An Upper Bound for the Mean Speed of Transition Fronts and Unbounded Front Widths for Fisher KPP Equations in Almost Periodic Media
Abstract
In this paper, we investigate transition fronts and spreading solutions of Fisher--KPP equations in one-dimensional almost periodic media, both on the real line and on the lattice. Let λ1 be the supremum of the spectrum of the linearized operator acting on L2(R) or ℓ2(Z), respectively, and let L(λ1) denote the spatial Lyapunov exponent at the spectral parameter λ1. We prove that, if L(λ1)>0, the global mean speed of any transition front is at most λ1/L(λ1). Moreover, if a solution of the Cauchy problem spreads faster than this bound, its transition width is unbounded along a sequence of times. This occurs for a class of initial data with slowly decaying exponential tails. This contrasts with periodic media, where pulsating fronts exist at every speed above the minimal speed. As applications, we consider almost Mathieu coefficients and a continuous quasiperiodic coefficient with two frequencies. Together with the results of Nadin--Rossi and Liang--Wang--Zhou--Zhou \citep{LWZZ24}, our results provide an almost complete picture of the admissible speeds of generalized transition fronts, while leaving the critical cases open.
Related
- Bistable fronts in periodic media: phase reduction and realization of circle flows
- Existence of multi-dimensional pulsating fronts for KPP equations: a new formulation approach
- Stationary bistable fronts on a periodic necklace graph: existence, stability and multiplicity
- Generalized Transition Fronts and Spectral Speed Asymptotics in Heterogeneous Cooperative Systems
- Lattice Green's functions for Poisson problems with two unbounded and one periodic direction
BibTeX
@misc{liang2026upper,
title = {{An Upper Bound for the Mean Speed of Transition Fronts and Unbounded Front Widths for Fisher KPP Equations in Almost Periodic Media}},
author = {Xing Liang and Linfeng Xu and Qi Zhou and Tao Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30760},
eprint = {2609.30760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30760},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Upper Bound for the Mean Speed of Transition Fronts and Unbounded Front Widths for Fisher KPP Equations in Almost Periodic Media
AU - Xing Liang
AU - Linfeng Xu
AU - Qi Zhou
AU - Tao Zhou
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30760
UR - https://arxiv.org/abs/2609.30760
AB - In this paper, we investigate transition fronts and spreading solutions of Fisher--KPP equations in one-dimensional almost periodic media, both on the real line and on the lattice. Let λ_1 be the supremum of the spectrum of the linearized operator acting on L^2(R) or ℓ^2(Z), respectively, and let L(λ_1) denote the spatial Lyapunov exponent at the spectral parameter λ_1. We prove that, if L(λ_1)>0, the global mean speed of any transition front is at most λ_1/L(λ_1). Moreover, if a solution of the Cauchy problem spreads faster than this bound, its transition width is unbounded along a sequence of times. This occurs for a class of initial data with slowly decaying exponential tails. This contrasts with periodic media, where pulsating fronts exist at every speed above the minimal speed. As applications, we consider almost Mathieu coefficients and a continuous quasiperiodic coefficient with two frequencies. Together with the results of Nadin--Rossi and Liang--Wang--Zhou--Zhou \citep{LWZZ24}, our results provide an almost complete picture of the admissible speeds of generalized transition fronts, while leaving the critical cases open.
ER -