  We study the large time behavior of solutions to a non-local diffusion
equation, $u_t=J*u-u$ with $J$ smooth, radially symmetric and compactly
supported, posed in $\mathbb{R}_+$ with zero Dirichlet boundary conditions. In
sets of the form $x\ge \xi t^{1/2}$, $\xi>0$, the outer region, the asymptotic
behavior is given by a multiple of the dipole solution for the local heat
equation, and the solution is $O(t^{-1})$. The proportionality constant is
determined from a conservation law, related to the asymptotic first momentum.
On compact sets, the inner region, after scaling the solution by a factor
$t^{3/2}$, it converges to a multiple of the unique stationary solution of the
problem that behaves as $x$ at infinity. The precise proportionality factor is
obtained through a matching procedure with the outer behavior. Since the outer
and the inner region do not overlap, the matching is quite involved. It has to
be done for the scaled function $t^{3/2}u(x,t)/x$, which takes into account
that different scales lead to different decay rates.
