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