This paper considers the parabolic-elliptic Keller-Segel system with density-dependent sublinear sensitivity and logistic source: \[ \begin{cases} u_{t}=\Delta u- \chi \nabla \cdot (u(u+1)^{\alpha-1} \nabla v)+ \lambda u- \mu u^{\kappa}, & x\in \Omega,\, t>0, \\ 0= \Delta v-v +u, & x\in \Omega,\, t>0, \end{cases} \] where \(\Omega:= B_{R}(0)\subset \mathbb{R}^{n}\) \((n\geq 3)\) is a ball with some \(R>0\) and \(\chi>0\), \(0<\alpha<1\), \(\lambda\in R\), \(\mu>0\) and \(\kappa>1\).
The main contribution of this paper is to find conditions for \(\alpha\) and \(\kappa\) such that there exist solutions that blow up in finite time in the case of weak-chemotactic sensitivity, that is, in the case \(0<\alpha<1\). The authors extend the results of M. Winkler [Z. Angew. Math. Phys. 69, No. 2, Paper No. 40, 25 p. (2018; Zbl 1395.35048)] in which discovered the conditions for \(\alpha=1\) and \(\kappa>1\). In the proof of the main results, the authors establish a key differential inequality (see Lemma 4.12).
The proof of the theorem is delicate and seem to be right. This is a great work. The paper is well organized with a complete list of relevant references. The presentation of the paper is very clear. No typos was detected. I think it is a good paper. However, the case is unknown whether the parabolic-parabolic Keller-Segel system with some same conditions appears blow up or not. This will arouse more interest to relevant researchers to study the above system extensively.