Finite-time blow-up in fully parabolic quasilinear Keller–Segel systems with supercritical exponents

We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller–Segel model (Figure presented.) in a ball Ω⊂Rn with n≥2. Previous results show that unbounded solutions exist for all m,q∈R with m-q<n-2n, which, however, are necessarily global in time if q≤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}≥1. In the present paper, we substantially extend these findings. Our main results for the two- and three-dimensional settings state that (⋆) admits solutions blowing up in finite time if (Formula presented.) 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.