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, $u_t=\nabla \cdot (D(u)\nabla u-S(u)\nabla v)$, $v_t=\Delta v-v+w$ and $w_t=\Delta w-w+u$, posed on a bounded smooth domain $\Omega \subset {\mathbb{R}}^n$, subjected to homogenerous Neumann boundary conditions, where nonlinear diffusion $D$ and sensitivity $S$ generalize the prototype $D(s)=(s+1{)}^{-\alpha }$ and $S(s)=(s+1{)}^{\beta -1}s$. 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 $\alpha +\beta <\min \{1+2/n,4/n\}$. While for the Jäger-Luckhaus variant of this model, namely the second equation replaced by $0=\Delta v-{\int }_{\Omega }w/|\Omega |+w$, Tao and Winkler [2023, preprint] announced that if $\alpha +\beta >4/n$ and $\beta >2/n$ for $n\ge 3$, with radial assumptions, the variant admits occurrence of finite-time blowup. We focus on the case $\beta <2/n$, and prove that $\beta <2/n$ for $n\ge 2$ is sufficient for global solvability of classical solutions; if $\alpha +\beta >4/n$ for $n\ge 4$, 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 $\alpha +\beta >4/n$ and $\beta <2/n$ for $n\ge 4$.

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Cited by in corpus (1)

• Finite-time blowup in a parabolic-parabolic-elliptic chemotaxis model involving indirect signal production