2021/02/27 by Yuya Tanaka, Tanaka, Yuya
Mathematics · Biochemistry, Genetics and Molecular Biology · #Mathematical Biology Tumor Growth #Gene Regulatory Network Analysis #Microtubule and mitosis dynamics
paper · pdf · doi:10.48550/arxiv.2103.00159
This paper deals with the quasilinear parabolic-elliptic Keller-Segel system with logistic source, ut=Δ(u+1)m - χ∇ ⋅ (u(u+1)α- 1 ∇ v) + λ(|x|) u - μ(|x|) uκ, 0=Δv - v + u, x∈Ω, tgt;0, where Ω:=BR(0)⊂ℝn (n≥3) is a ball with some R>0; m>0, χ>0, α>0 and κ≥1; λ and μ are spatially radial nonnegative functions. About this problem, Winkler (Z. Angew. Math. Phys.; 2018; 69; Art. 69, 40) found the condition for κ such that solutions blow up in finite time when m=α=1. In the case that m=1 and α∈(0,1) as well as λ and μ are constant, some conditions for α and κ such that blow-up occurs were obtained in a previous paper (Math. Methods Appl. Sci.; 2020; 43; 7372-7396). Moreover, in the case that m≥1 and α=1 Black, Fuest and Lankeit (arXiv:2005.12089[math.AP]) showed that there exists initial data such that solutions blow up in finite time under some conditions for m and κ. The purpose of the present paper is to give conditions for m≥1, α>0 and κ≥1 such that solutions blow up in finite time.