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Bramson correction and convergence to the critical wave for Fisher-KPP equations on ℤd
Abstract
We consider Fisher-KPP equations with nearest-neighbor diffusion on ℤd, d≥2, with nonzero finitely supported initial data. We prove a logarithmic delay of the front along each signed coordinate axis and show that the transition region has uniformly bounded width. On every fixed-width half-tube around an axis, the solution converges to translates of the minimal-speed lattice traveling wave, with a bounded phase. We also obtain an upper bound with a logarithmic correction in every direction. The proof uses weighted estimates, bounds for tilted random walks, and a product lower solution. Concavity of the reaction is not assumed.
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BibTeX
@misc{hu2026bramson,
title = {{Bramson correction and convergence to the critical wave for Fisher-KPP equations on $\mathbb Z^d$}},
author = {Yuanyang Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10143},
eprint = {2609.10143},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10143},
}
RIS
TY - UNPB
TI - Bramson correction and convergence to the critical wave for Fisher-KPP equations on ℤ^d
AU - Yuanyang Hu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10143
UR - https://arxiv.org/abs/2609.10143
AB - We consider Fisher-KPP equations with nearest-neighbor diffusion on ℤ^d, d≥2, with nonzero finitely supported initial data. We prove a logarithmic delay of the front along each signed coordinate axis and show that the transition region has uniformly bounded width. On every fixed-width half-tube around an axis, the solution converges to translates of the minimal-speed lattice traveling wave, with a bounded phase. We also obtain an upper bound with a logarithmic correction in every direction. The proof uses weighted estimates, bounds for tilted random walks, and a product lower solution. Concavity of the reaction is not assumed.
ER -