Geodesic
Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations
Izvestiya. Mathematics , Tome 43 (1994) no. 1, pp. 31-47.

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Sufficient conditions are obtained for the existence of a $k$-dimensional invariant manifold that attracts as $t\to\infty$ all solutions $u(t)$ of the evolution equation $\dot u=-Au+F(u)$ in a Hilbert space, where $A$ is a linear selfadjoint operator, semibounded from below, with compact resolvent, and $F$ is a uniformly Lipschitz (in suitable norms) nonlinearity; these conditions sharpen previously known conditions and cannot be improved. 
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A. V. Romanov. Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations. Izvestiya. Mathematics , Tome 43 (1994) no. 1, pp. 31-47. http://geodesic.mathdoc.fr/item/IM2_1994_43_1_a1/

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