Dimension of attractors and invariant sets in reaction diffusion equations
This paper studies reaction diffusion equations of the form NEWLINE\[NEWLINEu_t+\beta(x)u-\Delta u =f(x,u)NEWLINE\]NEWLINE on a possibly unbounded domain \(\Omega\subset\mathbb R^3\) with Dirichlet boundary condition. Under some assumptions it is proved here that any invariant set of the semilflow generated by the equation in \(H_0^1\) has finite Hausdorff dimension. The conditions on the nonlinearity \(f(x,u)\) are general and do not include a dissipativity assumption, so that the invariant set in question is not necessarily an attractor. When the nonlinearity satisfies a dissipativity condition and the invariant set is a global attractor, a more explicit bound on the dimension of this attractor is derived. An important component of the proofs of these results is the application of spectral-theoretic results like the Lieb-Thirring and Cwickel-Lieb-Rozenblum inequality.
scientific article; zbMATH DE number 6242607