Asymptotic Behavior for a nonlocal diffusion equation on the half line

Abstract

We study the large time behavior of solutions to a nonlocal diffusion equation, ut=J∗u−u with J smooth, radially symmetric and compactly supported, posed in R+ with zero Dirichlet boundary conditions. In the far-field scale, ξ1≤xt−1/2≤ξ2 with ξ1,ξ2>0, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, hence tu(x,t) is bounded above and below by positive constants in this region for large times. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. In compact sets, after scaling the solution by a factor t3/2, it converges to a multiple of the unique stationary solution of the problem that behaves as x at infinity. The precise proportionality factor is obtained through a matching procedure with the far-field limit. Finally, in the very far-field, x≥t1/2g(t) with g(t)→∞, the solution is proved to be of order o(t−1).