Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

Jianing Xie

doi:10.3934/dcdsb.2021216

Article Contents

2022, Volume 27, Issue 7: 4007-4022. Doi: 10.3934/dcdsb.2021216

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$ \mathbb{R}^4 $

Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production

Jianing Xie^,

School of Data Science and Artificial Intelligence, Dongbei University of Finance and Economics, Dalian, 116025, China

^* Corresponding author: Jianing Xie
^* Corresponding author: Jianing Xie

Received: February 28, 2021

Revised: June 30, 2021

Published: July 2022

The author is supported by National Science Foundation of China grant (No. 11572081), the Program Funded by Liaoning Province Education Administration (No. LN2021M37) and Dongbei University of Finance and Economics (No. DUFE2020Y20)

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Abstract

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype

$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t>0,\\ { }{ v_t = \Delta v- v +w},\quad x\in \Omega, t>0,\\ { }{\tau w_t+\delta w = u},\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} (*)$

in a smoothly bounded domain $ \Omega\subset\mathbb{R}^N(N\geq1) $ under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters $ \mu $ as well as $ \delta $ and $ \tau $ are positive. Based on an new energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever

$ \mu>\left\{ \begin{array}{ll} {0, \; \; \; {\rm{if}}\; \; N\leq4},\\ {\frac{(N-4)_{+}}{N-2}\max\{1,\lambda_{0}\},\; \; \; {\rm{if}}\; \; N\geq5}\\ \end{array} \right. $

and the initial data $ (u_0,v_0,w_0) $ are sufficiently regular. Here $ \lambda_0 $ is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.

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Mathematics Subject Classification: Primary: 92C17, 35K55, 35K59, 35K20.

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