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

TL;DR

This paper investigates how the critical conditions for blow-up in a chemotaxis system with indirect signal production depend on spatial dimension, revealing a switch in the dimension dependence of critical exponents between low and high dimensions.

Contribution

It identifies a new critical line for blow-up in high dimensions, showing a different dimension dependence than in low dimensions for a chemotaxis model with indirect attractant production.

Findings

Critical line for blow-up in high dimensions:  = m - 1 + 4/n

Different dimension dependence in low vs. high dimensions

Blow-up behavior linked to asymptotic diffusion and cross-diffusion rates

Abstract

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 \[ D(\xi) \simeq \xi^{m-1} \quad \mbox{and} \quad S(\xi) \simeq \xi^\sigma, \qquad \xi \simeq \infty, \] the identity \[ \sigma=m-1+\frac{4}{n} \qquad \qquad (n\ge 3), \] is shown to determine a critical line for the occurrence of blow-up. This considerably differs from low-dimensional cases, in which the relation \[ \sigma=m+\frac{2}{n} \qquad \qquad (n\le 2) \] is known to play a correspondingly pivotal role.