Directional delays and singular strong-barrier limits for Fisher--KPP propagation across a semipermeable interface
Abstract
We study Fisher--KPP propagation from compactly supported data across a planar semipermeable interface in N≥2 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/2IN(θ)(logβ)/β uniformly on each fixed receiving cone bounded away from the equator. Here IN 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'≤ f'(0). Concavity is used for strict resistance ordering and the strict bounds on the limiting equatorial delay.
Related
- Time Delay of KPP Fronts Across a Semipermeable Barrier
- Maximal directional singular integrals
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
- Uniform-in-time propagation of chaos for singular Lk interaction kernels with rates
- High frequency wave propagation for the viscoelastic wave equation with singular memory
BibTeX
@misc{wang2026directional,
title = {{Directional delays and singular strong-barrier limits for Fisher--KPP propagation across a semipermeable interface}},
author = {Yantao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21328},
eprint = {2610.21328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21328},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Directional delays and singular strong-barrier limits for Fisher--KPP propagation across a semipermeable interface
AU - Yantao Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21328
UR - https://arxiv.org/abs/2610.21328
AB - We study Fisher--KPP propagation from compactly supported data across a planar semipermeable interface in N≥2 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/2I_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'≤ f'(0). Concavity is used for strict resistance ordering and the strict bounds on the limiting equatorial delay.
ER -