Spatial Decay and Asymptotic Stability of Forced Waves at the Critical Speed for the Fisher-KPP Equation in a Shifting Environment
Abstract
In this paper, we investigate the spatial decay and asymptotic stability of forced waves for the Fisher-KPP equation in a shifting heterogeneous environment, focusing on the critical speed c*=2√(r+). Within the framework of regular variation, we establish a complete classification of critical wave tails in terms of the asymptotic behavior of the environmental deviation r+−r(z) as z→+∞. In each case, we derive precise spatial decay that characterize how the environmental tail modifies the leading-order decay of the wave at criticality. We further establish asymptotic stability estimates of forced waves. For initial data satisfying an exponentially weighted boundedness condition, we prove an O(t−1/2) convergence rate in the uniform norm. Under stronger spatial decay assumptions on the weighted initial data, this rate improves to O(t−1−σ/2), for σ∈[0,1], in both the uniform norm and a weighted norm. Moreover, no smallness assumption on the initial perturbations is needed for our results.
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BibTeX
@misc{tang2026spatial,
title = {{Spatial Decay and Asymptotic Stability of Forced Waves at the Critical Speed for the Fisher-KPP Equation in a Shifting Environment}},
author = {Zhibao Tang and Lina Wang and Shi-Liang Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30820},
eprint = {2610.30820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30820},
}
RIS
TY - UNPB
TI - Spatial Decay and Asymptotic Stability of Forced Waves at the Critical Speed for the Fisher-KPP Equation in a Shifting Environment
AU - Zhibao Tang
AU - Lina Wang
AU - Shi-Liang Wu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30820
UR - https://arxiv.org/abs/2610.30820
AB - In this paper, we investigate the spatial decay and asymptotic stability of forced waves for the Fisher-KPP equation in a shifting heterogeneous environment, focusing on the critical speed c_*=2√(r_+). Within the framework of regular variation, we establish a complete classification of critical wave tails in terms of the asymptotic behavior of the environmental deviation r_+−r(z) as z→+∞. In each case, we derive precise spatial decay that characterize how the environmental tail modifies the leading-order decay of the wave at criticality. We further establish asymptotic stability estimates of forced waves. For initial data satisfying an exponentially weighted boundedness condition, we prove an O(t^−1/2) convergence rate in the uniform norm. Under stronger spatial decay assumptions on the weighted initial data, this rate improves to O(t^−1−σ/2), for σ∈[0,1], in both the uniform norm and a weighted norm. Moreover, no smallness assumption on the initial perturbations is needed for our results.
ER -