On the Morse complex for semilinear parabolic equations

A Morse–Smale function on a compact Riemannian manifold can be used to define an associated Morse complex. Its homology is isomorphic to the singular homology of the manifold, which coincides with the singular homology of the Conley index of the manifold. In this paper, we consider an analogous Morse complex for isolated invariant sets of certain semilinear parabolic equations. It is shown that its homology is isomorphic to the singular homology of the Conley index of the isolated invariant set.