Front propagation through a perforated wall

Abstract: We consider a bistable reaction-diffusion equation $u_t=\Delta u +f(u)$ on $\mathbb{R}^N$ in the presence of an obstacle $K$, which is a wall of infinite span with many holes. More precisely, $K$ is a closed subset of $\mathbb{R}^N$ with smooth boundary such that its projection onto the $x_1$-axis is bounded and that $\mathbb{R}^N \setminus K$ is connected. Our goal is to study what happens when a planar traveling front coming from $x_1 = -\infty$ meets the wall $K$.We first show that there is clear dichotomy between "propagation" and "blocking". In other words, the traveling front either passes through the wall and propagates toward $x_1=+\infty$ (propagation) or is trapped around the wall (blocking), and that there is no intermediate behavior. This dichotomy holds for any type of walls of finite thickness. Next we discuss sufficient conditions for blocking and propagation. For blocking, assuming either that $K$ is periodic in $y:=(x_2,\ldots, x_N)$ or that the holes are localized within a bounded area, we show that blocking occurs if the holes are sufficiently narrow. For propagation, three different types of sufficient conditions for propagation will be presented, namely "walls with large holes", "small-capacity walls", and "parallel-blade walls". We also discuss complete and incomplete invasions.