The Conley index along heteroclinic trajectories of reaction–diffusion equations

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 McCord. This result has recently been generalized by Dancer to some reaction–diffusion equations by using finite-dimensional approximations. We extend McCordʼs result to reaction–diffusion equations. Additionally, an error in the original proof is corrected.