2024/10/17 by Mao, Xuan, Li, Yuxiang · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.13238
This paper is concerned with a quasilinear chemotaxis model with indirect signal production, ut = ∇⋅(D(u)∇ u - S(u)∇ v), vt = Δv - v + w and wt = Δw - w + u, posed on a bounded smooth domain Ω⊂\mathbb Rn, 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-1s. 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 - ∫Ωw/|Ω| + w, Tao and Winkler [2023, preprint] announced that if α+ β> 4/n and β>2/n for n≥3, 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≥2 is sufficient for global solvability of classical solutions; if α+ β> 4/n for n≥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 α+ β> 4/n and β< 2/n for n≥ 4.