Finite-dimensional limiting dynamics for dissipative parabolic equations

A. V. Romanov

Sbornik. Mathematics, Tome 191 (2000) no. 3, pp. 415-429

Résumé

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.