Existence and regularity for a nonlinear boundary flow problem of population genetics

In this paper we consider a heat equation with nonlinear boundary condition occurring in population genetics, the selection–migration problem for alleles in a region, considering flow of genes throughout the boundary. Such a problem determines a gradient flow in a convenient phase space and then the dynamics for large times depends heavily on the knowledge of the equilibrium solutions. We address the questions of the existence of a nontrivial equilibrium solution and its regularity.