Critical mass and stability of radial steady states for a flux-limited Keller–Segel system in the critical case

This paper deals with a flux-limited Keller–Segel system with critical blow-up exponent. We first show the existence of a number M > 0 with the property that all radial solutions remain bounded if the mass of initial data is smaller than M , and that there exists radial initial data with the mass bigger than M such that the corresponding solution blows up in finite time. We next consider behaviour of radial solutions having the critical mass M , and prove a result ensuring a certain stability property of steady states which were found in Kohatsu and Senba (2025 Nonlinear Anal. Real World Appl. 84 104308). The analysis is based on discovering a Lyapunov functional structure of the system, which is a generalisation of the one in a classical Keller–Segel system.