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Session 23. Nonlinear Evolution Equations and their Applications
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Long-time behavior for a simplified Keller-Rubinov model for Liesegang rings in the fast reaction limit |
Marcel Oliver, Jacobs University, Germany
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Liesegang rings are regular patterns in a chemical precipation
reaction which typically follow power laws in the spacing of width
and distance. Among several mathematical models, the
Keller--Rubinov model [3] is a reaction-diffusion equation
with a super-saturation threshold in the reaction term. Hilhorst
et al. [2,1] study the fast reaction limit in
which, among other assumptions, the reaction rate of the initial
step of the mechanism is assumed to be infinitely fast. As a
result, the Keller--Rubinov model reduces to a single
reaction-diffusion equation with a singular driving term and a
reaction term involving memory and thresholding.
As in [1], we consider the one-dimensional situation,
which we shall refer to as the HHMO model. Noting that the HHMO
model possesses a self-similar solution that is explicitly
expressable in terms of special fuctions, we present numerical
evidence as well as a mechanism derived from a simplifed version of
the system which indicate that the self-similar profile is
``super-attracting'' in the sense that solutions to the HHMO model
tend to the self-similar profile in a finite time. We then
re-interpret the self-similar profile as a precipitation density function.
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References- D. Hilhorst, R. van der Hout, M. Mimura, and
I. Ohnishi, A mathematical study of the one-dimensional
Keller and Rubinow model for Liesegang bands ,
J. Stat. Phys. 135, 2009, 107-132.
- D. Hilhorst, R. van der Hout, and L.A. Peletier,
The fast reaction limit for a reaction-diffusion system ,
J. Math. Anal. Appl. 199, 1996, 349-373.
- J.B. Keller and S.I. Rubinow, Recurrent
precipitation and Liesegang rings , J. Chem. Phys. 74, 1981,
5000-5007.
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