On the large time behavior of the solutions of a nonlocal ordinary differential equation with mass conservation

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal differential equation with mass conservation, demonstrating convergence to steady states using rearrangement theory.

Contribution

It provides new insights into the asymptotic behavior of nonlocal ODEs with mass conservation in multiple dimensions, including proof of convergence to steady states.

Findings

Solutions converge to steady states over time

Relatively compact solution orbits are established

Rearrangement theory is used to analyze solution behavior

Abstract

We consider an initial value problem for a nonlocal differential equation with a bistable nonlinearity in several space dimensions. The equation is an ordinary differential equation with respect to the time variable t, while the nonlocal term is expressed in terms of spatial integration. We discuss the large time behavior of solutions and prove, among other things, the convergence to steady-states. The proof that the solution orbits are relatively compact is based upon the rearrangement theory.