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

ABSTRACT

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.