Non-local diffusion problems with Neumann type boundary values

Abstract

In this paper, we study the non-local diffusion problemJ⁎u−u−ut=f(u) with non-local Neumann boundary condition. We first establish some integration by parts formulas similar to the Laplacian. Then the existence and uniqueness of the solution and the maximum principle are proved. We also investigate the properties of the Neumann condition for one equation and for a family of equations.