发表机构
University of Science and Technology of China; Nankai University; Anhui University（中国科学技术大学; 南开大学; 安徽大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在几乎周期介质中，Fisher-KPP方程过渡前沿的平均速度受限于谱参数与Lyapunov指数之比，并揭示快于该界传播时前沿宽度无界，与周期介质形成对比。
AI 中文摘要
本文研究了一维几乎周期介质中（实数轴和格点上的）Fisher--KPP方程的过渡前沿和传播解。设$\lambda_1$分别为作用在$L^2(\R)$或$\ell^2(\Z)$上的线性化算子谱的上确界，并令$L(\lambda_1)$表示谱参数$\lambda_1$处的空间Lyapunov指数。我们证明，若$L(\lambda_1)>0$，则任何过渡前沿的全局平均速度至多为$\lambda_1/L(\lambda_1)$。此外，若Cauchy问题的解以快于该界的速度传播，则其过渡宽度沿时间序列无界。这发生在一类具有缓慢衰减指数尾部的初始数据上。这与周期介质形成对比，在周期介质中，脉动前沿以高于最小速度的每个速度存在。作为应用，我们考虑了几乎Mathieu系数和具有两个频率的连续准周期系数。结合Nadin--Rossi和Liang--Wang--Zhou--Zhou的结果，我们的结果为广义过渡前沿的容许速度提供了几乎完整的图景，同时留下了临界情形未解决。
英文摘要
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 $\ell^2(\Z)$, 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.
Comments24 pages
