Finite-dimensional limiting dynamics for dissipative parabolic equations

For a broad class of semilinear parabolic equations with compact attractor $\mathscr A$ in a Banach space $E$ the problem of a description of the limiting phase dynamics (the dynamics on $\mathscr A$) of a corresponding system of ordinary differential equations in $\mathbb R^N$ 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 $\mathscr M\subset E$. 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 $\mathscr A$;
  
• b) the phase semiflow extends on $\mathscr A$ to a Lipschitz flow;
  
• c) the attractor $\mathscr A$ 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.