A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations

TL;DR

This paper extends the theory of minimal wave speeds in reaction-diffusion equations with periodic coefficients to measure-valued cases and identifies the optimal measure configuration for maximizing wave speed.

Contribution

It proves the existence of minimal wave speed for measure-valued coefficients and determines the optimal measure distribution to maximize this speed.

Findings

Existence of minimal wave speed for measure-valued periodic coefficients.

Maximum wave speed achieved by periodically arrayed Dirac delta functions.

Addresses a problem of ecological and mathematical interest from the 1980s.

Abstract

We consider the equation $u_t=u_{xx}+b(x)u(1-u),$ $x\in\mathbb R,$ where $b(x)$ is a nonnegative measure on $\mathbb R$ that is periodic in $x.$ In the case where $b(x)$ is a smooth periodic function, it is known that there exists a travelling wave with speed $c$ for any $c\geq c^*(b),$ where $c^*(b)$ is a certain positive number depending on $b.$ Such a travelling wave is often called a \lq\lq pulsating travelling wave" or a \lq\lq periodic travelling wave", and $c^*(b)$ is called the \lq\lq minimal speed". In this paper, we first extend this theory by showing the existence of the minimal speed $c^*(b)$ for any nonnegative measure $b$ with period $L.$ Next we study the question of maximizing $c^*(b)$ under the constraint $\int_{[0,L)}b(x)dx=\alpha L,$ where $\alpha$ is an arbitrarily given constant. This question is closely related to the problem studied by mathematical ecologists in late 1980's but its answer has not been known. We answer this question by proving that the maximum is attained by periodically arrayed Dirac's delta functions $\alpha L\sum_{k\in\mathbb Z}\delta(x+kL).$