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, < i> Localized spatial homogenization and large diffusion< /i> , SIAM J. Math. Anal. < b> 29< /b> (1998), 1361-1380] and [J.M. Arrieta, A.N. Carvalho and A. Rodr\'{\i}guez-Bernal, < i> Parabolic problems with nonlinear boundary conditions and critical nonlinearities< /i> , J. Differential Equations < b> 156< /b> (1999), 376-406].