Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions

Abstract

We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled nonlocal problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation.

Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. The supercritical case

“…(See, for instance, [7,8]. ) Another close relation to the heat operator with diffusivity a was found in [5,13], where it was proven that for bounded and integrable initial data, the asymptotic behavior as t tends to infinity of the solution u L to the equation without absorption,…”

Nonlocal heat equations in the Heisenberg group

“…The main arguments are basically the same of [8] or [9], but we write them here to make the paper self-contained.…”

“…At the present work we study a similar problems to the ones in [8] and [9], in the Heisenberg group. In order to do this we have to consider the results obtained in [20], the fact that H n is a homogeneous group and the harmonic analysis related to the action of the unitary group U (n) by automorphism on H n .…”

“…In the work [9] the authors prove that solutions of properly rescaled nonlocal Dirichlet problems of the equation (1.1) approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation in R n .…”

Quenching for a non-local diffusion equation with a singular absorption term and Neumann boundary condition

“…In [2,5], the authors studied the asymptotic behavior of the solutions for a non-local non-linear diffusion operator under blowing-up boundary conditions of Dirchlet or Neumann type (the non-local diffusion equation with Dirichlet boundary condition has studied in [4]). Motivated by the above works, the purpose of this paper is to study the quenching phenomenon for the following non-local diffusion equation with singular absorption term and Neumann boundary condition…”