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

Abstract

This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=Δu-χ\nabla \cdot (u\nabla v)+ξ\nabla\cdot(u \nabla w), \quad v_t=Δv-v+u, \quad w_t=Δw-w+u, \quad x \in Ω,\ 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.

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• Blow-up phenomena in a parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with superlinear logistic degradation