Abstract: The aggregation-diffusion equation is a nonlocal PDE driven by two competing effects: local repulsion modeled by nonlinear diffusion, and long-range attraction modeled by nonlocal interaction. I will talk about how this equation arises in modeling the collective motion of cells, and discuss several qualitative properties of its steady states and dynamical solutions.Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint work with Carrillo, Hittmeir and Volzone). In a recent work, we further investigate whether they are unique within the radial class, and show that for a given mass, uniqueness/non-uniqueness of steady states are determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint work with Delgadino and Yan.)

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