| Abstract: |
| 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 $\frac{S(u)}{D(u)} \sim u^{\frac{n}{2}}$ 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. |
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