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Propagation of bistable fronts through a perforated wall Matano, Hiroshi

Description

We consider a bistable reaction-diffusion equation on ${\bf R}^N$ in the presence of an obstacle $K$, which is a wall of infinite span with periodically arrayed holes. More precisely, $K$ is a closed subset of ${\bf R}^N$ with smooth surface such that its projection onto the $x_1$-axis is bounded, while it is periodic in the rest of variables $(x_2,\ldots, x_N)$. We assume that ${\bf R}^N \setminus K$ 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 {\it propagation} and {\it blocking}. In other words, the traveling front either completely penetrates through the wall or is totally blocked, and that there is no intermediate behavior. This dichotomy result will be proved by what we call a De Giorgi type lemma for the elliptic equation $\Delta v + f(v) = 0$ on ${\bf R}^N$. Then we will discuss sufficient conditions for blocking, and those for propagation. This is joint work with Henri Berestycki and Fran\c{c}ois Hamel. If time allows, I will also talk about the non-KPP monostable equation with the nonlinearity $u^p(1-u)$, $p>1$, and discuss briefly whether or not blocking can occur for the front solution.

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