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

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 (more precisely a “pulsating travelling wave”) with average speed $c$ if and only if $c\geq c^*(b),$ where $c^*(b)$ is a certain positive number depending on $b.$ This constant $c^*(b)$ is called the “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 positive constant. This question is closely related to the problem studied by mathematical ecologists in late 1980s 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 $\alpha L\sum _{k\in \mathbb Z}\delta (x+kL).$