ASYMPTOTIC BEHAVIOR FOR A ONE-DIMENSIONAL NONLOCAL DIFFUSION EQUATION IN EXTERIOR DOMAINS

We study the large time behavior of solutions to the nonlocal diffusion equation partial derivative(t)u = J * u - u in an exterior one-dimensional domain, with zero Dirichlet data on the complement. In the far field scale, xi 1 <= vertical bar x vertical bar t(-1/2) <= xi 2, xi 1, xi 2 > 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, vertical bar x vertical bar <= t(1/2)h(t), lim(t ->infinity) h(t) = 0, the solution scaled by a factor t(3/2) /(vertical bar x vertical bar + 1) converges to a stationary solution of the problem that behaves as b(+/-)x as x -> +/-infinity. The constants b(+/-) are obtained through a matching procedure with the far field limit. In the very far field, vertical bar x vertical bar >= t(1/2)g(t), g(t) -> infinity, the solution decays as o(t(-1)).