Geodesic
Finite-dimensional dynamics on attractors of non-linear parabolic equations
Izvestiya. Mathematics , Tome 65 (2001) no. 5, pp. 977-1001.

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We show that one-dimensional semilinear second-order parabolic equations have finite-dimensional dynamics on attractors. In particular, this is true for reaction-diffusion equations with convection on $(0,1)$. We obtain new topological criteria for a class of dissipative equations of parabolic type in Banach spaces to have finite-dimensional dynamics on invariant compact sets. The dynamics of these equations on an attractor $\mathcal A$ is finite-dimensional (can be described by an ordinary differential equation) if $\mathcal A$ can be embedded in a finite-dimensional $C^1$-submanifold of the phase space. 
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A. V. Romanov. Finite-dimensional dynamics on attractors of non-linear parabolic equations. Izvestiya. Mathematics , Tome 65 (2001) no. 5, pp. 977-1001. http://geodesic.mathdoc.fr/item/IM2_2001_65_5_a4/

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