Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation

Abstract

This paper deals with the quasilinear degenerate chemotaxis system with flux limitation \begin{align*} \begin{cases} u_t = \nabla\cdot\left(\dfrac{u^p \nabla u}{\sqrt{u^2 + |\nabla u|^2}} \right) -\chi \nabla\cdot\left( \dfrac{u^q\nabla v}{\sqrt{1 + |\nabla v|^2}}\right), &x\in \Omega,\ t>0, \\[1mm] 0 = \Delta v - \mu + u, &x\in \Omega,\ t>0, \end{cases} \end{align*} where $\Omega := B_R(0) \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) is a ball with some $R>0$, and $\chi >0$, $p,q\geq1$, $\mu := \frac 1{|\Omega|} \int_\Omega u_0$ and $u_0$ is an initial data of an unknown function $u$. Bellomo--Winkler (Trans.\ Amer.\ Math.\ Soc.\ Ser.\ B;2017;4;31--67) established existence of an initial data such that the corresponding solution blows up in finite time when $p=q=1$. This paper gives existence of blow-up solutions under some condition for $\chi$ and $u_0$ when $1\leq p\leq q$.