A Note on the formulation of the Neumann boundary condition for a nonlocal problem
The nonlocal diffusion equation \begin{equation*} \begin{cases} u_t(t,x) & = \int_Ω K(x,y) \, u(t,y) \, d\,y - \int_Ω K(y,x) \, u(t,x) \, d \, y u(0,x) & = u_0(x) \end{cases} \end{equation*} has been proposed as a model for some evolution processes with diffusion, including population models. However, in general, this formulation does not satisfy $ \int_Ω K(y,x) \, d \, y = 1$, as expected from its interpretation as a probability density. In this note, we propose a modification of the kernel, based on the idea of `reflection' at the boundary, a technique familiar in one dimensional problems. We show that a similar construction is possible in higher dimensions, with the new kernel satisfying the aforementioned integral equality and being also symmetric in certain special cases.