Global existence and uniform boundedness to a bi-attraction chemotaxis system with nonlinear indirect signal mechanisms

Chang-Jian Wang; Jia-Yue Zhu; Chang-Jian Wang; Jia-Yue Zhu

doi:10.3934/cam.2023036

Communications in Analysis and Mechanics

2023, Volume 15, Issue 4: 743-762. doi: 10.3934/cam.2023036

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Research article

Global existence and uniform boundedness to a bi-attraction chemotaxis system with nonlinear indirect signal mechanisms

Chang-Jian Wang ,
Jia-Yue Zhu ^,

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 08 August 2023 Revised: 17 October 2023 Accepted: 25 October 2023 Published: 02 November 2023
35K55, 35Q92, 35A01, 92C17

In this paper, we study the following quasilinear chemotaxis system

$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi \nabla \cdot (\varphi (u)\nabla v)-\xi \nabla \cdot (\psi(u)\nabla w)+f(u), \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta v-v+v_{1}^{\gamma_{1}}, \ 0 = \Delta v_{1}-v_{1}+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta w-w+w_{1}^{\gamma_{3}}, \ 0 = \Delta w_{1}-w_{1}+u^{\gamma_{4}}, \ &\ \ x\in \Omega, \ t>0, \end{array} \right. \end{equation*} $

in a smoothly bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \varphi(\varrho)\leq\varrho(\varrho+1)^{\theta-1}, $ $ \psi(\varrho)\leq\varrho(\varrho+1)^{l-1} $ and $ f(\varrho)\leq a \varrho-b\varrho^{s} $ for all $ \varrho\geq0, $ and the parameters satisfy $ a, b, \chi, \xi, \gamma_{2}, \gamma_{4} > 0, $ $ s > 1, $ $ \gamma_{1}, \gamma_{3}\geq1 $ and $ \theta, l\in \mathbb{R}. $ It has been proven that if $ s \geq\max\{ \gamma_{1}\gamma_{2}+\theta, \gamma_{3}\gamma_{4}+l\}, $ then the system has a nonnegative classical solution that is globally bounded. The boundedness condition obtained in this paper relies only on the power exponents of the system, which is independent of the coefficients of the system and space dimension $ n. $ In this work, we generalize the results established by previous researchers.
- bi-attraction chemotaxis system,
- nonlinear indirect signal,
- global boundedness
Citation: Chang-Jian Wang, Jia-Yue Zhu. Global existence and uniform boundedness to a bi-attraction chemotaxis system with nonlinear indirect signal mechanisms[J]. Communications in Analysis and Mechanics, 2023, 15(4): 743-762. doi: 10.3934/cam.2023036

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Abstract

In this paper, we study the following quasilinear chemotaxis system

$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi \nabla \cdot (\varphi (u)\nabla v)-\xi \nabla \cdot (\psi(u)\nabla w)+f(u), \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta v-v+v_{1}^{\gamma_{1}}, \ 0 = \Delta v_{1}-v_{1}+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta w-w+w_{1}^{\gamma_{3}}, \ 0 = \Delta w_{1}-w_{1}+u^{\gamma_{4}}, \ &\ \ x\in \Omega, \ t>0, \end{array} \right. \end{equation*} $

in a smoothly bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \varphi(\varrho)\leq\varrho(\varrho+1)^{\theta-1}, $ $ \psi(\varrho)\leq\varrho(\varrho+1)^{l-1} $ and $ f(\varrho)\leq a \varrho-b\varrho^{s} $ for all $ \varrho\geq0, $ and the parameters satisfy $ a, b, \chi, \xi, \gamma_{2}, \gamma_{4} > 0, $ $ s > 1, $ $ \gamma_{1}, \gamma_{3}\geq1 $ and $ \theta, l\in \mathbb{R}. $ It has been proven that if $ s \geq\max\{ \gamma_{1}\gamma_{2}+\theta, \gamma_{3}\gamma_{4}+l\}, $ then the system has a nonnegative classical solution that is globally bounded. The boundedness condition obtained in this paper relies only on the power exponents of the system, which is independent of the coefficients of the system and space dimension $ n. $ In this work, we generalize the results established by previous researchers.

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