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

We study well-posedness and long-time behaviour of aggregation-diffusion equations of the form $\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}=\mathrm{\Delta }{\rho }^m+\mathrm{\nabla }\cdot (\rho (\mathrm{\nabla }V+\mathrm{\nabla }W\ast \rho ))$ 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 $t\to \mathrm{\infty }$, i.e. the limit as $t\to \mathrm{\infty }$ is of the form $\alpha {\delta }_0+\widehat{\rho }dx$ with $\alpha \ge 0$ and $\widehat{\rho }\in L^1$. Finally, we give instances of $W\mathrm{\ne }0$ showing that partial mass concentration does happen in infinite time, i.e. $\alpha >0$.