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.