发表机构
S.I.S.S.A.（的里雅斯特高等研究学校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对入侵物种根除模型的努力函数，构造反例证明其相关假设不成立，并确定该函数仅为$C^1$类，修正了文献中的相关结论。
AI 中文摘要
在本注记中，我们研究“努力函数$E(\beta)$”的正则性性质。该函数可定义为控制量$\tilde{\alpha}(x) \geq 0$的最小$L^1$范数，使得偏微分方程$U_t = \Delta U + f(U) - \tilde{\alpha}(t,x) U$（其中$U \in [0,1]$）存在连接0与1的行波剖面$U(x - \beta t)$。上述偏微分方程用于模拟入侵物种的演化，控制量$\tilde{\alpha}(t,x)$表示物种清除的百分比。文献中对$f(U)$的性质及由此产生的努力函数$E(\beta)$通常假设一些条件，主要包括行波剖面具有简单结构，且努力函数可作为各向异性周长生成有限周长集合上的代价函数。本注记给出若干例子表明，在PDE具有非平凡动力学的有趣情形下，这些假设通常不成立；我们还证明，$E(\beta)$通常仅为$C^1$类函数。
英文摘要
In this note we study the regularity properties of the \emph{Effort Function $E(β)$}. This can be defined as the minimal $L^1$-norm of the control $\tilde α(x) \geq 0$ so that the PDE $$U_t = ΔU + f(U) - \tilde α(t,x) U, \quad U \in [0,1], %\ \tilde α\geq 0,$$ admits a traveling profile $U(x - βt)$ connecting $0$ to $1$. The above PDE models the evolution of an invasive species, where the control $\tilde α(t,x)$ describes the percentage amount of species removal. In the literature some conditions are assumed on the properties of $f(U)$ and (consequently) on the effort function $E(β)$, the main ones being that the traveling profile has a simple structure and that the effort function can be used as an anisotropic perimeter to generated a cost function over sets of finite perimeter. This note gives some examples showing that in the interesting cases, i.e. when the PDE has a non trivial dynamics, these assumptions are in general not satisfied. We also show that $E(β)$ is in general only $C^1$.
Comments17 pages, 5 figures
