On curved squeezing and Conley index

Abstract

We consider reaction-diffusion equations on a family of domains depending on a parameter $\eps> 0$. As $\eps\to 0$, the domains degenerate to a lower dimensional manifold. Using some abstract results introduced in the recent paper \cite{\rfa{CR2}} we show that there is a limit equation as $\eps\to 0$ and obtain various convergence and admissibility results for the corresponding semiflows. As a consequence, we also establish singular Conley index and homology index continuation results. Under an additional dissipativeness assumption, we also prove existence and upper-semicontinuity of global attractors. The results of this paper extend and refine earlier results of \cite{\rfa{CR1}} and \cite{\rfa{PRR}}.