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Finite-time blow-up in a repulsive chemotaxis-consumption system

Part of: Parabolic equations and systems Equations of mathematical physics and other areas of application Partial differential equations Physiological, cellular and medical topics

Published online by Cambridge University Press:  06 June 2022

Yulan Wang  and
Michael Winkler
Yulan Wang
Affiliation:
School of Science, Xihua University, 610039 Chengdu, China (wangyulan-math@163.com)
Michael Winkler
Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany (michael.winkler@math.uni-paderborn.de)
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Abstract

In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$, the chemotaxis system

\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]
is considered along with no-flux boundary conditions for $u$ and with prescribed constant positive Dirichlet boundary data for $v$. It is shown that if $D\in C^{3}([0,\infty ))$ is such that $0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$ for all $\xi >0$ with some ${K_D}>0$ and $\alpha >0$, then for all initial data from a considerably large set of radial functions on $\Omega$, the corresponding initial-boundary value problem admits a solution blowing up in finite time.

Keywords

Chemotaxisconsumption mechanismfinite-time blow-up

MSC classification

Primary: 35K65: Degenerate parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35Q92: PDEs in connection with biology and other natural sciences 92C17: Cell movement (chemotaxis, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 4 , August 2023 , pp. 1150 - 1166
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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