Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source
We consider the chemotaxis system: \begin{equation*} {\left\lbrace \begin{array}{ll} u_{t}=\nabla \cdot \big (\gamma (v) \nabla u-u \,\xi (v) \nabla v\big )+\mu \, u(1-u), & x\in \Omega , \ t>0, \\ v_{t}=\Delta v-uv, & x\in \Omega , \ t>0, \end{array}\right.} \end{equation*} under ho...
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Académie des sciences
2023-11-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.519/ |
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| author | Baghaei, Khadijeh |
| author_facet | Baghaei, Khadijeh |
| author_sort | Baghaei, Khadijeh |
| collection | DOAJ |
| description | We consider the chemotaxis system:
\begin{equation*}
{\left\lbrace \begin{array}{ll} u_{t}=\nabla \cdot \big (\gamma (v) \nabla u-u \,\xi (v) \nabla v\big )+\mu \, u(1-u), & x\in \Omega , \ t>0, \\ v_{t}=\Delta v-uv, & x\in \Omega , \ t>0, \end{array}\right.}
\end{equation*}
under homogeneous Neumann boundary conditions in a bounded domain $ \Omega \subset \mathbb{R}^{n}, n\ge 2,$ with smooth boundary. Here, the functions $\gamma (v)$ and $\xi (v)$ are as:
\begin{equation*}
\gamma (v)=(1+v)^{-k}\quad \text{and} \quad \xi (v)=-(1-\alpha )\gamma ^{\prime }(v),
\end{equation*}
where $k>0$ and $\alpha \in (0,1).$We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $ k\,(1-\alpha )<\frac{4}{n+5}$ and the initial value $ v_{0}$ and $\mu $ satisfy the following conditions:
\begin{align*}
0<\Vert v_{0}\Vert _{L^{\infty }(\Omega )}\le \Bigg [\frac{4\big [1-k \, \big (1-\alpha \big )\big ]}{k\, (n+1)\,(1-\alpha )}\Bigg ]^{\frac{1}{k}}-1,
\end{align*}
and
\begin{equation*}
\mu > \frac{k\,n\,(1-\alpha ) \Vert v_{0}\Vert _{L^{\infty }(\Omega )}}{(n+1)(1+\Vert v_{0}\Vert _{L^{\infty }(\Omega )})}.
\end{equation*}
This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023). |
| format | Article |
| id | doaj-art-56b734fffced448984ef7465bdfa8fd9 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-56b734fffced448984ef7465bdfa8fd92025-02-07T11:11:47ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-11-01361G101641165210.5802/crmath.51910.5802/crmath.519Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic sourceBaghaei, Khadijeh0Pasargad Institute for Advanced Innovative Solutions, No.30, Hakim Azam St., North Shiraz St., Mollasadra Ave., Tehran, IranWe consider the chemotaxis system: \begin{equation*} {\left\lbrace \begin{array}{ll} u_{t}=\nabla \cdot \big (\gamma (v) \nabla u-u \,\xi (v) \nabla v\big )+\mu \, u(1-u), & x\in \Omega , \ t>0, \\ v_{t}=\Delta v-uv, & x\in \Omega , \ t>0, \end{array}\right.} \end{equation*} under homogeneous Neumann boundary conditions in a bounded domain $ \Omega \subset \mathbb{R}^{n}, n\ge 2,$ with smooth boundary. Here, the functions $\gamma (v)$ and $\xi (v)$ are as: \begin{equation*} \gamma (v)=(1+v)^{-k}\quad \text{and} \quad \xi (v)=-(1-\alpha )\gamma ^{\prime }(v), \end{equation*} where $k>0$ and $\alpha \in (0,1).$We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $ k\,(1-\alpha )<\frac{4}{n+5}$ and the initial value $ v_{0}$ and $\mu $ satisfy the following conditions: \begin{align*} 0<\Vert v_{0}\Vert _{L^{\infty }(\Omega )}\le \Bigg [\frac{4\big [1-k \, \big (1-\alpha \big )\big ]}{k\, (n+1)\,(1-\alpha )}\Bigg ]^{\frac{1}{k}}-1, \end{align*} and \begin{equation*} \mu > \frac{k\,n\,(1-\alpha ) \Vert v_{0}\Vert _{L^{\infty }(\Omega )}}{(n+1)(1+\Vert v_{0}\Vert _{L^{\infty }(\Omega )})}. \end{equation*} This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023).https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.519/ |
| spellingShingle | Baghaei, Khadijeh Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source Comptes Rendus. Mathématique |
| title | Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source |
| title_full | Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source |
| title_fullStr | Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source |
| title_full_unstemmed | Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source |
| title_short | Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source |
| title_sort | boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.519/ |
| work_keys_str_mv | AT baghaeikhadijeh boundednessofclassicalsolutionstoachemotaxisconsumptionsystemwithsignaldependentmotilityandlogisticsource |