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In this paper we derive the limiting PDES for parabolic problems where localized large diffusion touches the boundary of the domain. Hence the limit problem reflects the interaction of large diffusion and boundary conditions. The limit problem is a PDE coupled with a system of ODEs through some nonlocal terms. We study the well posedness and dissipativity of the (nonstandard) limit problem as well as the continuity of the dynamics as diffusion becomes large. (C) 2019 Elsevier Inc. All rights reserved.