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, ∂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⁡tu(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→∞⁡∫R2u(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).