On this talk we will focus on the family of aggregation-diffusion equations

$$\frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho)\nabla (U'(\rho) + V) \right).$$

Here, $\mathrm{m}(s)$ represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. $U$ corresponds to the diffusive potential and includes all the porous medium cases, i.e. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$.

Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. Furthermore, since this problem has a formal gradient-flow structure, we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu.

The talk presents joint work with Prof. J.A. Carrillo and Prof. D.  Gómez-Castro.