GROW-UP RATE AND REFINED ASYMPTOTICS FOR ATWO-DIMENSIONAL PATLAK—KELLER—SEGEL MODEL IN A DISK

摘要

We consider a special case of the Patlak–Keller–Segel system in a disc, which arisesin the modeling of chemotaxis phenomena. For a critical value of the total mass, the solutions areknown 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 asymptoticestimates of the solutions. Unlike in most of the similar, recently studied, grow-up problems, therate is neither polynomial nor exponential. In fact, the maximum of the density behaves like e's./ztfor large time. In particular, our study provides a rigorous proof of a behavior suggested by Sire andChavanis [Phys. Rev. E (3), 66 (2002), 046133] on the basis of formal arguments.