Partial Differential Equations
Global existence and boundedness of classical solutions for a chemotaxis model with logistic source
Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 585-591.

We consider the chemotaxis system:

{ut=Δu(uχ(v)v)+f(u),xΩ,t>0,vt=Δvv+ug(u),xΩ,t>0,
under homogeneous Neumann boundary conditions in a bounded domain ΩRn, n1, with smooth boundary and function f is assumed to generalize the logistic source:
f(u)=aubu2,u0, with a>0,b>0.
Moreover, χ(s) and g(s) are nonnegative smooth functions and satisfy:
χ(s)ϱ(1+ϑs)k,s0, with some ϱ>0,ϑ>0 and k>1,
g(s)h0(1+hs)δ,s0,withh0>0,h0,δ0.
We prove that for all positive values of ϱ, a and b, classical solutions to the above system are uniformly-in-time bounded. This result extends a recent result by C. Mu, L. Wang, P. Zheng and Q. Zhang (2013) [13], which asserts the global existence and boundedness of classical solutions on condition that 0a<2b and ϱ be sufficiently small.

On considère le système de la chimiotaxie :

{ut=Δu(uχ(v)v)+f(u),xΩ,t>0,vt=Δvvug(u),xΩ,t>0,
avec conditions de Neumann homogènes dans un domaine borné ΩRn, n1, de frontière régulière ; on suppose que f est une généralisation dʼune source logistique :
f(u)=aubu2,u0, avec a>0,b>0.
De plus, χ(s) et g(s) sont des fonctions positives ou nulles régulières vérifiant :
χ(s)ϱ(1+ϑs)k,s0, avec ϱ>0,ϑ>0,k>1,
g(s)h0(1+hs)δ,s0, avec h0>0,h0,δ0.
On démontre que, pour toute valeur positive de ϱ, a et b, les solutions classiques du système ci-dessus sont uniformément bornées en temps. Ce résultat étend un résultat récent de C. Mu, L. Wang, P. Zheng et Q. Zhang (2013) [13], qui établit lʼexistence globale et des bornes des solutions classiques sous les conditions 0a<2b et ϱ assez petit.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.07.027
Baghaei, Khadijeh 1; Hesaaraki, Mahmoud 2

1 Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
2 Department of Mathematics, Sharif University of Technology, Tehran, Iran
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Baghaei, Khadijeh; Hesaaraki, Mahmoud. Global existence and boundedness of classical solutions for a chemotaxis model with logistic source. Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 585-591. doi : 10.1016/j.crma.2013.07.027. http://www.numdam.org/articles/10.1016/j.crma.2013.07.027/

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