A free boundary problem describing S-K-T competition ecological model with cross-diffusion

In this paper we investigate a free boundary problem describing S-K-T competition ecological model with two competing species and with cross-diffusion and self-diffusion in one space dimension, where one species is made up of two groups separated by a free boundary, and the other has a single group. The system under consideration is strongly coupled and the coefficients of the equations are allowed to be discontinuous. We first show the global existence and uniqueness of the solutions for the corresponding diffraction problem by approximation method, Galerkin method and Schauder fixed point theorem, and then prove the local existence of the solutions for the free boundary problem by Schauder fixed point theorem.