Semilinear elliptic equations in thin domains with reaction terms concentrating on boundary

In this paper we analyze the behavior of a family of steady state solutions of a semilinear reaction–diffusion equation with homogeneous Neumann boundary condition, posed in a two-dimensional thin domain with reaction terms concentrated in a narrow oscillating neighborhood of the boundary. We assume that the domain, and therefore, the oscillating boundary neighborhood, degenerates into an interval as a small parameter ϵ goes to zero. Our main result is that this family of solutions converges to the solution of a one-dimensional limit equation capturing the geometry and oscillatory behavior of the open sets where the problem is established.