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

Youshan Tao, Michael Winkler

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

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

Abstract

In bounded -dimensional domains with , 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 and of diffusion and cross-diffusion are suitably regular functions of the population density, essentially exhibiting asymptotic behavior of the form the identity is shown to determine a critical line for the occurrence of blow-up. This considerably differs from low-dimensional cases, in which the relation is known to play a correspondingly pivotal role.

Paper Structure

This paper contains 11 sections, 21 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Local existence and extensibility
  3. Blow-up. Proof of Theorem \ref{['theo13']}
  4. A cooperative parabolic system satisfied by cumulated densities
  5. Construction of exploding subsolutions: a general template
  6. Subsolution properties: inner region
  7. Subsolution properties: outer region
  8. Conclusion. Proof of Theorem \ref{['theo13']}
  9. Global existence for subcritical $\sigma$
  10. Criticality of the line $\sigma=m-1+\frac{4}{n}$. Proof of Proposition \ref{['prop23']}
  11. Criticality of the line $\sigma=\frac{2}{n}$. Proof of Proposition \ref{['prop25']}

Key Result

Theorem 1.1

Let $n\ge 3, R>0$ and $\Omega=B_R\equiv B_R(0)\subset\mathbb{R}^n$, let and assume an that and with some $\xi_0>0$, $K_D>0$, $k_S>0$, $m\in\mathbb{R}$ and $\sigma\in\mathbb{R}$ satisfying Then for any $T^\star>0, M_\star>0$ and $M^\star>M_\star$, one can find $M^{(u)}\in C^0([0,R])$ and $M^{(w)} \in C^0([0,R])$ such that that and that whenever with as well as the corresponding solution $

Theorems & Definitions (22)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Definition 3.5
  • Lemma 3.6
  • ...and 12 more