On curved squeezing and Conley index

Abstract

We consider reaction-diffusion equations on a family of domains depending on a parameter $\varepsilon > 0$. As $\varepsilon\to 0$, the domains degenerate to a lower dimensional manifold. Using some abstract results introduced in the recent paper [M.C. Carbinatto and K.P. Rybakowski, Localized singularities and Conley index, Topol. Methods Nonlinear Anal. 37 (2011), 1-36] we show that there is a limit equation as $\varepsilon\to 0$ 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 [M.C. Carbinatto and K.P. Rybakowski, Conley index continuation and thin domain problems, Topol. Methods Nonlinear Anal. 16 (2000), 201-251] and [M. Prizzi, M. Rinaldi and K.P. Rybakowski, Curved thin domains and parabolic equations, Studia Math. 151 (2002), 109-140].