Global solvability and unboundedness in a fully parabolic quasilinear chemotaxis model with indirect signal production

This paper is concerned with a quasilinear chemotaxis model with indirect signal production, given by $u_t=\nabla \cdot \left(D\right(u)\nabla u-S(u\left)\nabla v\right)$, $v_t=\Delta v-v+w$ and $w_t=\Delta w-w+u$, posed in a bounded smooth domain $\Omega \subset R^n$, subjected to homogeneous Neumann boundary conditions. Here, the nonlinear diffusion D and sensitivity S generalize the prototypes $D\left(s\right)={(s+1)}^{-\alpha }$ and $S\left(s\right)={(s+1)}^{\beta -1}s$. Ding and Wang (2019) [8] showed that the system possesses a globally bounded classical solution if $\alpha +\beta <\min \{1+2/n,4/n\}$. In contrast, for the Jäger-Luckhaus variant of this model, in which the second equation is replaced by $0=\Delta v-{\int }_{\Omega }w/\left|\Omega \right|+w$, Tao and Winkler (2025) [36] established that if $\alpha +\beta >4/n$ and $\beta >2/n$ for $n\ge 3$, with radial assumptions, the variant admits finite-time blow-up solutions. We focus on the case $\beta <2/n$ and prove that the assumption $\beta <2/n$ for $n\ge 2$ is sufficient for global solvability of classical solutions. Furthermore, if $\alpha +\beta >4/n$ for $n\ge 4$, then radially symmetric initial data with large negative energy lead to blow-up happening in finite or infinite time. Both of these results imply that the system allows infinite-time blow-up if $\alpha +\beta >4/n$ and $\beta <2/n$ for $n\ge 4$.