Finite-dimensional dynamics on attractors of non-linear parabolic equations

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.