In the higher dimensional setting, critical mass phenomena are known to occur in the quasilinear Keller--Segel system for a variety of different diffusion rates $D(u)$ and taxis sensitivity functions $S(u)$ being critical in the sense that $S(u)/D(u) \sim u^{n/2}$ for large $u$, where $n$ denotes the space dimension. The most famous example is the two-dimensional minimal Keller--Segel system given by $D(u)=1$ and $S(u)=u$, for which the mass $4\pi$ (or $8\pi$ in the radially symmetric case) distinguishes between boundedness and the possibility of blow-up.

In this talk, based on a recent joint work with Xinru Cao, it is shown that this is no longer the case for one-dimensional domains: Solutions of the quasilinear system with $D(u)=(u+1)^{m−1}$ and $S(u)=u(u+1)^m$ for (many) $m \in \mathbb R$ emanating from initial data with arbitrary large mass are globally bounded. Accordingly, the absence of a critical mass phenomenon appears to be a general property of the one-dimensional setting and is not limited to the case $m=0$ already studied in the literature.