of Semilinear Parabolic Equations, Heidelberg, Springer (1981).
G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, Oxford (1979).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 439–447, September, 2000.
Rights and permissions
About this article
Cite this article
Romanov, A.V. Three counterexamples in the theory of inertial manifolds. Math Notes 68, 378–385 (2000). https://doi.org/10.1007/BF02674562
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02674562