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

Abstract

This paper is concerned with a quasilinear chemotaxis model with indirect signal production, $u_t = \nabla\cdot(D(u)\nabla u - S(u)\nabla v)$, $v_t = Δv - v + w$ and $w_t = Δw - w + u$, posed on a bounded smooth domain $Ω\subset\mathbb R^n$, subjected to homogenerous Neumann boundary conditions, where nonlinear diffusion $D$ and sensitivity $S$ generalize the prototype $D(s) = (s+1)^{-α}$ and $S(s) = (s+1)^{β-1}s$. Ding and Wang [M.Ding and W.Wang, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684.] showed that the system possesses a globally bounded classical solution if $α+ β<\min\{1+2/n,4/n\}$. While for the Jäger-Luckhaus variant of this model, namely the second equation replaced by $0=Δv - \int_Ωw/|Ω| + w$, Tao and Winkler [2023, preprint] announced that if $α+ β> 4/n$ and $β>2/n$ for $n\geq3$, with radial assumptions, the variant admits occurrence of finite-time blowup. We focus on the case $β<2/n$, and prove that $β< 2/n$ for $n\geq2$ is sufficient for global solvability of classical solutions; if $α+ β> 4/n$ for $n\geq4$, then radially symmetric initial data with large negative energy enforce blowup happening in finite or infinite time, both of which imply that the system allows infinite-time blowup if $α+ β> 4/n$ and $β< 2/n$ for $n\geq 4$.