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arXiv · 2610.21328

Directional delays and singular strong-barrier limits for Fisher--KPP propagation across a semipermeable interface

Yantao Wang
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Abstract

We study Fisher--KPP propagation from compactly supported data across a planar semipermeable interface in $N\ge2$ dimensions, with unit diffusion and $f'(0)=1$. For every finite resistance, the receiving front has speed $2$ and logarithmic correction $-(N+2)\log t/2$. Its phase is determined by a positive directional amplitude on the closed receiving hemisphere. This amplitude is jointly continuous in resistance and direction and is Hölder continuous in direction with exponent $1/3$. It also determines a threshold-independent signed arrival-time difference. Under concavity, the amplitude decreases strictly with resistance. As the resistance $β$ tends to infinity, the amplitude is asymptotic to $2(4π)^{-N/2}I_N(θ)(\logβ)/β$ uniformly on each fixed receiving cone bounded away from the equator. Here $I_N$ is a weighted input moment of the closed Neumann donor problem. Although this moment diverges logarithmically near the equator, the equatorial amplitude tends to half the amplitude of the even Neumann donor solution. The strong-resistance and equatorial limits therefore do not commute. The proof combines critical derivative moments, an integrated Robin trace estimate, Cartesian leading-edge limits, and matching through the wave displacement. The phase and asymptotic formulas require only the smooth KPP assumptions, without the additional restriction $f'\le f'(0)$. Concavity is used for strict resistance ordering and the strict bounds on the limiting equatorial delay.

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Yantao Wang. 2026-10-15. Directional delays and singular strong-barrier limits for Fisher--KPP propagation across a semipermeable interface. https://arxiv.org/abs/2610.21328

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