Stability of solutions to aggregation equation in bounded domains

TL;DR

This paper investigates the existence, stability, and instability of solutions to a nonlinear aggregation equation in bounded domains with Neumann boundary conditions, focusing on how the integral operator influences solution behavior.

Contribution

It provides new conditions for local and global existence and characterizes stability and instability of constant solutions in bounded domains.

Findings

Conditions for local and global existence established

Criteria for stability of constant solutions derived

Instability conditions identified

Abstract

We consider the aggregation equation $u_t= \div(\nabla u-u\nabla \K(u))$ in a bounded domain $\Omega\subset \R^d$ with supplemented the Neumann boundary condition and with a nonnegative, integrable initial datum. Here, $\K=\K(u)$ is an integral operator. We study the local and global existence of solutions and we derive conditions which lead us to either the stability or instability of constant solutions.