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 $\alpha {\delta }_0+\hat{\rho }dx$ with $\alpha \ge 0$ and $\hat{\rho }\in L^1$. Finally, we give instances of $W\ne 0$ showing that partial mass concentration does happen in infinite time, i.e. $\alpha >0$.