English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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 otherproperties, $K(\theta) \sim k \, (1-\theta)^\sigma$ as $\theta \to 1$ forconstants $k>0$ and $\sigma \in (0, \log_2 3 -1)$, $H$ denotes the Heavisidefunction, and $x \in [0,\infty)$. This equation arises from areaction-diffusion equation describing Liesegang precipitation band patternsunder certain simplifying assumptions. We argue that the integral equation isan analytically tractable paradigm for the clustering of precipitation ringsobserved in the full model. This problem is nontrivial as the right hand sidefails a Lipschitz condition so that classical contraction mapping arguments donot apply.  Our results are the following. Solutions to the integral equation, whichinitially feature a sequence of relatively open intervals on which $\omega$ ispositive ("rings") or negative ("gaps") break down beyond a finite interval$[0,x^*]$ in one of two possible ways. Either the sequence of rings accumulatesat $x^*$ ("non-degenerate breakdown") or the solution cannot be continued pastone of its zeroes at all ("degenerate breakdown"). Moreover, we show thatdegenerate breakdown is possible within the class of kernels considered.Finally, we prove existence of generalized solutions which extend the integralequation past the point of breakdown.

Breakdown of Liesegang precipitation bands in a simplified fast reaction limit of the Keller-Rubinow model