| Abstract: |
| We consider a bistable reaction-diffusion equation on ${\mathbb R}^N\setminus K$, where $K$ represents an obstacle that can be regarded as an infinite wall of finite thickness with periodically arrayed holes. More precisely, $K$ is a set with smooth surface whose projection onto the $x_1$-axis is bounded, while it is periodic in the rest of variables $\tilde x:=(x_2,\ldots,x_N)$. We assume that ${\mathbb R}^N\setminus K$ is connected.
Our goal is to study what happens when a planar traveling front coming from $x_1=+\infty$ encounters the wall $K$. We first show that there is clear dichotomy between {\it blocking} and {\it propagation}, and that there is no intermediate behavior of solutions. To prove this result, we first establish a De Giorgi type theorem for the elliptic equation $\Delta u+f(u)=0$ on ${\mathbb R}^N$, which may be of interest in its own right. Then we will discuss sufficient conditions for propagation and those for blocking. |
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