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Three counterexamples in the theory of inertial manifolds. (English. Russian original) Zbl 0984.35034
Math. Notes 68, No. 3, 378-385 (2000); translation from Mat. Zametki 68, No. 3, 439-447 (2000).
The author considers a dissipative semilinear parabolic equation \(\dot u=-Au+F(u)\), where \(A\) is sectorial and has compact resolvent (also \(\text{Re} \sigma(A)\geq 0)\) and, for certain \(\alpha [0,1)\), \(F\) is globally Lipschitz from \(X^\alpha\) into \(X\) (additionally \(F\in C^1(X^\alpha,X)\) on \(X^1\); \(X\) being a Banach space).
In a sequence of lemmas there are given the necessary conditions concerning the existence of a \(C^1\) inertial manifold, especially in the case when the manifold is normally hyperbolic or absolutely normally hyperbolic.
In this context the author presents three interesting examples: (i) (pseudodifferential) parabolic equations without inertial \(C^1\)-manifold, (ii) reaction-diffusion equation in bounded subdomains of \(\mathbb{R}^n\) without inertial manifold absolutely normally hyperbolic on the stationary set, (iii) reaction-diffusion equations in a cube \((0,\pi)^3\) without inertial manifold normally hyperbolic at stationary points.

35B42 Inertial manifolds
35K90 Abstract parabolic equations
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
Full Text: DOI
[1] I. D. Chueshov, ”Global attractors in nonlinear problems of mathematical physics,”Uspekhi Mat. Nauk [Russian Math. Surveys],48, No. 3, 135–162 (1993). · Zbl 0805.58042
[2] 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. · Zbl 0642.35062
[3] M. W. Hirsch, C. C. Pugh, and M. Shub, ”Invariant manifolds,” in:Lecture Notes in Math, Vol. 583, Springer, New York (1977). · Zbl 0355.58009
[4] 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). · Zbl 0802.35085 · doi:10.1512/iumj.1993.42.42048
[5] 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). · Zbl 0674.35049 · doi:10.1090/S0894-0347-1988-0943276-7
[6] 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).
[7] D. Henry,Geometric Theory of Semilinear Parabolic Equations, Heidelberg, Springer (1981). · Zbl 0456.35001
[8] G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, Oxford (1979). · Zbl 0423.10001
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