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

Khadijeh Baghaei; Ali Khelghati

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

Khadijeh Baghaei ¹ ; Ali Khelghati 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, Volume 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 :

Khadijeh Baghaei ¹ ; Ali Khelghati 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

@article{CRMATH_2017__355_6_633_0,
     author = {Khadijeh Baghaei and Ali Khelghati},
     title = {Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {633--639},
     publisher = {Elsevier},
     volume = {355},
     number = {6},
     year = {2017},
     doi = {10.1016/j.crma.2017.04.009},
     language = {en},
}

TY  - JOUR
AU  - Khadijeh Baghaei
AU  - Ali Khelghati
TI  - Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 633
EP  - 639
VL  - 355
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2017.04.009
LA  - en
ID  - CRMATH_2017__355_6_633_0
ER  -

%0 Journal Article
%A Khadijeh Baghaei
%A Ali Khelghati
%T Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant
%J Comptes Rendus. Mathématique
%D 2017
%P 633-639
%V 355
%N 6
%I Elsevier
%R 10.1016/j.crma.2017.04.009
%G en
%F CRMATH_2017__355_6_633_0

Khadijeh Baghaei; Ali Khelghati. Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 633-639. doi : 10.1016/j.crma.2017.04.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.009/

Bibliographie
Cité par

[1] K. Baghaei; A. Khelghati Global existence and boundedness of classical solutions for a chemotaxismodel with consumption of chemoattractant and logistic source, Math. Methods Appl. Sci. (2016) | DOI

[2] D. Horstmann; G. Wang Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., Volume 12 (2001), pp. 159-177

[3] E.F. Keller; L.A. Segel Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., Volume 26 (1970), pp. 399-415

[4] J. Lankeit; Y. Wang Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, 2016 (pp. 1–20) | arXiv

[5] T. Li; A. Suen; M. Winkler; C. Xue Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 4, pp. 721-746

[6] T. Nagai; T. Seneba; K. Yoshida Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Volume 40 (1997), pp. 411-433

[7] K. Osaki; A. Yagi Finite dimensional attractors for one-dimensional Keller–Segel equations, Funkc. Ekvacioj, Volume 44 (2001), pp. 349-367

[8] Y. Tao Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., Volume 381 (2011), pp. 521-529

[9] Y. Tao; M. Winkler Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., Volume 252 (2012), pp. 2520-2543

[10] M. Winkler Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differ. Equ., Volume 248 (2010), pp. 2889-2905

[11] M. Winkler Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., Volume 100 (2013), pp. 748-767

[12] Q. Zhang Boundedness in chemotaxis systems with rotational flux terms, Math. Nachr. (2016), pp. 1-12 | DOI

[13] Q. Zhang; Y. Li Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant, J. Math. Phys., Volume 56 (2015) no. 8

[14] P. Zheng; C. Mu Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source, Math. Nachr., Volume 288 (2015), pp. 710-720

Cité par Sources :

Commentaires - Politique