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arXiv · 1804.07485

A counterexample to the Liouville property of some nonlocal problems

Julien Brasseur·Jérôme Coville
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Abstract

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,( u(y)-u(x) )\mathrm{d}y+f(u(x))=0, \quad x\in\R^N\setminus K,$$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 $u(x)\to1$ 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.

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Julien Brasseur, Jérôme Coville. 2018-04-20. A counterexample to the Liouville property of some nonlocal problems. https://arxiv.org/abs/1804.07485

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