A note on the $8\ddot{I}€$ problem of Jäger-Luckhaus system

Abstract

We show that for any nonnegative, radially symmetric and continuous initial datum with critical mass $8\ddot{I}€$, Jäger-Luckhaus system in the unit disk, known as a parabolic-elliptic Keller-Segel model, admits a globally bounded classical solution. Moreover, it is asserted that the spatial constant equilibrium $8$ is globally and exponentially asymptotically stable.

convergence rates are further deduced