Localized singularities and the Conley index

Abstract

We establish some abstract convergence and Conley index continuation principles for families of singularly perturbed semilinear parabolic equations and apply them to reaction-diffusion equations with nonlinear boundary conditions and localized large diffusion. This extends and refines previous results of [A. Rodriguez-Bernal, Localized spatial homogenization and large diffusion, SIAM J. Math. Anal. 29 (1998), 1361-1380] and [J.M. Arrieta, A.N. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999), 376-406].