Singular limit of a competition–diffusion system with large interspecific interaction

We consider a competition–diffusion system for two competing species; the density of the first species satisfies a parabolic equation together with an inhomogeneous Dirichlet boundary condition whereas the second one either satisfies a parabolic equation with a homogeneous Neumann boundary condition, or an ordinary differential equation. Under the situation where the two species spatially segregate as the interspecific competition rate becomes large, we show that the resulting limit problem turns out to be a free boundary problem. We focus on the singular limit of the interspecific reaction term, which involves a measure located on the free boundary.