Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 81, pp. 1-10.

Title: Finite-time blow-up in a quasilinear fully parabolic attraction-repulsion chemotaxis system with density-dependent sensitivity

Authors: Yutaro Chiyo (Tokyo Univ. of Science, Tokyo, Japan)
         Takeshi Uemura (Tokyo Univ. of Science, Tokyo, Japan)
         Tomomi Yokota (Tokyo Univ. of Science, Tokyo, Japan)

Abstract:
This article concerns the quasilinear fully parabolic attraction-repulsion
chemotaxis system
$$\displaylines{
u_t=\nabla \cdot ((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2} \nabla v
+ \xi u(u+1)^{p-2}\nabla w),\quad x \in \Omega,\; t>0,\cr
v_t=\Delta v+\alpha u-\beta v, \quad x \in \Omega,\; t>0,\cr
w_t=\Delta w+\gamma u-\delta w, \quad x \in \Omega,\; t>0
}$$
with homogeneous Neumann boundary conditions,
where $\Omega \subset \mathbb{R}^n$ $(n \in \{2,3\})$ is an open ball,
$m, p \in \mathbb{R}$,
$\chi, \xi, \alpha, \beta, \gamma, \delta >0$ are constants.
The main result asserts finite-time blow-up of solutions to this system
with some positive initial data when $\chi\alpha-\xi\gamma>0$, $p \ge 2$ and
$p-m >2/n$.

Submitted May 1, 2025. Published August 06, 2025.
Math Subject Classifications: 35B44, 35K59, 35Q92, 92C17.
Key Words: Finite-time blow-up; quasilinear; attraction-repulsion; chemotaxis.