Comptes Rendus
Partial differential equations
Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source
[Existence globale et bornes pour les solutions classiques d'un système quasi linéaire, parabolique–elliptique, de chimiotaxie avec source logistique]
Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 913-917.

Nous considérons le système quasi linéaire, parabolique–elliptique, de chimiotaxie

{ut=(D(u)uχuv)+g(u),xΩ,t>0,0=Δvv+u,xΩ,t>0,
avec des conditions au bord homogènes de Neumann, dans un domaine lisse, borné ΩRn, n1. Nous supposons que les fonctions D et
D(s)>0pours0,D(s)CDsm1pours>0,
g(0)0,g(s)absγ,s>0
pour certaines constantes CD>0, m1, a0, b>0 et γ>2.

Nous démontrons que les solutions classiques du système ci-dessus sont uniformément bornées en temps, sans restriction sur m et b. Ceci étend un résultat récent de Wang et al. (2014) [16], qui borne les solutions pour γ>2 sous la condition b>b, où b=0 si m22n et b=(2m)n2(2m)nχ si m<22n.

We consider the quasilinear parabolic–elliptic chemotaxis system

{ut=(D(u)uχuv)+g(u),xΩ,t>0,0=Δvv+u,xΩ,t>0,
under homogeneous Neumann boundary conditions in a smooth bounded domain ΩRn,n1. We assume that the functions D and g are smooth and satisfy
D(s)>0fors0,D(s)CDsm1fors>0,g(0)0,g(s)absγ,s>0
with some constants CD>0,m1,a0,b>0 and γ>2.

We prove that the classical solutions to the above system are uniformly in-time-bounded without any restrictions on m and b. This result extends one of the recent results by Wang et al. (2014) [16], which assert the boundedness of solutions for γ>2 under the condition b>b with b=0 for m22n and b=(2m)n2(2m)nχ for m<22n.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.08.006

Ali Khelghati 1 ; Khadijeh Baghaei 1

1 Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697, Tehran, Iran
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Ali Khelghati; Khadijeh Baghaei. Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source. Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 913-917. doi : 10.1016/j.crma.2015.08.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.08.006/

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