Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism
In this paper, we study a quasilinear chemotaxis model with a nonlinear indirect consumption mechanism$ \begin{equation*} \left\{ \begin{array}{ll} v_{1t} = \nabla \cdot\big(\psi(v_{1})\nabla v_{1}-\chi \phi(v_{1})\nabla v_{2}\big)+\lambda_{1}v_{1}-\lambda_{2}v_{1}^{\beta},\ &\ \ x\in \Omega, \...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-08-01
|
| Series: | Electronic Research Archive |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2024219 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832590851640918016 |
|---|---|
| author | Zihan Zheng Juan Wang Liming Cai |
| author_facet | Zihan Zheng Juan Wang Liming Cai |
| author_sort | Zihan Zheng |
| collection | DOAJ |
| description | In this paper, we study a quasilinear chemotaxis model with a nonlinear indirect consumption mechanism$ \begin{equation*} \left\{ \begin{array}{ll} v_{1t} = \nabla \cdot\big(\psi(v_{1})\nabla v_{1}-\chi \phi(v_{1})\nabla v_{2}\big)+\lambda_{1}v_{1}-\lambda_{2}v_{1}^{\beta},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] v_{2t} = \Delta v_{2}-w^{\theta}v_{2}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w-w+v_{1}^{\alpha}, \ &\ \ x\in \Omega, \ t>0 ,\\[2.5mm] \end{array} \right. \end{equation*} $in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \chi, \; \lambda_{1}, \; \lambda_{2}, \; \theta > 0, \; 0 < \alpha\leq\frac{1}{\theta}, \; \beta\geq 2, \; $ $ \psi $, and $ \phi $ are nonlinear functions that satisfy $ \psi(s)\geq a_{0}(s+1)^{r_{1}} $ and $ 0\leq\phi(s)\leq b_{0}s(s+1)^{r_{2}} $ for all $ s\geq 0 $ with $ a_{0}, b_{0} > 0 $ and $ r_{1}, r_{2}\in \mathbb{R}. $ It has been proven that if $ r_{1} > 2r_{2}+1, $ then the problem admits a global and bounded classical solution for some appropriate nonnegative initial data. |
| format | Article |
| id | doaj-art-e446e86576664216836b206ef843cdae |
| institution | Kabale University |
| issn | 2688-1594 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj-art-e446e86576664216836b206ef843cdae2025-01-23T07:51:27ZengAIMS PressElectronic Research Archive2688-15942024-08-013284796480810.3934/era.2024219Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanismZihan Zheng0Juan Wang1Liming Cai2School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, ChinaSchool of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, ChinaSchool of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, ChinaIn this paper, we study a quasilinear chemotaxis model with a nonlinear indirect consumption mechanism$ \begin{equation*} \left\{ \begin{array}{ll} v_{1t} = \nabla \cdot\big(\psi(v_{1})\nabla v_{1}-\chi \phi(v_{1})\nabla v_{2}\big)+\lambda_{1}v_{1}-\lambda_{2}v_{1}^{\beta},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] v_{2t} = \Delta v_{2}-w^{\theta}v_{2}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w-w+v_{1}^{\alpha}, \ &\ \ x\in \Omega, \ t>0 ,\\[2.5mm] \end{array} \right. \end{equation*} $in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \chi, \; \lambda_{1}, \; \lambda_{2}, \; \theta > 0, \; 0 < \alpha\leq\frac{1}{\theta}, \; \beta\geq 2, \; $ $ \psi $, and $ \phi $ are nonlinear functions that satisfy $ \psi(s)\geq a_{0}(s+1)^{r_{1}} $ and $ 0\leq\phi(s)\leq b_{0}s(s+1)^{r_{2}} $ for all $ s\geq 0 $ with $ a_{0}, b_{0} > 0 $ and $ r_{1}, r_{2}\in \mathbb{R}. $ It has been proven that if $ r_{1} > 2r_{2}+1, $ then the problem admits a global and bounded classical solution for some appropriate nonnegative initial data.https://www.aimspress.com/article/doi/10.3934/era.2024219chemotaxis systemglobal boundednessnonlinear indirect signal |
| spellingShingle | Zihan Zheng Juan Wang Liming Cai Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism Electronic Research Archive chemotaxis system global boundedness nonlinear indirect signal |
| title | Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism |
| title_full | Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism |
| title_fullStr | Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism |
| title_full_unstemmed | Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism |
| title_short | Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism |
| title_sort | global boundedness in a keller segel system with nonlinear indirect signal consumption mechanism |
| topic | chemotaxis system global boundedness nonlinear indirect signal |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024219 |
| work_keys_str_mv | AT zihanzheng globalboundednessinakellersegelsystemwithnonlinearindirectsignalconsumptionmechanism AT juanwang globalboundednessinakellersegelsystemwithnonlinearindirectsignalconsumptionmechanism AT limingcai globalboundednessinakellersegelsystemwithnonlinearindirectsignalconsumptionmechanism |