Spatial segregation for a competition-diffusion system with inhomogeneous Dirichlet boundary conditions

We consider competition-diffusion systems for two species when there is large inter-species competition and the inhomogeneous species densities are specified on the boundary of the spatial domain. We discuss the limit of solutions of this system in the large competition limit, showing convergence to a free-boundary problem on a finite time interval using integral estimates. Moreover, in the special case when the diffusion coefficients of the two species are the same, convergence on unbounded time intervals can be established using a "blow-up" method, which enables aspects of the long-time behaviour of the competition-diffusion system with sufficiently large competition to be deduced from that of solutions of the limiting problem.