A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production

In bounded $n$-dimensional domains with $n\ge 3$, this manuscript considers an initial-boundary problem for a quasilinear chemotaxis system with indirect attractant production, as arising, inter alia, in the modeling of effects due to phenotypical heterogeneity in microbial populations. Under the assumption that the rates $D$ and $S$ of diffusion and cross-diffusion are suitably regular functions of the population density, essentially exhibiting asymptotic behavior of the form $$D(\xi )\simeq {\xi }^{m-1}andS(\xi )\simeq {\xi }^{\sigma },\xi \simeq \mathrm{\infty },$$ the identity $$\sigma =m-1+\frac{4}{n}(n\ge 3),$$ is shown to determine a critical line for the occurrence of blow-up. This considerably differs from low-dimensional cases, in which the relation $$\sigma =m+\frac{2}{n}(n\le 2)$$ is known to play a correspondingly pivotal role.