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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-11-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.519/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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). |
|---|---|
| ISSN: | 1778-3569 |