We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on $0<x<L(t)$. A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with $L(t)>L_{\text{c}}$ will eventually spread as $t\rightarrow \infty$, whereas solutions where $L(t)\ngtr L_{\text{c}}$ will vanish as $t\rightarrow \infty$. In one dimension it is well known that the critical length is $L_{\text{c}}=\unicode[STIX]{x1D70B}/2$. In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate $L_{\text{c}}$ as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how $L_{\text{c}}$ depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

中文翻译：

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高维传播-消失二分法的关键长度

我们考虑经典Fisher-Kolmogorov方程的扩展，称为“Fisher-Stefan”模型，它是一个移动边界问题$0<x<L(t)$. Fisher-Stefan 模型的一个关键特性是“传播-消失二分法”，其中具有$L(t)>L_{\text{c}}$最终会传播为$t\rightarrow \infty$，而解决方案$L(t)\ngtr L_{\text{c}}$将消失为$t\rightarrow \infty$. 众所周知，在一维中，临界长度为$L_{\text{c}}=\unicode[STIX]{x1D70B}/2$. 在这项工作中，我们重新制定了更高维度的 Fisher-Stefan 模型并计算$L_{\文本{c}}$作为径向对称坐标系中空间维度的函数。我们的结果表明如何$L_{\文本{c}}$取决于问题的维度，控制偏微分方程的数值解与我们的计算一致。