Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case

We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, partial derivative(t)u = J * u - u, where J is a smooth, radially symmetric kernel with support B-d(0) subset of R-2. The problem is set in an exterior two-dimensional domain which excludes a hole H, and with zero Dirichlet data on H. In the far field scale, xi(1) <= vertical bar x vertical bar t(-1/2) <= xi(2) with xi(1), xi(2) > 0, the scaled function log t u(x,t) behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by J. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic 'logarithmic momentum' of the solution, lim(t ->infinity) integral(R2) u(x, t) log vertical bar x vertical bar dx. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, vertical bar x vertical bar <= t(1/2)h(t) with lim(t ->infinity)h(t) = 0, the scaled function t(logt)(2)u(x,t)/ log vertical bar x vertical bar converges to a multiple of phi(x)/ log vertical bar x vertical bar, where phi is the unique stationary solution of the problem that behaves as log vertical bar x vertical bar when vertical bar x vertical bar -> infinity. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, vertical bar x vertical bar >= t(1/2)g(t) with g(t) -> infinity, the solution is proved to be of order o((t logt)(-1)). (C) 2015 Elsevier Inc. All rights reserved.