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We consider a special case of the Patlak-Keller-Segel system in a disc, whicharises in the modelling of chemotaxis phenomena. For a critical value of thetotal mass, the solutions are known to be global in time but with densitybecoming unbounded, leading to a phenomenon of mass-concentration in infinitetime. We establish the precise grow-up rate and obtain refined asymptoticestimates of the solutions. Unlike in most of the similar, recently studied,grow-up problems, the rate is neither polynomial nor exponential. In fact, themaximum of the density behaves like $e^{\sqrt{2t}}$ for large time. Inparticular, our study provides a rigorous proof of a behaviour suggested bySire and Chavanis [Phys. Rev. E, 2002] on the basis of formal arguments.

Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk