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

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

{ut=(ξuχuv),xΩ,t>0,vt=Δvuv,xΩ,t>0,
sous des conditions de Neumann homogènes au bord, supposé lisse, d'un domaine borné ΩRn, n1. 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 :

v0L(Ω)<{1χξ2(n+1)[π+2arctan((1ξ)22(n+1)ξ)],si0<ξ<1,πχ2(n+1),siξ=1,1χξ2(n+1)[π2arctan((ξ1)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<v0L(Ω)16(n+1)χ. Comme 16(n+1)χ<πχ2(n+1) ou, plus précisément, limnπχ2(n+1)16(n+1)χ=+, ces résultats se déduisent des nôtres.

In this paper, we study the chemotaxis system:

{ut=(ξuχuv),xΩ,t>0,vt=Δvuv,xΩ,t>0,
under homogeneous Neumann boundary conditions in a bounded domain ΩRn,n1, 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:

v0L(Ω)<{1χξ2(n+1)[π+2arctan((1ξ)22(n+1)ξ)],if0<ξ<1,πχ2(n+1),ifξ=1,1χξ2(n+1)[π2arctan((ξ1)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<v0L(Ω)16(n+1)χ. Because of 16(n+1)χ<πχ2(n+1), or more precisely, limnπχ2(n+1)16(n+1)χ=+, our results extend the recent results.

Reçu le :
Accepté le :
Publié le :
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/

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