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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 converge 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.
@article{AIHPC_2022__39_6_1413_0,
author = {Darbenas, Zymantas and van der Hout, Rein and Oliver, Marcel},
title = {Long-time asymptotics of solutions to the {Keller{\textendash}Rubinow} model for {Liesegang} rings in the fast reaction limit},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1413--1458},
volume = {39},
number = {6},
year = {2022},
doi = {10.4171/aihpc/34},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/34/}
}
TY - JOUR AU - Darbenas, Zymantas AU - van der Hout, Rein AU - Oliver, Marcel TI - Long-time asymptotics of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit JO - Annales de l'I.H.P. Analyse non linéaire PY - 2022 SP - 1413 EP - 1458 VL - 39 IS - 6 UR - http://geodesic.mathdoc.fr/articles/10.4171/aihpc/34/ DO - 10.4171/aihpc/34 LA - en ID - AIHPC_2022__39_6_1413_0 ER -
%0 Journal Article %A Darbenas, Zymantas %A van der Hout, Rein %A Oliver, Marcel %T Long-time asymptotics of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit %J Annales de l'I.H.P. Analyse non linéaire %D 2022 %P 1413-1458 %V 39 %N 6 %U http://geodesic.mathdoc.fr/articles/10.4171/aihpc/34/ %R 10.4171/aihpc/34 %G en %F AIHPC_2022__39_6_1413_0
Darbenas, Zymantas; van der Hout, Rein; Oliver, Marcel. Long-time asymptotics of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit. Annales de l'I.H.P. Analyse non linéaire, Tome 39 (2022) no. 6, pp. 1413-1458. doi: 10.4171/aihpc/34
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