An integro-PDE model with variable motility

This paper is concerned with a nonlocal reaction-diffusion-mutation model. It involves the spatial variable and a trait variable which govern the spatial diffusion of species. By establishing comparison principle and constructing monotone iterative sequence, we have proved the existence of solution to Cauchy problem. Then, based on the quasi-elementary solution, auxiliary equation and method of successive improvement of upper and lower solutions, the solutions are shown to be unique, bounded and globally stable.