This talk is concerned with the asymptotic behaviour of two types of Aggregation-Diffusion PDEs. On the first part of the talk, I will discuss about the asymptotic behaviour of the family of Aggregation-Diffusion equations

$$

\partial_t\rho=\Delta\rho^m+\mbox{div}(\rho\nabla (V+W\ast\rho )),

$$

for $0

$$

\partial_t \rho = \mbox{div}(\mathrm{m}(\rho) \nabla (U'(\rho) + V))

$$

where $\mathrm{m}(s)$ is a saturation term, i.e. bounded and with compact support, $U$ is a convex diffusive potential, and $V$ is regular enough. Taking advantage of a regularised version of this problem, we study existence and long-time behaviour in a general setting, showing the appearance of kinks.

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