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This paper is concerned with the spatially periodic Fisher-KPP equation$u_t=(d(x)u_x)_x+(r(x)-u)u$, $x\in \mathbb{R}$, where $d(x)$ and $r(x)$ areperiodic functions with period $L>0$. We assume that $r(x)$ has positive meanand $d(x)>0$. It is known that there exists a positive number $c^*_d(r)$,called the minimal wave speed, such that a periodic traveling wave solutionwith average speed $c$ exists if and only if $c \geq c^*_d(r)$. In theone-dimensional case, the minimal speed $c^*_d(r)$ coincides with the``spreading speed'', that is, the asymptotic speed of the propagating front ofa solution with compactly supported initial data. In this paper, we study theminimizing problem for the minimal speed $c^*_d(r)$ by varying $r(x)$ under acertain constraint, while $d(x)$ arbitrarily. We have been able to obtain anexplicit form of the minimizing function $r(x)$. Our result provides the firstcalculable example of the minimal speed for spatially periodic Fisher-KPPequations as far as the author knows.

Determining the optimal coefficient of the spatially periodic Fisher-KPP equation that minimizes the spreading speed