Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions

Abstract

The motivations to study the problem considered in this paper come from the theory of composite materials, where the heat diffusion properties can change from one part of the domain to another. Mathematically, this leads to a nonlinear second-order parabolic equation for which the diffusion coefficient becomes large in a subdomain Ω 0 ⊂Ω . The equation is supplemented by a nonlinear boundary condition on ∂Ω and an initial condition. The authors determine the form of the limit problem (the so-called shadow system), which involves an evolution equation for the averages of the density over Ω 0 . The main results include global-in-time existence of solutions and upper semicontinuity of the associated global attractors when the system approaches the shadow system.