Logarithmic Bramson correction for Fisher--KPP equations on homogeneous trees
Abstract
In this paper, we study the Fisher--KPP equation ∂t u = αΔTTq+1u+βu(1−u), on the homogeneous (q+1)-regular tree TTq+1, with q≥2, for nontrivial, compactly supported radial initial data. We first obtain an explicit representation of the fundamental solution of the corresponding linearized problem, together with sharp two-sided pointwise estimates. These estimates identify the propagation threshold β>α(√q−1)2 and determine the associated critical speed c∗ and decay rate λ∗. In this regime, we prove that, for every fixed θ∈(0,1), the outermost θ-level set satisfies κθ(t) = c∗ t−3/2λ∗log t+O(1), t→∞. Therefore, while the branching geometry affects the threshold for propagation and the values of the critical parameters, the classical Bramson factor 3/2 persists. Our results therefore extend classical propagation and logarithmic-delay phenomena, previously known for one-dimensional continuous and discrete Fisher--KPP models, to homogeneous trees.
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BibTeX
@misc{alvarez2026logarithmic,
title = {{Logarithmic Bramson correction for Fisher--KPP equations on homogeneous trees}},
author = {Edagardo Álvarez and Marina Murillo-Arcila and Rodrigo Ponce and Juan C. Pozo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33805},
eprint = {2609.33805},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33805},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logarithmic Bramson correction for Fisher--KPP equations on homogeneous trees
AU - Edagardo Álvarez
AU - Marina Murillo-Arcila
AU - Rodrigo Ponce
AU - Juan C. Pozo
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33805
UR - https://arxiv.org/abs/2609.33805
AB - In this paper, we study the Fisher--KPP equation ∂_t u = αΔ_TT_q+1u+βu(1−u), on the homogeneous (q+1)-regular tree TT_q+1, with q≥2, for nontrivial, compactly supported radial initial data. We first obtain an explicit representation of the fundamental solution of the corresponding linearized problem, together with sharp two-sided pointwise estimates. These estimates identify the propagation threshold β>α(√q−1)^2 and determine the associated critical speed c_∗ and decay rate λ_∗. In this regime, we prove that, for every fixed θ∈(0,1), the outermost θ-level set satisfies κ_θ(t) = c_∗ t−3/2λ_∗log t+O(1), t→∞. Therefore, while the branching geometry affects the threshold for propagation and the values of the critical parameters, the classical Bramson factor 3/2 persists. Our results therefore extend classical propagation and logarithmic-delay phenomena, previously known for one-dimensional continuous and discrete Fisher--KPP models, to homogeneous trees.
ER -