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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.

##### MSC:
 35B42 Inertial manifolds 35K90 Abstract parabolic equations 37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
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##### References:
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