A variational problem associated with the minimal speed of traveling waves for spatially periodic KPP type equations

Abstract

We consider a variational problem associated with the minimal speed of pulsating traveling waves of the equation $u_t=u_{xx}+b(x)(1-u)u$, $x\in{\mathbb R},\ t>0$, where the coefficient $b(x)$ is nonnegative and periodic in $x\in{\mathbb R}$ with a period $L>0$. It is known that there exists a quantity $c^*(b)>0$ such that a pulsating traveling wave with the average speed $c>0$ exists if and only if $c\geq c^*(b)$. The quantity $c^*(b)$ is the so-called minimal speed of pulsating traveling waves. In this paper, we study the problem of maximizing $c^*(b)$ by varying the coefficient $b(x)$ under some constraints. We prove the existence of the maximizer under a certain assumption of the constraint and derive the Euler--Lagrange equation which the maximizer satisfies under $L^2$ constraint $\int_0^L b(x)^2dx=\beta$. The limit problems of the solution of this Euler--Lagrange equation as $L\rightarrow0$ and as $\beta\rightarrow0$ are also considered. Moreover, we also consider the variational problem in a certain class of step functions under $L^p$ constraint $\int_0^L b(x)^pdx=\beta$ when $L$ or $\beta$ tends to infinity.