The authors consider the parabolic-elliptic chemotaxis model with flux limitation \[ \begin{cases} u_t = \Delta u - \nabla \cdot (u (1 + |\nabla v|^2)^\frac{\alpha-2}{2}) \nabla v), \\ 0 = \Delta v - \mu + u, \quad \mu = \frac{1}{|\Omega|} \int_\Omega u_0, \end{cases} \] complemented with Neumann boundary and initial conditions, in smooth, bounded domains \(\Omega \subset \mathbb R^n\), \(n \ge 3\). In [M. Winkler, Indiana Univ. Math. J. 71, No. 4, 1437–1465 (2022; Zbl 1501.35094)] it has been shown that all solutions are global in time if \(\alpha < \frac{n-1}{n}\) but when \(\alpha > \frac{n-1}{n}\), for each \(\mu > 0\) one can find initial data \(u_0\) with \(\frac{1}{|\Omega|} \int_\Omega u_0 = \mu\) such that the corresponding solution blows up in finite time.
The authors show that a critical mass phenomenon detected in [M. Winkler, Math. Ann. 373, No. 3–4, 1237–1282 (2019; Zbl 1416.35049)] for \(\alpha = 2\) also holds in the regime \(\alpha \in [\frac{n}{n-1}, 2)\). That is, their main result states that if \(\mu\) is large enough and \(\Omega\) is a ball, all radially symmetric initial data strictly more concentrated than the average mass lead to finite-time blow-up, while in contrast for sufficiently small \(\mu\), the homogeneous steady state \((\mu, \mu)\) is locally asymptotically stable with respect to the \(L^\infty\) topology.
The blow-up proof relies on an intricate comparison argument for the transformed quantity \(\int_0^{\xi^\frac1n} \rho^{n-1} u(\rho, t) \,\mathrm d\rho\), while the stability result is obtained by semigroup arguments.