发表机构
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh（爱丁堡大学数学学院与麦克斯韦数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对退化Fisher-KPP方程，采用几何去奇异性方法，证明了波速略高于最小波速时，行波前解会从尖锐过渡为光滑。
AI 中文摘要
我们研究一类退化Fisher-KPP方程$u_t=(u^n u_x)_x+u(1-u^n)$的行波前解，其中$n$为正整数。在最小波速$c_{\rm min}=\tfrac{1}{\sqrt{1+n}}$处，这些方程存在尖锐前解，对应行波相空间中的显式异宿轨道。我们展示当波速增加到$c=\tfrac{1}{\sqrt{1+n}}+\varepsilon$（$\varepsilon$足够小）时，该尖锐前解如何被扰动。我们在前缘的退化平衡点附近采用动机明确的几何去奇异性（亦称爆破）方法，通过分析所得的定向和重标度图表，构造连接相关渐近态的奇异异宿轨道，为从尖锐到光滑行波的过渡提供了一个简单的几何证明。
英文摘要
We study travelling front solutions of a family of degenerate Fisher-KPP equations $u_t=(u^n u_x)_x+u(1-u^n)$, where $n$ is a positive integer. At the minimal wave speed $c_{\rm min}=\tfrac{1}{\sqrt{1+n}}$, these equations admit sharp front solutions, corresponding to an explicit heteroclinic orbit in the travelling wave phase-space. We show how this sharp front is perturbed when the wave speed is increased to $c=\tfrac{1}{\sqrt{1+n}}+\varepsilon$, with $\varepsilon$ sufficiently small. We use a well-motivated geometric desingularisation (also known as blow-up) near the degenerate equilibrium at the leading edge. By analysing the resulting directional and rescaling charts, we construct a singular heteroclinic orbit connecting the relevant asymptotic states, providing a simple geometric proof of the transition from sharp to smooth travelling fronts.
