A note on the $8\pi$ problem of J\"ager-Luckhaus system

TL;DR

This paper proves that the Jäger-Luckhaus Keller-Segel system with critical mass admits a globally bounded solution in the unit disk, with the equilibrium at 8 being globally and exponentially stable.

Contribution

It establishes global boundedness and stability results for the critical mass case of the Jäger-Luckhaus system in a bounded domain.

Findings

Existence of globally bounded classical solutions at critical mass 8π.

Exponential stability of the spatial constant equilibrium 8.

Global stability holds for radially symmetric, continuous initial data.

Abstract

We show that for any nonnegative, radially symmetric and continuous initial datum with critical mass $8\pi$, J\"ager-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.