Continuity of attractors for parabolic problems with localized large diffusion

In this paper we study the continuity of asymptotics of semilinear parabolic problems of the formut−div(p(x)∇u)+λu=f(u) in a bounded smooth domain Ω⊂RnΩ⊂Rn with Dirichlet boundary conditions when the diffusion coefficient pp becomes large in a subregion Ω0Ω0 which is interior to the physical domain ΩΩ. We prove, under suitable assumptions, that the family of attractors behave upper and lower semicontinuously as the diffusion blows up in Ω0Ω0.