This paper is concerned with the spatially periodic Fisher-KPP equation [Formula: see text], [Formula: see text], where d(x) and r(x) are periodic functions with period [Formula: see text]. We assume that r(x) has positive mean and [Formula: see text]. It is known that there exists a positive number [Formula: see text], called the minimal wave speed, such that a periodic traveling wave solution with average speed c exists if and only if [Formula: see text]. In the one-dimensional case, the minimal speed [Formula: see text] coincides with the "spreading speed", that is, the asymptotic speed of the propagating front of a solution with compactly supported initial data. In this paper, we study the minimizing problem for the minimal speed [Formula: see text] by varying r(x) under a certain constraint, while d(x) arbitrarily. We have been able to obtain an explicit form of the minimizing function r(x). Our result provides the first calculable example of the minimal speed for spatially periodic Fisher-KPP equations as far as the author knows.

中文翻译：

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确定空间分布的Fisher-KPP方程的最佳系数，该系数可使扩展速度最小。

本文涉及空间周期性Fisher-KPP方程[公式：参见文本]，[公式：参见文本]，其中d（x）和r（x）是周期为周期的周期函数[公式：参见文本]。我们假设r（x）具有正平均值，并且[公式：参见文本]。已知存在一个正数[公式：参见文本]，称为最小波速，这样，当且仅当[公式：参见文本]时，才存在具有平均速度c的周期性行波解。在一维情况下，最小速度[公式：参见文本]与“传播速度”一致，即具有紧密支持的初始数据的解决方案的传播前沿的渐近速度。在本文中，我们研究了通过在一定约束下任意改变r（x）而任意改变d（x）来实现最小速度的最小化问题。我们已经能够获得最小化函数r（x）的显式形式。据作者所知，我们的结果提供了空间周期性Fisher-KPP方程的最小速度的第一个可计算示例。