Conley index continuation and thin domain problems

Given ε> 0 and a bounded Lipschitz domain Ω in RM× RN let Ω_ε:=\(x,ε y) (x,y)∈ Ω\ be the ε-{ squeezed domain\/}. Consider the reaction-diffusion equation uₜ = Δ u + f(u) ( E_ε) on Ω_ε with Neumann boundary condition. Here f is an appropriate nonlinearity such that ( E_ε) generates a (local) semiflow π_ ε on H¹(Ω_ε). It was proved by Prizzi and Rybakowski (J. Differential Equations, to appear), generalizing some previous results of Hale and Raugel, that there are a closed subspace H¹ₛ(Ω) of H¹(Ω), a closed subspace L²ₛ(Ω) of L²(Ω) and a sectorial operator A₀ on L²ₛ(Ω) such that the semiflow π₀ defined on H¹ₛ(Ω) by the abstract equation u+A₀u= f(u) is the limit of the semiflows π_ε as ε→ 0⁺. In this paper we prove a singular Conley index continuation principle stating that every isolated invariant set K₀ of π₀ can be continued to a nearby family K_ε of isolated invariant sets of π_ε with the same Conley index. We present various applications of this result to problems like connection lifting or resonance bifurcation.