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

Abstract

We examine the possibility of finite-time blow-up of solutions to the fully parabolic quasilinear Keller--Segel model \begin{align}\tag{$\star$}\label{prob:star} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u(u+1)^{q-1}\nabla v) & \text{in $\Omega \times (0,T)$}, \\ v_t = Δv - v + u & \text{in $\Omega \times (0,T)$} \end{cases} \end{align} 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 \eqref{prob:star} admits solutions blowing up in finite time if \begin{align*} m-q<\frac{n-2}{n} \quad \text{and} \quad \begin{cases} q < 2m & \text{if } n = 2, \\ q < 2m - \frac23 \text{ or } m > \frac23 & \text{if } n = 3, \end{cases} \end{align*} 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 $u$.

23 pages, incorporated reviewer comments

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