In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form$\underset{{\mathbb{R}}^N\setminus K}{\int }J(x-y)(u(y)-u(x))\mathrm{d}y+f(u(x))=0,x\in {\mathbb{R}}^N\setminus K,$ where $K\subset {\mathbb{R}}^N$ is a bounded compact set, called an “obstacle”, and f is a bistable nonlinearity. When K is convex, it is known that solutions ranging in $[0,1]$ and satisfying $u(x)\to 1$ as $|x|\to \infty$ must be identically 1 in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data f and J for which this property fails.

中文翻译：

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一些非本地问题对Liouville属性的反例

在本文中，我们构造了以下形式的一些非局部反应扩散方程的Liouville性质的反例$\underset{{\mathbb{[R}}^{ñ}\setminus ķ}{\int }Ĵ（X-ÿ）（ü（ÿ）-ü（X））\mathrm{d}ÿ+F（ü（X））=0，X\in {\mathbb{[R}}^{ñ}\setminus ķ，$ 哪里 $ķ\subset {\mathbb{[R}}^{ñ}$是有界紧集，称为“障碍”，f是双稳态非线性。当K为凸时，已知解的范围为$[0，\mathrm{1个}]$ 并令人满意 $ü（X）\to \mathrm{1个}$ 如 $|X|\to \infty$在整个空间中必须等于1。我们构造了一个简单连接的（非星形）障碍物的非平凡族，以及构造失败的数据f和J。