Blow‐up in a parabolic–elliptic Keller–Segel system with density‐dependent sublinear sensitivity and logistic source

This paper deals with the parabolic–elliptic Keller–Segel system with density‐dependent sublinear sensitivity and logistic source,u t = Δ u − χ ∇ · ( u ( u + 1 ) α − 1 ∇ v ) + λ u − μ 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 and χ >0, 0< α <1, λ ∈ R , μ >0, and κ >1. In the case α =1, Winkler (Z. Angew. Math. Phys.; 2018; 69: 40) discovered the condition for κ such that solutions blow up in finite time. The purpose of the present paper is to find conditions for α and κ such that there exist solutions that blow up in finite time in the case of weak‐chemotactic sensitivity, that is, in the case 0< α <1.