A counterexample to the Liouville property of some nonlocal problems

TL;DR

This paper constructs a counterexample demonstrating that the Liouville property does not hold for certain nonlocal reaction-diffusion equations with specific obstacle shapes, challenging previous assumptions about solution behavior.

Contribution

It introduces new nontrivial obstacle configurations where the Liouville property fails, expanding understanding of solution behaviors in nonlocal PDEs.

Findings

Counterexamples for nonlocal reaction-diffusion equations with non-starshaped obstacles.

Demonstration that the Liouville property can fail under certain geometric conditions.

Identification of specific conditions where solutions are not identically constant.

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.