Explicit propagation reversal bounds for bistable differential equations on trees

TL;DR

This paper derives explicit bounds for propagation reversal in bistable reaction-diffusion equations on trees, using a piecewise linear caricature to obtain closed-form formulas and analyze wave stability.

Contribution

It provides the first explicit, closed-form bounds for pinning and reversal phenomena in bistable equations on trees, specifically for the McKean caricature.

Findings

Explicit pinning region formulas derived

Stable pinned waves constructed and analyzed

Unbounded pinning region for McKean's caricature

Abstract

In this paper we provide explicit description of the pinning region and propagation reversal phenomenon for the bistable reaction diffusion equation on regular biinfinite trees. In contrast to the general existence results for smooth bistabilities, the closed-form formulas are enabled by the choice of the piecewise linear McKean's caricature. We construct exact pinned waves and show their stability. The results are qualitatively similar to the propagation reversal results for smooth bistabilities. Major exception consists in the unboundedness of the pinning region in the case of the bistable McKean's caricature. Consequently, the propagation reversal also occurs for arbitrarily large diffusion.