Remarks on finite-time blow-up in a fully parabolic attraction-repulsion chemotaxis system via reduction to the Keller-Segel system

TL;DR

This paper investigates finite-time blow-up phenomena in a fully parabolic attraction-repulsion chemotaxis system, extending known results from Keller-Segel models and establishing blow-up in higher dimensions through a novel transformation.

Contribution

It introduces a transformation that simplifies the analysis of the chemotaxis system, proving finite-time blow-up in dimensions greater than three, which was previously unresolved.

Findings

Finite-time blow-up established for dimensions n ≥ 4.

Transformation reduces the system to a more analyzable form.

Extends blow-up results beyond the known case n=3.

Abstract

This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\Delta u-\chi\nabla \cdot (u\nabla v)+\xi \nabla\cdot(u \nabla w), \quad v_t=\Delta v-v+u, \quad w_t=\Delta w-w+u, \quad x \in \Omega,\ t>0 \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\Omega$ is an open ball in $\mathbb{R}^n$ ($n \ge 3$), $\chi, \xi>0$ are constants. When $w=0$, finite-time blow-up in the corresponding Keller-Segel system has already been obtained. However, finite-time blow-up in the above attraction-repulsion chemotaxis system has not yet been established except for the case $n=3$. This paper provides an answer to this open problem by using a transformation which leads to a system presenting structural advantages respect to the original.