Partial differential equations
Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1641-1652.

We consider the chemotaxis system:

 $\left\{\begin{array}{cc}{u}_{t}=\nabla ·\left(\gamma \left(v\right)\nabla u-u\phantom{\rule{0.166667em}{0ex}}\xi \left(v\right)\nabla v\right)+\mu \phantom{\rule{0.166667em}{0ex}}u\left(1-u\right),\hfill & x\in \Omega ,\phantom{\rule{4pt}{0ex}}t>0,\hfill \\ {v}_{t}=\Delta v-uv,\hfill & x\in \Omega ,\phantom{\rule{4pt}{0ex}}t>0,\hfill \end{array}\right\$

under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset {ℝ}^{n},n\ge 2,$ with smooth boundary. Here, the functions $\gamma \left(v\right)$ and $\xi \left(v\right)$ are as:

 $\gamma \left(v\right)={\left(1+v\right)}^{-k}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\xi \left(v\right)=-\left(1-\alpha \right){\gamma }^{\prime }\left(v\right),$

where $k>0$ and $\alpha \in \left(0,1\right).$

We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $k\phantom{\rule{0.166667em}{0ex}}\left(1-\alpha \right)<\frac{4}{n+5}$ and the initial value ${v}_{0}$ and $\mu$ satisfy the following conditions:

 $\begin{array}{c}\hfill 0<\parallel {v}_{0}{\parallel }_{{L}^{\infty }\left(\Omega \right)}\le {\left[\frac{4\left[1-k\phantom{\rule{0.166667em}{0ex}}\left(1-\alpha \right)\right]}{k\phantom{\rule{0.166667em}{0ex}}\left(n+1\right)\phantom{\rule{0.166667em}{0ex}}\left(1-\alpha \right)}\right]}^{\frac{1}{k}}-1,\end{array}$

and

 $\mu >\frac{k\phantom{\rule{0.166667em}{0ex}}n\phantom{\rule{0.166667em}{0ex}}\left(1-\alpha \right)\parallel {v}_{0}{\parallel }_{{L}^{\infty }\left(\Omega \right)}}{\left(n+1\right)\left(1+\parallel {v}_{0}{\parallel }_{{L}^{\infty }\left(\Omega \right)}\right)}.$

This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023).

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.519

1 Pasargad Institute for Advanced Innovative Solutions, No.30, Hakim Azam St., North Shiraz St., Mollasadra Ave., Tehran, Iran
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Khadijeh Baghaei. Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1641-1652. doi : 10.5802/crmath.519. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.519/

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