Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production

Ya Tian; Dan Li; Chunlai Mu

doi:10.1016/j.crma.2019.05.010

Partial differential equations

Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production
[Stabilisation d'un modèle tridimensionnel de croissance chimiotaxique avec production d'attracteur indirecte]

Présenté par : the Editorial Board

Ya Tian ¹ ; Dan Li ² ; Chunlai Mu ³

¹ Key Lab of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
² School of Mathematics, South China University of Technology, Guangzhou 510641, China
³ College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 513-519.

Résumé
Abstract

Cette Note traite du système de croissance chimiotaxique : $u_{t} = Δ u - \nabla (u \nabla v) + μ u (1 - u)$ , $v_{t} = Δ v - v + w$ , $τ w_{t} + δ w = u$ dans un domaine borné lisse $Ω \subset R^{3}$ avec une condition de flux zéro au bord et où μ, δ et τ sont des paramètres positifs donnés. Nous montrons que la solution $(u, v, w)$ se stabilise exponentiellement vers la solution constante stationnaire $(1, 1 / δ, 1 / δ)$ en norme $L^{\infty} (Ω)$ lorsque t tend vers l'infini, pourvu que $μ > 0$ et que toute donnée initiale positive ou nulle suffisamment lisse satisfasse $u_{0} ≢ 0$ . Ces hypothèses relaxent la condition $μ > 1 / 8 δ^{2}$ de [8].

This paper deals with the chemotaxis-growth system: $u_{t} = Δ u - \nabla \cdot (u \nabla v) + μ u (1 - u)$ , $v_{t} = Δ v - v + w$ , $τ w_{t} + δ w = u$ in a smooth bounded domain $Ω \subset R^{3}$ with zero-flux boundary conditions, where μ, δ, and τ are given positive parameters. It is shown that the solution $(u, v, w)$ exponentially stabilizes to the constant stationary solution $(1, \frac{1}{δ}, \frac{1}{δ})$ in the norm of $L^{\infty} (Ω)$ as $t \to \infty$ provided that $μ > 0$ and any given nonnegative and suitably smooth initial data $(u_{0}, v_{0}, w_{0})$ fulfills $u_{0} ≢ 0$ , which extends the condition $μ > \frac{1}{8 δ^{2}}$ in [8].

Reçu le : 2018-12-11
Accepté le : 2019-06-06
Publié le : 2019-06-01

DOI : 10.1016/j.crma.2019.05.010

Affiliations des auteurs :

Ya Tian ¹ ; Dan Li ² ; Chunlai Mu ³

@article{CRMATH_2019__357_6_513_0,
     author = {Ya Tian and Dan Li and Chunlai Mu},
     title = {Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {513--519},
     publisher = {Elsevier},
     volume = {357},
     number = {6},
     year = {2019},
     doi = {10.1016/j.crma.2019.05.010},
     language = {en},
}

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Ya Tian; Dan Li; Chunlai Mu. Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 513-519. doi : 10.1016/j.crma.2019.05.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.05.010/

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