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

Abstract

We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, ∂tu = J ∗ u − u, where J is a smooth, radially symmetric kernel with support Bd(0) ⊂ R2. 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, ξ1 ≤ |x|t−1/2 ≤ ξ2 with ξ1, ξ2 > 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, limt→∞ R2 u(x,t) log |x| dx. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, |x| ≤ t1/2h(t) with limt→∞ h(t) = 0, the scaled function t(log t)2u(x,t)/ log |x| converges to a multiple of φ(x)/ log |x|, where φ is the unique stationary solution of the problem that behaves as log |x| when |x| → ∞. The proportionality constant 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((tlog t)−1).