Sublinear Power-Law Growth of Level Set Diameters in Heterogeneous F-KPP Equations
Abstract
For every α∈(1/4,1), we construct a heterogeneous medium ξ such that the diameter of the level sets of the solution to the F-KPP equation is of order tα along a sequence of times tending to infinity. This is a natural continuation of a series of recent works establishing logarithmic lower bounds as well as linear asymptotics for the diameters of the respective level sets in random and deterministic heterogeneous media. Both results assume supξ> 2infξ; we also show that the diameter can diverge in the critical regime supξ=2infξ. A key ingredient is an estimate for branching Brownian motion with fixed time horizon t and branching rate $1$ on the negative half-line and 2+t−η on the positive half-line. This estimate exhibits a saturation transition: for η<1/2 the polynomial and logarithmic corrections to the extremal position depend on η, whereas for η≥1/2 the extremal position becomes independent of η, up to an additive constant.
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BibTeX
@misc{drewitz2026sublinear,
title = {{Sublinear Power-Law Growth of Level Set Diameters in Heterogeneous F-KPP Equations}},
author = {Alexander Drewitz and Pascal Hanigk and Lisa Hartung},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03332},
eprint = {2611.03332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03332},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sublinear Power-Law Growth of Level Set Diameters in Heterogeneous F-KPP Equations
AU - Alexander Drewitz
AU - Pascal Hanigk
AU - Lisa Hartung
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03332
UR - https://arxiv.org/abs/2611.03332
AB - For every α∈(1/4,1), we construct a heterogeneous medium ξ such that the diameter of the level sets of the solution to the F-KPP equation is of order t^α along a sequence of times tending to infinity. This is a natural continuation of a series of recent works establishing logarithmic lower bounds as well as linear asymptotics for the diameters of the respective level sets in random and deterministic heterogeneous media. Both results assume supξ> 2infξ; we also show that the diameter can diverge in the critical regime supξ=2infξ. A key ingredient is an estimate for branching Brownian motion with fixed time horizon t and branching rate $1$ on the negative half-line and 2+t^−η on the positive half-line. This estimate exhibits a saturation transition: for η<1/2 the polynomial and logarithmic corrections to the extremal position depend on η, whereas for η≥1/2 the extremal position becomes independent of η, up to an additive constant.
ER -