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

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 $(\cosθ,\sinθ)$ -- with average speed $c$ if and only if $c\geq c^*(θ,b),$ where $c^*(θ,b)$ is a certain positive number depending on $b.$ Moreover, the quantity $w(θ;{b})=\min_{|θ-ϕ|<\fracπ{2}}c^*(ϕ;{b})/\cos(θ-ϕ)$ is called the spreading speed. This theory can be extended by showing the existence of the minimal speed $c^*(θ,b)$ for any nonnegative measure $b$ with period $L.$ We then study the question of maximizing $c^*(θ,b)$ under the constraint $\int_{[0,L)}b(x)dx=αL,$ where $α$ is an arbitrarily given positive constant. We prove that the maximum is attained by periodically arrayed Dirac's delta functions $h(x)=αL\sum_{k\in\mathbb Z}δ(x+kL)$ for any direction $θ$. Based on these results, for the case that $b=h$ we also show the monotonicity of the spreading speedsin $θ$ 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.