We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller–Segel model in a ball \(\Omega \subset \mathbb {R}^n\) with \(n\ge 2\) . Previous results show that unbounded solutions exist for all \(m, q \in \mathbb {R}\) with \(m-q<\frac{n-2}{n}\) , which, however, are necessarily global in time if \(q \le 0\) . It is expected that finite-time blow-up is possible whenever \(q > 0\) but in the fully parabolic setting this has so far only been shown when \(\max \{m, q\} \ge 1\) . In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that ( \(\star \) ) admits solutions blowing up in finite time if \(\begin{aligned} m-q<\frac{n-2}{n} \quad \text {and} \quad {\left\{ \begin{array}{ll} q< 2m & \text {if } n = 2, \\ q < 2m - \frac{2}{3} \text { or } m > \frac{2}{3} & \text {if } n = 3, \end{array}\right. } \end{aligned}\) that is, also for certain m, q with \(\max \{m, q\} < 1\) . As a key new ingredient in our proof, we make use of (singular) pointwise upper estimates for u.
