Global existence and boundedness of classical solutions for a chemotaxis model with logistic source

Khadijeh Baghaei; Mahmoud Hesaaraki

doi:10.1016/j.crma.2013.07.027

Partial Differential Equations

Global existence and boundedness of classical solutions for a chemotaxis model with logistic source

Presented by: the Editorial Board

Khadijeh Baghaei ¹; Mahmoud Hesaaraki ²

¹ Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
² Department of Mathematics, Sharif University of Technology, Tehran, Iran

Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 585-591.

Abstract (Original language)
Résumé

We consider the chemotaxis system:

{\begin{matrix} u_{t} = Δ u - \nabla \cdot (u χ (v) \nabla v) + f (u), & x \in Ω, t > 0, \\ v_{t} = Δ v - v + u g (u), & x \in Ω, t > 0, \end{matrix}

under homogeneous Neumann boundary conditions in a bounded domain

Ω \subset R^{n}

,

n ⩾ 1

, with smooth boundary and function f is assumed to generalize the logistic source:

f (u) = a u - b u^{2}, u ⩾ 0, with a > 0, b > 0 .

Moreover,

χ (s)

and

g (s)

are nonnegative smooth functions and satisfy:

χ (s) ⩽ \frac{ϱ}{{(1 + ϑ s)}^{k}}, s ⩾ 0, with some ϱ > 0, ϑ > 0 and k > 1,

g (s) ⩽ \frac{h_{0}}{{(1 + h s)}^{δ}}, s ⩾ 0, with h_{0} > 0, h ⩾ 0, δ ⩾ 0 .

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

0 ⩽ a < 2 b

and ϱ be sufficiently small.

On considère le système de la chimiotaxie :

{\begin{matrix} u_{t} = Δ u - \nabla \cdot (u χ (v) \nabla v) + f (u), & x \in Ω, t > 0, \\ v_{t} = Δ v - v - u g (u), & x \in Ω, t > 0, \end{matrix}

avec conditions de Neumann homogènes dans un domaine borné

Ω \subset R^{n}

,

n \geq 1

, de frontière régulière ; on suppose que f est une généralisation dʼune source logistique :

f (u) = a u - b u^{2}, u \geq 0, avec a > 0, b > 0 .

De plus,

χ (s)

et

g (s)

sont des fonctions positives ou nulles régulières vérifiant :

χ (s) \leq \frac{ϱ}{{(1 + ϑ s)}^{k}}, s \geq 0, avec ϱ > 0, ϑ > 0, k > 1,

g (s) \leq \frac{h_{0}}{{(1 + h s)}^{δ}}, s \geq 0, avec h_{0} > 0, h \geq 0, δ \geq 0 .

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

0 \leq a < 2 b

et ϱ assez petit.

Received: 2013-03-12
Accepted: 2013-07-31
Published online: 2013-08-01

DOI: 10.1016/j.crma.2013.07.027

Author's affiliations:

Khadijeh Baghaei ¹; Mahmoud Hesaaraki 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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     author = {Khadijeh Baghaei and Mahmoud Hesaaraki},
     title = {Global existence and boundedness of classical solutions for a chemotaxis model with logistic source},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {585--591},
     publisher = {Elsevier},
     volume = {351},
     number = {15-16},
     year = {2013},
     doi = {10.1016/j.crma.2013.07.027},
     language = {en},
}

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Khadijeh Baghaei; Mahmoud Hesaaraki. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.07.027/

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