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

This paper deals with the quasilinear parabolic-elliptic Keller-Segel system with logistic source, {u(t) = Delta( u + 1)(m) - chi del center dot (u( u + 1)(alpha-1)del v) + lambda(vertical bar x vertical bar)u - mu(vertical bar x vertical bar)u(kappa), x epsilon Omega, t > 0, 0 = Delta v - v + u, x epsilon Omega, t > 0, where Omega := B-R(0) subset of R-n (n >= 3) is a ball with some R > 0; m > 0, chi > 0, alpha > 0 and kappa >= 1; lambda and mu are continuous nonnegative functions. About this problem, Winkler (2018) found the condition for. such that solutions blow up in finite time when m = alpha = 1. In the case that m = 1 and alpha epsilon (0, 1) as well as lambda and mu are constants, some conditions for alpha and kappa such that blow-up occurs were obtained in a previous paper (Tanaka and Yokota, 2020). Moreover, in the case that m >= 1 and alpha = 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 kappa. The purpose of the present paper is to give conditions for m >= 1, alpha > 0 and kappa >= 1 such that solutions blow up in finite time. (C) 2021 Elsevier Ltd. All rights reserved.