Global existence and boundedness of classical solutions in a quasilinear parabolic–elliptic chemotaxis system with logistic source

Ali Khelghati; Khadijeh Baghaei

doi:10.1016/j.crma.2015.08.006

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]

Présenté par : the Editorial Board

Ali Khelghati ¹ ; Khadijeh Baghaei ¹

¹ Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697, Tehran, Iran

Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 913-917.

Résumé
Abstract

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

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

avec des conditions au bord homogènes de Neumann, dans un domaine lisse, borné

Ω \subset R^{n}

n \geq 1

. Nous supposons que les fonctions D et

D (s) > 0 pour s \geq 0, D (s) \geq C_{D} s^{m - 1} pour s > 0,

g (0) \geq 0, g (s) \leq a - b s^{γ}, s > 0

pour certaines constantes

C_{D} > 0

m \geq 1

a \geq 0

b > 0

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 $m \geq 2 - \frac{2}{n}$ et $b^{⁎} = \frac{(2 - m) n - 2}{(2 - m) n} χ$ si $m < 2 - \frac{2}{n}$ .

We consider the quasilinear parabolic–elliptic chemotaxis system

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

under homogeneous Neumann boundary conditions in a smooth bounded domain

Ω \subset R^{n}, n \geq 1

. We assume that the functions D and g are smooth and satisfy

\begin{matrix} D (s) > 0 for s \geq 0, D (s) \geq C_{D} s^{m - 1} for s > 0, \\ g (0) \geq 0, g (s) \leq a - b s^{γ}, s > 0 \end{matrix}

with some constants

C_{D} > 0, m \geq 1, a \geq 0, 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 $m \geq 2 - \frac{2}{n}$ and $b^{⁎} = \frac{(2 - m) n - 2}{(2 - m) n} χ$ for $m < 2 - \frac{2}{n}$ .

Reçu le : 2014-08-29
Accepté le : 2015-07-25
Publié le : 2015-10-01

DOI : 10.1016/j.crma.2015.08.006

Affiliations des auteurs :

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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     author = {Ali Khelghati and Khadijeh Baghaei},
     title = {Global existence and boundedness of classical solutions in a quasilinear parabolic{\textendash}elliptic chemotaxis system with logistic source},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {913--917},
     publisher = {Elsevier},
     volume = {353},
     number = {10},
     year = {2015},
     doi = {10.1016/j.crma.2015.08.006},
     language = {en},
}

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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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