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

Abstract

This paper deals with the parabolic–elliptic Keller–Segel system with density‐dependent sublinear sensitivity and logistic source, where is a ball with some R>0 and χ>0, 0<α<1, , 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.