Abstract

Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller–Segel model [V. Calvez and J. A. Carrillo, J. Math. Pures Appl. (9), 86 (2006), pp. 155–175]. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyze the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller–Segel model for general initial data. This is a joint work with J.A. Carrillo (Imperial College, London) and Bruno Volzone (Napoli Parthenope).