Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source
Abstract This paper is concerned with a chemotaxis system with singular sensitivity and logistic source, {ut=Δu−χ1∇⋅(uw∇w)+μ1u−μ1uα,x∈Ω,t>0,υt=Δv−χ2∇⋅(υw∇w)+μ2υ−μ2υβ,x∈Ω,t>0,wt=Δw−(u+υ)w,x∈Ω,t>0, $$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi _{1}\nabla \cdot (\frac{u}{w}\nab...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-09-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | http://link.springer.com/article/10.1186/s13660-019-2187-3 |
| _version_ | 1828439199752126464 |
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| author | Ting Huang Lu Yang Yongjie Han |
| author_facet | Ting Huang Lu Yang Yongjie Han |
| author_sort | Ting Huang |
| collection | DOAJ |
| description | Abstract This paper is concerned with a chemotaxis system with singular sensitivity and logistic source, {ut=Δu−χ1∇⋅(uw∇w)+μ1u−μ1uα,x∈Ω,t>0,υt=Δv−χ2∇⋅(υw∇w)+μ2υ−μ2υβ,x∈Ω,t>0,wt=Δw−(u+υ)w,x∈Ω,t>0, $$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi _{1}\nabla \cdot (\frac{u}{w}\nabla w)+\mu _{1}u-\mu _{1}u^{\alpha }, &x\in \varOmega , t>0, \\ \upsilon _{t}=\Delta v-\chi _{2}\nabla \cdot ( \frac{\upsilon }{w}\nabla w)+\mu _{2}\upsilon -\mu _{2}\upsilon ^{\beta } , &x\in \varOmega , t>0, \\ w_{t}=\Delta w-(u+\upsilon )w, &x\in \varOmega , t>0, \end{cases}\displaystyle \end{aligned}$$ under the homogeneous Neumann boundary conditions and for widely arbitrary positive initial data in a bounded domain Ω⊂Rn(n≥1) $\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)$ with smooth boundary, where χi $\chi _{i}$, μi>0 $\mu _{i}>0$ (i=1,2) $(i=1, 2)$ and α, β>1 $\beta >1$. It is proved that there exists a global classical solution if max{χ1,χ2}<2n,min{μ1,μ2}>n−2n,α=β=2 $\max \{\chi _{1}, \chi _{2}\}<\sqrt{\frac{2}{n}}, \min \{\mu _{1}, \mu _{2}\}>\frac{n-2}{n}, \alpha =\beta =2$ for n≥2 $n\geq 2$ or any χi>0 $\chi _{i}>0$ (i=1,2) $(i=1,2)$, μi>0 $\mu _{i}>0 $ (i=1,2) $(i=1,2)$, α, β>1 $\beta >1$ for n=1 $n=1$. |
| first_indexed | 2024-12-10T20:19:22Z |
| format | Article |
| id | doaj.art-9753fbed93184eec96c93fe6dad09257 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-10T20:19:22Z |
| publishDate | 2019-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-9753fbed93184eec96c93fe6dad092572022-12-22T01:35:06ZengSpringerOpenJournal of Inequalities and Applications1029-242X2019-09-012019111710.1186/s13660-019-2187-3Global solutions to a two-species chemotaxis system with singular sensitivity and logistic sourceTing Huang0Lu Yang1Yongjie Han2School of Science, Xihua UniversitySchool of Science, Xihua UniversitySchool of Science, Xihua UniversityAbstract This paper is concerned with a chemotaxis system with singular sensitivity and logistic source, {ut=Δu−χ1∇⋅(uw∇w)+μ1u−μ1uα,x∈Ω,t>0,υt=Δv−χ2∇⋅(υw∇w)+μ2υ−μ2υβ,x∈Ω,t>0,wt=Δw−(u+υ)w,x∈Ω,t>0, $$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi _{1}\nabla \cdot (\frac{u}{w}\nabla w)+\mu _{1}u-\mu _{1}u^{\alpha }, &x\in \varOmega , t>0, \\ \upsilon _{t}=\Delta v-\chi _{2}\nabla \cdot ( \frac{\upsilon }{w}\nabla w)+\mu _{2}\upsilon -\mu _{2}\upsilon ^{\beta } , &x\in \varOmega , t>0, \\ w_{t}=\Delta w-(u+\upsilon )w, &x\in \varOmega , t>0, \end{cases}\displaystyle \end{aligned}$$ under the homogeneous Neumann boundary conditions and for widely arbitrary positive initial data in a bounded domain Ω⊂Rn(n≥1) $\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)$ with smooth boundary, where χi $\chi _{i}$, μi>0 $\mu _{i}>0$ (i=1,2) $(i=1, 2)$ and α, β>1 $\beta >1$. It is proved that there exists a global classical solution if max{χ1,χ2}<2n,min{μ1,μ2}>n−2n,α=β=2 $\max \{\chi _{1}, \chi _{2}\}<\sqrt{\frac{2}{n}}, \min \{\mu _{1}, \mu _{2}\}>\frac{n-2}{n}, \alpha =\beta =2$ for n≥2 $n\geq 2$ or any χi>0 $\chi _{i}>0$ (i=1,2) $(i=1,2)$, μi>0 $\mu _{i}>0 $ (i=1,2) $(i=1,2)$, α, β>1 $\beta >1$ for n=1 $n=1$.http://link.springer.com/article/10.1186/s13660-019-2187-3ChemotaxisGlobal existenceSingular sensitivityLogistic source |
| spellingShingle | Ting Huang Lu Yang Yongjie Han Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source Journal of Inequalities and Applications Chemotaxis Global existence Singular sensitivity Logistic source |
| title | Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source |
| title_full | Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source |
| title_fullStr | Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source |
| title_full_unstemmed | Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source |
| title_short | Global solutions to a two-species chemotaxis system with singular sensitivity and logistic source |
| title_sort | global solutions to a two species chemotaxis system with singular sensitivity and logistic source |
| topic | Chemotaxis Global existence Singular sensitivity Logistic source |
| url | http://link.springer.com/article/10.1186/s13660-019-2187-3 |
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