Partial differential equations
Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 161-168.

This paper deals with the chemotaxis system with nonlinear signal secretion

 $\left\{\begin{array}{cc}{u}_{t}=\nabla ·\left(D\left(u\right)\nabla u-S\left(u\right)\nabla v\right),\hfill & x\in \Omega ,\phantom{\rule{1em}{0ex}}t>0,\hfill \\ {v}_{t}=\Delta v-v+g\left(u\right),\hfill & x\in \Omega ,\phantom{\rule{1em}{0ex}}t>0,\hfill \end{array}\right\$

under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset {ℝ}^{n}$ ($n\ge 2$). The diffusion function $D\left(s\right)\in {C}^{2}\left(\left[0,\infty \right)\right)$ and the chemotactic sensitivity function $S\left(s\right)\in {C}^{2}\left(\left[0,\infty \right)\right)$ are given by $D\left(s\right)\ge {C}_{d}{\left(1+s\right)}^{-\alpha }$ and $0 for all $s\ge 0$ with ${C}_{d},{C}_{s}>0$ and $\alpha ,\beta \in ℝ$. The nonlinear signal secretion function $g\left(s\right)\in {C}^{1}\left(\left[0,\infty \right)\right)$ is supposed to satisfy $g\left(s\right)\le {C}_{g}{s}^{\gamma }\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}s\ge 0$ with ${C}_{g},\gamma >0$. Global boundedness of solution is established under the specific conditions:

 $0<\gamma \le 1\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\alpha +\beta

The purpose of this work is to remove the upper bound of the diffusion condition assumed in [9], and we also give the necessary constraint $\alpha +\beta <1+\frac{1}{n}$, which is ignored in [9, Theorem 1.1].

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.148
Classification: 35K35,  35A01,  35B44,  35B35,  92C17
Xu Pan 1; Liangchen Wang 1

1. School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China
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Xu Pan; Liangchen Wang. Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 161-168. doi : 10.5802/crmath.148. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.148/

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