Critical mass phenomena in higher dimensional quasilinear Keller-Segel systems with indirect signal production

TL;DR

This paper investigates the critical mass phenomena and global existence versus blow-up in higher-dimensional quasilinear Keller-Segel systems with indirect signal production, revealing conditions under which solutions remain bounded or blow up.

Contribution

It establishes new thresholds for global existence and blow-up based on the parameter m and initial mass, highlighting a critical mass phenomenon in higher dimensions.

Findings

Global solutions exist for certain m and initial mass conditions.

Solutions blow up when m and initial mass exceed specific thresholds.

Identifies a critical mass in the case m=2-2/n.

Abstract

In this paper, we deal with quasilinear Keller--Segel systems with indirect signal production, $$\begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u) - \nabla \cdot (u \nabla v), &x \in \Omega,\ t> 0,\\ 0 = \Delta v - \mu(t) + w, &x \in \Omega,\ t> 0,\\ w_t + w = u, &x \in \Omega,\ t> 0, \end{cases}$$ complemented with homogeneous Neumann boundary conditions and suitable initial conditions, where $\Omega\subset\mathbb R^n$ $(n\ge3)$ is a bounded smooth domain, $m\ge1$ and $$\mu(t) := \frac{1}{|\Omega|} w(\cdot, t) \qquad\mbox{for}\ t>0.$$ We show that in the case $m\ge2-\frac{2}{n}$, there exists $M_c>0$ such that if either $m>2-\frac{2}{n}$ or $\int_\Omega u_0 <M_c$, then the solution exists globally and remains bounded, and that in the case $m\le2-\frac{2}{n}$, if either $m<2-\frac{2}{n}$ or $M>2^\frac{n}{2}n^{n-1}\omega_n$, then there exist radially symmetric initial data such that $\int_\Omega u_0 = M$ and the solution blows up in finite or infinite time, where the blow-up time is infinite if $m=2-\frac2n$. In particular, if $m=2-\frac{2}{n}$ there is a critical mass phenomenon in the sense that $\inf\left\{M > 0 : \exists u_0 \text{ with } \int_\Omega u_0 = M \text{ such that the corresponding solution blows up in infinite time}\right\}$ is a finite positive number.