Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Ω 0 which is interior to the physical domain Ω ⊂ R n . We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Ω 0 and converges uniformly to a continuous and positive function in Ω 1 = Ω ¯ ∖ Ω 0 .