Boundary fluxes for nonlocal diffusion

Abstract

We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition.

How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

“…As in [7] and [8], existence and uniqueness will be a consequence of Banach's fixed point theorem. We follow closely the ideas of those works in our proof, so we will only outline the main arguments.…”

“…For example, a comparison principle holds for both equations when G is nonnegative and the asymptotic behavior of their solutions as t → ∞ is similar, see [8].…”

“…Several properties of solutions to (1.3) have been recently investigated in [8] in the case G = G 2 for different choices of g.…”

The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles

“…Nonlocal evolution equations of the form u t (t, x) = J * u − u(t, x), and variations of it, have been recently widely used to model diffusion processes, see [1], [2], [5], [7], [16], [17], [18], [20], [29] and [30].…”

The first non-zero Neumann p-fractional eigenvalue

“…Moreover, this kind of process can be used to describe some random flow in a closed domain with free action on the boundary, and they are always connected to the Neumann boundary problems. As it was pointed in [4,12] the idea of s−process in which its jumps from Ω to the complement of Ω are suppressed, are related to the Neumann non-local evolution equation…”