On the $\omega$-limit set of a nonlocal differential equation: application of rearrangement theory

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal differential equation with conserved integral, using rearrangement theory to characterize the omega-limit set and establish its uniqueness.

Contribution

The authors apply rearrangement theory to analyze the omega-limit set of a nonlocal differential equation, showing it contains only a single element under certain conditions.

Findings

The omega-limit set of solutions is a singleton.

Rearrangement theory links higher-dimensional equations to one-dimensional cases.

The method characterizes long-term solution behavior in nonlocal equations.

Abstract

We study the $\omega$-limit set of solutions of a nonlocal ordinary differential equation, where the nonlocal term is such that the space integral of the solution is conserved in time. Using the monotone rearrangement theory, we show that the rearranged equation in one space dimension is the same as the original equation in higher space dimensions. In many cases, this property allows us to characterize the $\omega$-limit set for the nonlocal differential equation. More precisely, we prove that the $\omega$-limit set only contains one element.