Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source
This paper is devoted to studying the following quasilinear parabolic-elliptic-elliptic chemotaxis system \begin{equation*} \begin{cases} u_{t}=\nabla\cdot(\varphi(u)\nabla u-\psi(u)\nabla v)+au-bu^{\gamma},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0=\Delta v-v+w^{\gamma_{1}}, \ &\ \ x\in \...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2023-04-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10109 |
| _version_ | 1797352368104275968 |
|---|---|
| author | Chang-Jian Wang Ya-Jie Zhu Xin-Cai Zhu |
| author_facet | Chang-Jian Wang Ya-Jie Zhu Xin-Cai Zhu |
| author_sort | Chang-Jian Wang |
| collection | DOAJ |
| description | This paper is devoted to studying the following quasilinear parabolic-elliptic-elliptic chemotaxis system
\begin{equation*}
\begin{cases}
u_{t}=\nabla\cdot(\varphi(u)\nabla u-\psi(u)\nabla v)+au-bu^{\gamma},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm]
0=\Delta v-v+w^{\gamma_{1}}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm]
0=\Delta w-w+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0 ,
\end{cases}
\end{equation*}
with homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^{n}(n\geq 1),$ where $a,b,\gamma_{2}>0, \gamma_{1}\geq 1, \gamma>1 $ and the functions $\varphi,\psi\in C^{2}([0,\infty)$ satisfy
$\varphi(s)\geq a_{0}(s+1)^{\alpha}$ and $|\psi(s)|\leq b_{0}s(1+s)^{\beta-1}$ for all $s\geq 0$ with $a_{0},b_{0}>0$ and $\alpha,\beta \in \mathbb{R}.$ It is proved that if $\gamma-\beta\geq \gamma_{1}\gamma_{2} ,$ the classical solution of system would be globally bounded. Furthermore, a specific
model for $\gamma_{1}=1,\gamma_{2}=\kappa$ and $\gamma=\kappa+1$ with $\kappa>0$ is considered. If $\beta\leq 1$ and $b>0$ is large enough, there exist $C_{\kappa},\mu_{1},\mu_{2}>0$ such that the solution$(u,v,w)$ satisfies
\begin{align*}
\left\|u(\cdot,t)-\left(\frac{b}{a}\right)^{\frac{1}{\kappa}}\right\|_{L^{\infty}(\Omega)}+\left\|v(\cdot,t)-\frac{b}{a}\right\|_{L^{\infty}(\Omega)}+\left\|w(\cdot,t)-\frac{b}{a}\right\|_{L^{\infty}(\Omega)}
\leq
\begin{cases}
C_{\kappa}\mbox{e}^{-\mu_{1}t}, \ &\ \ \mbox{if} \ \kappa \in (0,1], \\[2.5mm]
C_{\kappa}\mbox{e}^{-\mu_{2}t}, \ &\ \ \mbox{if} \ \kappa \in (1,\infty),
\end{cases}
\end{align*}
for all $t\geq 0.$ The above results generalize some existing results. |
| first_indexed | 2024-03-08T13:15:27Z |
| format | Article |
| id | doaj.art-8928434681064cba84ba730be441bff2 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-03-08T13:15:27Z |
| publishDate | 2023-04-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-8928434681064cba84ba730be441bff22024-01-18T08:28:07ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752023-04-0120231112110.14232/ejqtde.2023.1.1110109Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic sourceChang-Jian Wang0Ya-Jie Zhu1Xin-Cai Zhu2School of Mathematics and Statistics, Xinyang Normal University, Xinyang, P.R. ChinaCollege of Teacher Education, Xinyang Normal University, Xinyang, P.R. ChinaSchool of Mathematics and Statistics, Xinyang Normal University Xinyang, P.R. ChinaThis paper is devoted to studying the following quasilinear parabolic-elliptic-elliptic chemotaxis system \begin{equation*} \begin{cases} u_{t}=\nabla\cdot(\varphi(u)\nabla u-\psi(u)\nabla v)+au-bu^{\gamma},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0=\Delta v-v+w^{\gamma_{1}}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0=\Delta w-w+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0 , \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^{n}(n\geq 1),$ where $a,b,\gamma_{2}>0, \gamma_{1}\geq 1, \gamma>1 $ and the functions $\varphi,\psi\in C^{2}([0,\infty)$ satisfy $\varphi(s)\geq a_{0}(s+1)^{\alpha}$ and $|\psi(s)|\leq b_{0}s(1+s)^{\beta-1}$ for all $s\geq 0$ with $a_{0},b_{0}>0$ and $\alpha,\beta \in \mathbb{R}.$ It is proved that if $\gamma-\beta\geq \gamma_{1}\gamma_{2} ,$ the classical solution of system would be globally bounded. Furthermore, a specific model for $\gamma_{1}=1,\gamma_{2}=\kappa$ and $\gamma=\kappa+1$ with $\kappa>0$ is considered. If $\beta\leq 1$ and $b>0$ is large enough, there exist $C_{\kappa},\mu_{1},\mu_{2}>0$ such that the solution$(u,v,w)$ satisfies \begin{align*} \left\|u(\cdot,t)-\left(\frac{b}{a}\right)^{\frac{1}{\kappa}}\right\|_{L^{\infty}(\Omega)}+\left\|v(\cdot,t)-\frac{b}{a}\right\|_{L^{\infty}(\Omega)}+\left\|w(\cdot,t)-\frac{b}{a}\right\|_{L^{\infty}(\Omega)} \leq \begin{cases} C_{\kappa}\mbox{e}^{-\mu_{1}t}, \ &\ \ \mbox{if} \ \kappa \in (0,1], \\[2.5mm] C_{\kappa}\mbox{e}^{-\mu_{2}t}, \ &\ \ \mbox{if} \ \kappa \in (1,\infty), \end{cases} \end{align*} for all $t\geq 0.$ The above results generalize some existing results.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10109chemotaxis systemnonlinear indirect secretionglobal boundednesslong time behavior |
| spellingShingle | Chang-Jian Wang Ya-Jie Zhu Xin-Cai Zhu Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source Electronic Journal of Qualitative Theory of Differential Equations chemotaxis system nonlinear indirect secretion global boundedness long time behavior |
| title | Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source |
| title_full | Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source |
| title_fullStr | Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source |
| title_full_unstemmed | Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source |
| title_short | Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source |
| title_sort | long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source |
| topic | chemotaxis system nonlinear indirect secretion global boundedness long time behavior |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10109 |
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