Partial mass concentration for fast-diffusions with non-local aggregation terms

Abstract

We study well-posedness and long-time behaviour of aggregation-diffusion equations of the form $\frac{\partial \rho}{\partial t} = \Delta \rho^m + \nabla \cdot( \rho (\nabla V + \nabla W \ast \rho))$ in the fast-diffusion range, $0<m<1$, and $V$ and $W$ regular enough. We develop a well-posedness theory, first in the ball and then in $\mathbb R^d$, and characterise the long-time asymptotics in the space $W^{-1,1}$ for radial initial data. In the radial setting and for the mass equation, viscosity solutions are used to prove partial mass concentration asymptotically as $t \to \infty$, i.e. the limit as $t \to \infty$ is of the form $\alpha \delta_0 + \widehat \rho \, dx$ with $\alpha \geq 0$ and $\widehat \rho \in L^1$. Finally, we give instances of $W \ne 0$ showing that partial mass concentration does happen in infinite time, i.e. $\alpha > 0$.