Blow-up in a quasilinear parabolic–elliptic Keller–Segel system with logistic source

Abstract

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,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>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 continuous nonnegative functions. About this problem, Winkler (2018) 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 constants, some conditions for α and κ such that blow-up occurs were obtained in a previous paper (Tanaka and Yokota, 2020). Moreover, in the case that m≥1 and α=1 Black et al. (2021) showed that there exist initial data such that the corresponding solution blows 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.