Nonlocal interaction PDEs with nonlinear diffusion coupling arise in many contexts in biology and social sciences, in which a PDE formulation can be seen as a multi-scale limit of a system of interacting particles. We are interested in those systems which can be formulated within the Wasserstein gradient flow theory. We shall quickly review the existence theory in a measure sense, and focus on the asymptotic behavior in some particular cases in which nonlocal attraction (concentration phenomena) competes with nonlinear diffusion (large time decay). We shall then focus on the two-species case and prove existence of segregated stationary solutions for a degenerate system with quadratic diffusion. The results are in collaboration with M. Burger (WWU Muester) and S. Fagioli (University of L'Aquila). We shall also propose some open problems to investigate in the future.
More detail can easily be written here using Markdown and $\rm \LaTeX$ math code.