Asymptotic behavior for a one-dimensional nonlocal diffusion equation in exterior domains

Abstract

We study the long time behavior of solutions to the nonlocal diffusion equation ∂tu = J ∗ u − u in an exterior one-dimensional domain, with zero Dirichlet data on the complement. In the far field scale, ξ1 ≤ |x|t −1/2 ≤ ξ2, ξ1, ξ2 > 0, this behavior is given by a multiple of the dipole solution for the local heat equation with a diffusivity determined by J. However, the proportionality constant is not the same on R+ and R−: it is given by the asymptotic first moment of the solution on the corresponding half line, which can be computed in terms of the initial data. In the near field scale, |x| ≤ t 1/2h(t), limt→∞ h(t) = 0, the solution scaled by a factor t 3/2 /(|x| + 1) converges to a stationary solution of the problem that behaves as b ±x as x → ±∞. The constants b ± are obtained through a matching procedure with the far field limit. In the very far field, |x|≥t 1/2 g(t), g(t) → ∞, the solution has order o(t −1 ).