Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production

Xu Pan; Liangchen Wang

doi:10.5802/crmath.148

Équations aux dérivées partielles

Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production

Xu Pan ¹ ; Liangchen Wang ¹

¹ School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China

Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 161-168.

Résumé

This paper deals with the chemotaxis system with nonlinear signal secretion

\{\begin{matrix} u_{t} = \nabla \cdot (D (u) \nabla u - S (u) \nabla v), & x \in Ω, t > 0, \\ v_{t} = Δ v - v + g (u), & x \in Ω, t > 0, \end{matrix}

under homogeneous Neumann boundary conditions in a bounded domain $Ω \subset ℝ^{n}$ ( $n \geq 2$ ). The diffusion function $D (s) \in C^{2} ([0, \infty))$ and the chemotactic sensitivity function $S (s) \in C^{2} ([0, \infty))$ are given by $D (s) \geq C_{d} {(1 + s)}^{- α}$ and $0 < S (s) \leq C_{s} s {(1 + s)}^{β - 1}$ for all $s \geq 0$ with $C_{d}, C_{s} > 0$ and $α, β \in ℝ$ . The nonlinear signal secretion function $g (s) \in C^{1} ([0, \infty))$ is supposed to satisfy $g (s) \leq C_{g} s^{γ} for all s \geq 0$ with $C_{g}, γ > 0$ . Global boundedness of solution is established under the specific conditions:

0 < γ \leq 1 and α + β < min \{1 + \frac{1}{n}, 1 + \frac{2}{n} - γ\} .

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 $α + β < 1 + \frac{1}{n}$ , which is ignored in [9, Theorem 1.1].

Reçu le : 2020-09-01
Révisé le : 2020-10-09
Accepté le : 2020-12-10
Publié le : 2021-03-17

DOI : 10.5802/crmath.148

Classification : 35K35, 35A01, 35B44, 35B35, 92C17

Affiliations des auteurs :

Xu Pan ¹ ; Liangchen Wang ¹

¹ School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2021__359_2_161_0,
     author = {Xu Pan and Liangchen Wang},
     title = {Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {161--168},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.148},
     language = {en},
}

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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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