The Conley index along heteroclinic trajectories of reaction-diffusion equations (Q763749)

It is well known that hyperbolic equilibria of reaction-diffusion equations have the homotopy Conley index of a pointed sphere, the dimension of which is the Morse index of the equilibrium. A similar result concerning the homotopy Conley index along heteroclinic solutions of ordinary differential equations under the assumption that the respective stable and unstable manifolds intersect transversally, is due to \textit{C. McCord} [Trans. Am. Math. Soc. 307, No. 1, 195--203 (1988; Zbl 0646.34056)]. This result has recently been generalized by \textit{E. N. Dancer} [Bull. Aust. Math. Soc. 80, No. 3, 510--520 (2009; Zbl 1181.35137)] to some reaction-diffusion equations by using finite-dimensional approximations. In this paper the author extends McCord's result to reaction-diffusion equations. Additionally, an error in the original proof is pointed out and corrected.