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Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production

  • *Corresponding author: Yuya Tanaka

    *Corresponding author: Yuya Tanaka 

The second author is supported by JSPS KAKENHI Grant Number 21K03278

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  • This paper deals with finite-time blow-up of solutions to the quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production,

    $ \begin{align*} \begin{cases} u_t = \Delta u^m - \chi \nabla \cdot (u^\alpha \nabla v) + \lambda u - \mu u^\kappa, \quad &x \in \Omega, \ t>0, \\ 0 = \Delta v - \overline{M_\ell}(t) + u^\ell, \quad &x \in \Omega, \ t>0, \end{cases} \end{align*} $

    where $ \Omega: = B_R(0) \subset \mathbb{R}^n \ (n \in \mathbb{N}) $ be a ball with some $ R>0 $ and $ m\ge1 $, $ \chi>0 $, $ \alpha\ge1 $, $ \lambda>0 $, $ \mu>0 $, $ \kappa>1 $, $ \ell>0 $ as well as $ \overline{M_\ell}(t) $ is the average of $ u^\ell $ over $ \Omega $. As to the corresponding system with nondegenerate diffusion, finite-time blow-up has been obtained under the condition that $ \alpha-\ell>\max\left\{\overline{m} +\frac{2}{n}\kappa, \kappa\right\} $, where $ \overline{m}: = \max\{m,0\} $ in a previous paper [26], which is based a work by Fuest [7]. The purpose of this paper is to establish finite-time blow-up for the above degenerate chemotaxis system within a concept of weak solutions with a moment inequality leading to blow-up.

    Mathematics Subject Classification: Primary: 35B44; Secondary: 35K65, 92C17.

    Citation:

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