Two-species nonlocal cross-diffusion models with free boundaries

A class of two-species nonlocal cross-diffusion models with free boundaries in one space dimension is investigated. In the models, the two species exist initially in (−∞,s10] and (−∞,s20], respectively, and then spread into the right space. The spreading fronts and nonlocal cross-diffusion of the species are described by the free boundaries and integral diffusion operators, respectively. By introducing some parameterized ODE problems and by applying the contraction mapping theorem and deriving some estimates, we give the global existence and uniqueness of solutions for the models. These results are applied to the nonlocal cross-diffusion prey-predator and competition models.