Z. Darbenas and M. Oliver,
Breakdown of Liesegang precipitation bands
in a simplified fast reaction limit of the Keller-Rubinow model,
Nonlinear Differ. Equ. Appl. 28 (2021), 4, doi:10.1007/s00030-020-00663-7.
Abstract:
We study solutions to the integral equation
\[
\omega(x)
= \Gamma - x^2 \int_{0}^1
K(\theta) \, H(\omega(x\theta)) \, \mathrm d \theta
\]
where \(\Gamma>0\), \(K\) is a weakly degenerate kernel satisfying, among
other properties, \(K(\theta) \sim k \, (1-\theta)^\sigma\) as
\(\theta \to 1\) for constants \(k>0\) and \(\sigma \in (0, \log_2 3 -1)\),
\(H\) denotes the Heaviside function, and \(x \in [0,\infty)\). This
equation arises from a reaction-diffusion equation describing
Liesegang precipitation band patterns under certain simplifying
assumptions. We argue that the integral equation is an analytically
tractable paradigm for the clustering of precipitation rings observed
in the full model. This problem is nontrivial as the right hand side
fails a Lipshitz condition so that classical contraction mapping
arguments do not apply.
Our results are the following. Solutions to the integral equation,
which initially feature a sequence of relatively open intervals on
which \(\omega\) is positive ("rings") or negative ("gaps") break
down beyond a finite interval \([0,x^*]\) in one of two possible ways.
Either the sequence of rings accumulates at \(x^*\) ("non-degenerate
breakdown") or the solution cannot be continued past one of its
zeroes at all ("degenerate breakdown"). Moreover, we show that
degenerate breakdown is possible within the class of kernels
considered. Finally, we prove existence of generalized solutions
which extend the integral equation past the point of breakdown.
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