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

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 $m \geq 2 - \frac{2}{n}$ 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/

Bibliographie
Cité par

[1] N.D. Alikakos $L^{p}$ bounds of solutions of reaction–diffusion equations, Commun. Partial Differ. Equ., Volume 4 (1979), pp. 827-868

[2] X. Cao Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., Volume 412 (2014), pp. 181-188

[3] X. Cao; S. Zheng Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, Math. Methods Appl. Sci., Volume 37 (2014), pp. 2326-2330

[4] T. Hillen; K.J. Painter A user's guide to PDE models for chemotaxis, J. Math. Biol., Volume 58 (2009), pp. 183-217

[5] D. Horstmann; G. Wang Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., Volume 12 (2001), pp. 159-177

[6] D. Horstmann; M. Winkler Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., Volume 215 (2005), pp. 52-107

[7] E.F. Keller; L.A. Segel Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., Volume 26 (1970), pp. 399-415

[8] J. Lankeit Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ., Volume 258 (2015), pp. 1158-1191

[9] J. Lankeit Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst., Ser. B, Volume 20 (2015), pp. 1499-1527

[10] T. Nagai; T. Senba; K. Yoshida Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Volume 40 (1997), pp. 411-433

[11] K. Osaki; A. Yagi Finite dimensional attractors for one-dimensional Keller–Segel equations, Funkc. Ekvacioj, Volume 44 (2001), pp. 441-469

[12] K. Osaki; A. Yagi Global existence of a chemotaxis-growth system in $R^{2}$ , Adv. Math. Sci. Appl., Volume 12 (2002), pp. 587-606

[13] Y. Tao; M. Winkler Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Differ. Equ., Volume 252 (2012), pp. 692-715

[14] J.I. Tello; M. Winkler A chemotaxis system with logistic source, Commun. Partial Differ. Equ., Volume 32 (2007), pp. 849-877

[15] R. Temam Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997

[16] L. Wang; C. Mu; P. Zheng On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Differ. Equ., Volume 256 (2014), pp. 1847-1872

[17] M. Winkler Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., Volume 348 (2008), pp. 708-729

[18] M. Winkler Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differ. Equ., Volume 248 (2010), pp. 2889-2905

[19] M. Winkler Does ‘volume-filling effect’ always prevent chemotactic collapse?, Math. Methods Appl. Sci., Volume 33 (2010), pp. 12-24

[20] M. Winkler Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 1516-1537

[21] M. Winkler Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., Volume 384 (2011), pp. 261-272

[22] M. Winkler Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., Volume 100 (2013), pp. 748-767

[23] M. Winkler Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differ. Equ., Volume 257 (2014), pp. 1056-1077

[24] M. Winkler How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., Volume 24 (2014), pp. 809-855

[25] M. Winkler; K.C. Djie Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA, Volume 72 (2010), pp. 1044-1064

[26] P. Zheng; C. Mu; X. Hu Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., Volume 35 (2015), pp. 2299-2323

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