  We consider the Keller--Rubinow model for Liesegang rings in one spatial
dimension in the fast reaction limit as introduced by Hilhorst, van der Hout,
Mimura, and Ohnishi in 2007. Numerical evidence suggests that solutions to this
model converge, independent of the initial concentration, to a universal
profile for large times in parabolic similarity coordinates. For the
concentration function, the notion of convergence appears to be similar to
attraction to a stable equilibrium point in phase space. The reaction term,
however, is discontinuous so that it can only convergence in a much weaker,
averaged sense. This also means that most of the traditional analytical tools
for studying the long-time behavior fail on this problem.
  In this paper, we identify the candidate limit profile as the solution of a
certain one-dimensional boundary value problem which can be solved explicitly.
We distinguish two nontrivial regimes. In the first, the transitional regime,
precipitation is restricted to a bounded region in space. We prove that the
concentration converges to a single asymptotic profile. In the second, the
supercritical regime, we show that the concentration converges to one of a
one-parameter family of asymptotic profiles, selected by a solvability
condition for the one-dimensional boundary value problem. Here, our convergence
result is only conditional: we prove that if convergence happens, either
pointwise for the concentration or in an averaged sense for the precipitation
function, then the other field converges likewise; convergence in concentration
is uniform, and the asymptotic profile is indeed the profile selected by the
solvability condition. A careful numerical study suggests that the actual
behavior of the equation is indeed the one suggested by the theorem.
