Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system
This article is concerned with the model $$\displaylines{ u_t=\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\quad x\in \Omega,\; t>0,\cr 0=\Delta v+\alpha u-\beta v,\quad x\in\Omega,\; t>0,\cr 0=\Delta w+\gamma u-\delta w,\quad x\in\Omega,\; t>0 }$$ with homogeneou...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-06-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/146/abstr.html |
| _version_ | 1818766082082603008 |
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| author | Yuhuan Li Ke Lin Chunlai Mu |
| author_facet | Yuhuan Li Ke Lin Chunlai Mu |
| author_sort | Yuhuan Li |
| collection | DOAJ |
| description | This article is concerned with the model
$$\displaylines{
u_t=\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\quad
x\in \Omega,\; t>0,\cr
0=\Delta v+\alpha u-\beta v,\quad x\in\Omega,\; t>0,\cr
0=\Delta w+\gamma u-\delta w,\quad x\in\Omega,\; t>0
}$$
with homogeneous Neumann boundary conditions in a bounded domain
$\Omega\subset \mathbb{R}^{n}\;(n=2,3)$. Under the critical condition
$\chi \alpha-\xi \gamma=0$, we show that the system possesses a unique
global solution that is uniformly bounded in time. Moreover, when $n=2$,
by some appropriate smallness conditions on the initial data, we
assert that this solution converges to
($\bar{u}_0$, $\frac{\alpha}{\beta}\bar{u}_0$,
$\frac{\gamma}{\delta}\bar{u}_0$) exponentially,
where $\bar{u}_0:=\frac{1}{|\Omega|}\int_{\Omega}u_0$. |
| first_indexed | 2024-12-18T08:28:20Z |
| format | Article |
| id | doaj.art-89a3b4409356498880b54376e506c647 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-18T08:28:20Z |
| publishDate | 2015-06-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-89a3b4409356498880b54376e506c6472022-12-21T21:14:31ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912015-06-012015146,113Asymptotic behavior for small mass in an attraction-repulsion chemotaxis systemYuhuan Li0Ke Lin1Chunlai Mu2 Sichuan Normal Univ., Chengdu, China Chongqing Univ., Chongqing, China Chongqing Univ., Chongqing, China This article is concerned with the model $$\displaylines{ u_t=\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\quad x\in \Omega,\; t>0,\cr 0=\Delta v+\alpha u-\beta v,\quad x\in\Omega,\; t>0,\cr 0=\Delta w+\gamma u-\delta w,\quad x\in\Omega,\; t>0 }$$ with homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset \mathbb{R}^{n}\;(n=2,3)$. Under the critical condition $\chi \alpha-\xi \gamma=0$, we show that the system possesses a unique global solution that is uniformly bounded in time. Moreover, when $n=2$, by some appropriate smallness conditions on the initial data, we assert that this solution converges to ($\bar{u}_0$, $\frac{\alpha}{\beta}\bar{u}_0$, $\frac{\gamma}{\delta}\bar{u}_0$) exponentially, where $\bar{u}_0:=\frac{1}{|\Omega|}\int_{\Omega}u_0$.http://ejde.math.txstate.edu/Volumes/2015/146/abstr.htmlChemotaxisattraction-repulsionboundednessconvergence |
| spellingShingle | Yuhuan Li Ke Lin Chunlai Mu Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system Electronic Journal of Differential Equations Chemotaxis attraction-repulsion boundedness convergence |
| title | Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system |
| title_full | Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system |
| title_fullStr | Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system |
| title_full_unstemmed | Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system |
| title_short | Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system |
| title_sort | asymptotic behavior for small mass in an attraction repulsion chemotaxis system |
| topic | Chemotaxis attraction-repulsion boundedness convergence |
| url | http://ejde.math.txstate.edu/Volumes/2015/146/abstr.html |
| work_keys_str_mv | AT yuhuanli asymptoticbehaviorforsmallmassinanattractionrepulsionchemotaxissystem AT kelin asymptoticbehaviorforsmallmassinanattractionrepulsionchemotaxissystem AT chunlaimu asymptoticbehaviorforsmallmassinanattractionrepulsionchemotaxissystem |