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

TL;DR

This paper investigates a variational problem to maximize the minimal speed of pulsating traveling waves in periodic KPP equations by optimizing the coefficient function under various constraints, deriving conditions for optimality and analyzing limiting cases.

Contribution

It introduces a new variational framework for maximizing the minimal wave speed in periodic KPP equations, including existence results and Euler-Lagrange equations for the optimal coefficient.

Findings

Existence of a maximizer under certain constraints.

Derivation of Euler-Lagrange equations for the optimal coefficient.

Analysis of limit cases as parameters tend to zero or infinity.

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.