Stability of solutions to aggregation equation in bounded domains

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.