Blow-up in a quasilinear parabolic–elliptic Keller–Segel system with logistic source. (February 2022) - Tanaka, Yuya
Abstract: This paper deals with the quasilinear parabolic–elliptic Keller–Segel system with logistic source, u t = Δ ( u + 1 ) m − χ ∇ ⋅ ( u ( u + 1 ) α − 1 ∇ v ) + λ ( | x | ) u − μ ( | x | ) u κ, x ∈ Ω, t > 0, 0 = Δ v − v + u, x ∈ Ω, t > 0, where Ω ≔ B R ( 0 ) ⊂ R n ( 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.