Conley index continuation and thin domain problems

Abstract

Given $\varepsilon> 0$ and a bounded Lipschitz domain $\Omega$ in $\mathbb R^M\times \mathbb R^N$ let $\Omega_\varepsilon:=\{(x,\varepsilon y)\mid (x,y)\in \Omega\}$ be the $\varepsilon$-{\it squeezed domain\/}. Consider the reaction-diffusion equation $$ u_t = \Delta u + f(u) \leqno(\widetilde E_\varepsilon) $$ on $\Omega_\varepsilon$ with Neumann boundary condition. Here $f$ is an appropriate nonlinearity such that $(\widetilde E_\varepsilon)$ generates a (local) semiflow $\widetilde\pi_ \varepsilon$ on $H^1(\Omega_\varepsilon)$. 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 $H^1_s(\Omega)$ of $H^1(\Omega)$, a closed subspace $L^2_s(\Omega)$ of $L^2(\Omega)$ and a sectorial operator $A_0$ on $L^2_s(\Omega)$ such that the semiflow $\pi_0$ defined on $H^1_s(\Omega)$ by the abstract equation $$\dot u+A_0u=\widehat f(u)$$ is the limit of the semiflows $\widetilde\pi_\varepsilon$ as $\varepsilon\to 0^+$. In this paper we prove a singular Conley index continuation principle stating that every isolated invariant set $K_0$ of $\pi_0$ can be continued to a nearby family $\widetilde K_\varepsilon$ of isolated invariant sets of $\widetilde \pi_\varepsilon$ with the same Conley index. We present various applications of this result to problems like connection lifting or resonance bifurcation.