Global weak solution for a singular two component reaction-diffusion system

We study a singular Reaction-Diffusion system motivated by a dedicated diffusive predator-prey model system devised in the spatially homogeneous case by Courchamp and Sugihara [2]. The reactive part features a functional response to predation and a singular numerical functional response to predation specifically designed for modeling the introduction of greedy predators into a fragile or insular environment. Under some circumstances this may lead to finite time quenching of the solution, that is finite time extinction for both species. The aim of this work is to derive a suitable notion of global (in time) weak solution and to prove that such global weak solutions do exist. The existence part is achieved by approximating the reactive part by a more classical and non singular one and then passing to the limit in the resulting Reaction-Diffusion system. Our first result shows that this limiting process supply global weak solutions. In the case of equidiffusivities such global weak solutions satisfy a suitable free boundary value problem.