Partial differential equations
Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant
Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 633-639.

In this paper, we study the chemotaxis system:

 ${ut=∇⋅(ξ∇u−χu∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,$
under homogeneous Neumann boundary conditions in a bounded domain $Ω⊂Rn,n≥1$, 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:

 $‖v0‖L∞(Ω)<{1χξ2(n+1)[π+2arctan⁡((1−ξ)22(n+1)ξ)],if0<ξ<1,πχ2(n+1),ifξ=1,1χξ2(n+1)[π−2arctan⁡((ξ−1)22(n+1)ξ)],ifξ>1.$
In the case of $ξ=1$, the recent results show that the classical solutions are global and bounded provided that $0<‖v0‖L∞(Ω)≤16(n+1)χ$. Because of $16(n+1)χ<πχ2(n+1)$, or more precisely, $limn→∞⁡πχ2(n+1)16(n+1)χ=+∞$, our results extend the recent results.

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

 ${ut=∇⋅(ξ∇u−χu∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,$
sous des conditions de Neumann homogènes au bord, supposé lisse, d'un domaine borné $Ω⊂Rn$, $n≥1$. 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 :

 $‖v0‖L∞(Ω)<{1χξ2(n+1)[π+2arctan⁡((1−ξ)22(n+1)ξ)],si0<ξ<1,πχ2(n+1),siξ=1,1χξ2(n+1)[π−2arctan⁡((ξ−1)22(n+1)ξ)],siξ>1.$
Dans le cas $ξ=1$, des résultats récents montrent que les solutions classiques sont globales et bornées dès que $0<‖v0‖L∞(Ω)≤16(n+1)χ$. Comme $16(n+1)χ<πχ2(n+1)$ ou, plus précisément, $limn→∞⁡πχ2(n+1)16(n+1)χ=+∞$, ces résultats se déduisent des nôtres.

Accepted:
Published online:
DOI: 10.1016/j.crma.2017.04.009

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
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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/

[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

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