Maximizing the spreading speed of KPP fronts in two-dimensional stratified media

TL;DR

This paper investigates the maximal spreading speed of KPP fronts in two-dimensional periodic media, showing that optimal speeds are achieved by periodic arrays of Dirac delta functions and analyzing the effects of media periodicity on front propagation.

Contribution

It introduces a method to maximize spreading speed using Dirac delta functions in periodic media and extends the theory of pulsating traveling waves to measure-valued coefficients.

Findings

Maximum spreading speed is attained by periodic Dirac delta arrays.

Spreading speeds are monotonic in the propagation direction for the optimal media.

Asymptotic behavior of spreading fronts varies with the period size L.

Abstract

We consider the equation $u_t=u_{xx}+u_{yy}+b(x)f(u)+g(u)$, $(x,y)\in\mathbb R^2$ with monostable nonliearity, 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, there exists a pulsating travelling wave that propagates in the direction $(\cos\theta,\sin\theta)$ -- with average speed $c$ if and only if $c\geq c^*(\theta,b),$ where $c^*(\theta,b)$ is a certain positive number depending on $b.$ Moreover, the quantity $w(\theta;{b})=\min_{|\theta-\phi|<\frac{\pi}{2}}c^*(\phi;{b})/\cos(\theta-\phi)$ is called the spreading speed. This theory can be extended by showing the existence of the minimal speed $c^*(\theta,b)$ for any nonnegative measure $b$ with period $L.$ We then study the question of maximizing $c^*(\theta,b)$ under the constraint $\int_{[0,L)}b(x)dx=\alpha L,$ where $\alpha$ is an arbitrarily given positive constant. We prove that the maximum is attained by periodically arrayed Dirac's delta functions $h(x)=\alpha L\sum_{k\in\mathbb Z}\delta(x+kL)$ for any direction $\theta$. Based on these results, for the case that $b=h$ we also show the monotonicity of the spreading speedsin $\theta$ and study the asymptotic shape of spreading fronts for large $L$ and small $L$ . Finally, we show that for general 2-dimensional periodic equation $u_t=u_{xx}+u_{yy}+b(x,y)f(u)+g(u)$, $(x,y)\in\mathbb R^2$, the similar conclusions do not hold.