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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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