Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems

TL;DR

This paper investigates conditions under which solutions to certain chemotaxis models with nonlinear diffusion and logistic growth blow up in finite time, extending parameter ranges for blow-up in higher dimensions.

Contribution

It provides a unified approach connecting existing blow-up detection methods and pointwise estimates, leading to extended parameter conditions for finite-time blow-up in Keller-Segel systems.

Findings

Extended parameter ranges for blow-up in dimensions three and above.

Identification of initial data leading to blow-up under new conditions.

Bridging of methods for blow-up detection and solution estimates.

Abstract

We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - \frac1{|\Omega|} \int_\Omega u + u \end{cases} \tag{JL} \end{align*} and \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + \lambda u - \mu u^{1+\kappa}, \\ 0 = \Delta v - v + u, \end{cases}\tag{PE} \end{align*} where $\lambda$ and $\mu$ are given spatially radial nonnegative functions and $m, \kappa > 0$ are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $m,\kappa$ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $\lambda, \mu > 0$, we find that there are initial data which lead to blow-up in (JL) if \begin{alignat*}{2} 0 \leq \kappa &< \min\left\{\frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&&\qquad\text{if } m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq \kappa&<\min\left\{\frac{1}{2},\frac{n-1}n-\frac{m}2\right\} &&\qquad \text{if } m\in\left(0,\frac{2}{n}\right), \end{alignat*} and in (PE) if $m \in [1, \frac{2n-2}{n})$ and \begin{align*} 0 \leq \kappa < \min\left\{\frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\}. \end{align*}