Finite-dimensional limiting dynamics for dissipative parabolic equations

For a broad class of semilinear parabolic equations with compact attractor A in a Banach space E the problem of a description of the limiting phase dynamics (the dynamics on A) of a corresponding system of ordinary differential equations in RN is solved in purely topological terms. It is established that the limiting dynamics for a parabolic equation is finite-dimensional if and only if its attractor can be embedded in a sufficiently smooth finite-dimensional submanifold M subset of E. Some other criteria are obtained for the finite dimensionality of the limiting dynamics: a) the vector field of the equation satisfies a Lipschitz condition on A; b) the phase semiflow extends on A to a Lipschitz flow; c) the attractor A has a finite-dimensional Lipschitz Cartesian structure. It is also shown that the vector field of a semilinear parabolic equation is always Holder on the attractor