Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant

Baghaei, Khadijeh; Khelghati, Ali

doi:10.1016/j.crma.2017.04.009

Partial differential equations

Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant
[Les solutions classiques d'un modèle de chimiotaxie avec consommation de chimioattracteurs sont bornées]

Présenté par : the Editorial Board

Baghaei, Khadijeh ¹ ; Khelghati, Ali ²

¹ School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
² Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

Comptes Rendus. Mathématique, Tome 355 (2017) no. 6, pp. 633-639.

Résumé
Abstract

Dans cette Note, nous étudions le système de chimiotaxie suivant :

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

sous des conditions de Neumann homogènes au bord, supposé lisse, d'un domaine borné

Ω \subset R^{n}

n \geq 1

. Ici, ξ et χ sont des constantes positives.

Nous montrons que les solutions classiques du système ci-dessus sont uniformément bornées en temps, pourvu que :

{‖ v_{0} ‖}_{L^{\infty} (Ω)} < {\begin{matrix} \frac{1}{χ} \sqrt{\frac{ξ}{2 (n + 1)}} [π + 2 \arctan (\frac{(1 - ξ)}{2} \sqrt{\frac{2 (n + 1)}{ξ}})], & si 0 < ξ < 1, \\ \frac{π}{χ \sqrt{2 (n + 1)}}, & si ξ = 1, \\ \frac{1}{χ} \sqrt{\frac{ξ}{2 (n + 1)}} [π - 2 \arctan (\frac{(ξ - 1)}{2} \sqrt{\frac{2 (n + 1)}{ξ}})], & si ξ > 1 . \end{matrix}

Dans le cas

ξ = 1

, des résultats récents montrent que les solutions classiques sont globales et bornées dès que

0 < {‖ v_{0} ‖}_{L^{\infty} (Ω)} \leq \frac{1}{6 (n + 1) χ}

. Comme

\frac{1}{6 (n + 1) χ} < \frac{π}{χ \sqrt{2 (n + 1)}}

ou, plus précisément,

\lim_{n \to \infty} \frac{\frac{π}{χ \sqrt{2 (n + 1)}}}{\frac{1}{6 (n + 1) χ}} = + \infty

, ces résultats se déduisent des nôtres.

In this paper, we study the chemotaxis system:

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

under homogeneous Neumann boundary conditions in a bounded domain

Ω \subset R^{n}, n \geq 1

, with smooth boundary. Here, ξ and χ are some positive constants.

We prove that the classical solutions to the above system are uniformly in-time-bounded provided that:

{‖ v_{0} ‖}_{L^{\infty} (Ω)} < {\begin{matrix} \frac{1}{χ} \sqrt{\frac{ξ}{2 (n + 1)}} [π + 2 \arctan (\frac{(1 - ξ)}{2} \sqrt{\frac{2 (n + 1)}{ξ}})], & if 0 < ξ < 1, \\ \frac{π}{χ \sqrt{2 (n + 1)}}, & if ξ = 1, \\ \frac{1}{χ} \sqrt{\frac{ξ}{2 (n + 1)}} [π - 2 \arctan (\frac{(ξ - 1)}{2} \sqrt{\frac{2 (n + 1)}{ξ}})], & if ξ > 1 . \end{matrix}

In the case of

ξ = 1

, the recent results show that the classical solutions are global and bounded provided that

0 < {‖ v_{0} ‖}_{L^{\infty} (Ω)} \leq \frac{1}{6 (n + 1) χ}

. Because of

\frac{1}{6 (n + 1) χ} < \frac{π}{χ \sqrt{2 (n + 1)}}

, or more precisely,

\lim_{n \to \infty} \frac{\frac{π}{χ \sqrt{2 (n + 1)}}}{\frac{1}{6 (n + 1) χ}} = + \infty

, our results extend the recent results.

Reçu le : 2016-10-26
Accepté le : 2017-04-19
Publié le : 2017-05-09

DOI : 10.1016/j.crma.2017.04.009

Affiliations des auteurs :

Baghaei, Khadijeh ¹ ; Khelghati, Ali ²

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     title = {Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {633--639},
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     doi = {10.1016/j.crma.2017.04.009},
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Baghaei, Khadijeh; Khelghati, Ali. Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant. Comptes Rendus. Mathématique, Tome 355 (2017) no. 6, pp. 633-639. doi : 10.1016/j.crma.2017.04.009. http://www.numdam.org/articles/10.1016/j.crma.2017.04.009/

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