Liouville type results for a nonlocal obstacle problem

TL;DR

This paper investigates Liouville-type theorems for solutions to nonlocal reaction-diffusion equations set outside a bounded obstacle, establishing conditions under which solutions exhibit specific qualitative behaviors at infinity.

Contribution

It extends Liouville results to nonlocal equations with obstacles, including convex and slightly non-convex sets, under asymptotic conditions.

Findings

Liouville-type results for convex obstacles

Robustness of results under relaxed obstacle conditions

Qualitative behavior of solutions at infinity

Abstract

This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact "obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.