Grow-Up Rate and Refined Asymptotics for a Two-Dimensional Patlak–Keller–Segel Model in a Disk

Abstract

We consider a special case of the Patlak–Keller–Segel system in a disc, which arises in the modeling of chemotaxis phenomena. For a critical value of the total mass, the solutions are known to be global in time but with density becoming unbounded, leading to a phenomenon of mass-concentration in infinite time. We establish the precise grow-up rate and obtain refined asymptotic estimates of the solutions. Unlike in most of the similar, recently studied, grow-up problems, the rate is neither polynomial nor exponential. In fact, the maximum of the density behaves like $e^{\sqrt{2t}}$ for large time. In particular, our study provides a rigorous proof of a behavior suggested by Sire and Chavanis [Phys. Rev. E (3), 66 (2002), 046133] on the basis of formal arguments.