Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

Hu, Xuegang; Wang, Liangchen; Mu, Chunlai; Li, Ling

doi:10.1016/j.crma.2016.12.005

Partial differential equations

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion
[Existence de solution bornée pour les modèles tri-dimensionnels de chimio-haptotaxie avec diffusion non-linéaire]

Présenté par : the Editorial Board

Hu, Xuegang ¹ ; Wang, Liangchen ¹ ; Mu, Chunlai ² ; Li, Ling ¹

¹ Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China
² College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China

Comptes Rendus. Mathématique, Tome 355 (2017) no. 2, pp. 181-186.

Résumé
Abstract

Nous considérons le système quasi-linéaire de chimio-haptotaxie

{\begin{matrix} u_{t} = \nabla \cdot (D (u) \nabla u) - χ \nabla \cdot (u \nabla v) - ξ \nabla \cdot (u \nabla w) \\ + μ u (1 - u - w), & x \in Ω, t > 0, \\ v_{t} = Δ v - v + u, & x \in Ω, t > 0, \\ w_{t} = - v w, & x \in Ω, t > 0, \end{matrix}

sous des conditions aux limites de Neumann homogènes, dans un domaine borné et lisse

Ω \subset R^{3}

. Ici

χ > 0

ξ > 0

μ > 0

D (u) \geq c_{D} u^{m - 1}

pour tout

u > 0

et un

c_{D} > 0

, et

D (u) > 0

pour tout

u \geq 0

. Lorsque le quotient

χ / μ

est assez petit, nous montrons que le système possède une unique solution globale classique, qui est uniformément bornée. Notre résultat est sans restriction sur m.

The quasilinear chemotaxis–haptotaxis system

{\begin{matrix} u_{t} = \nabla \cdot (D (u) \nabla u) - χ \nabla \cdot (u \nabla v) - ξ \nabla \cdot (u \nabla w) \\ + μ u (1 - u - w), & x \in Ω, t > 0, \\ v_{t} = Δ v - v + u, & x \in Ω, t > 0, \\ w_{t} = - v w, & x \in Ω, t > 0, \end{matrix}

is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain

Ω \subset R^{3}

. Here

χ > 0

ξ > 0

and

μ > 0

D (u) \geq c_{D} u^{m - 1}

for all

u > 0

with some

c_{D} > 0

and

D (u) > 0

for all

u \geq 0

. It is shown that if the ratio

\frac{χ}{μ}

is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.

Reçu le : 2016-07-10
Accepté le : 2016-12-09
Publié le : 2016-12-27

DOI : 10.1016/j.crma.2016.12.005

Affiliations des auteurs :

Hu, Xuegang ¹ ; Wang, Liangchen ¹ ; Mu, Chunlai ² ; Li, Ling ¹

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     author = {Hu, Xuegang and Wang, Liangchen and Mu, Chunlai and Li, Ling},
     title = {Boundedness in a three-dimensional chemotaxis{\textendash}haptotaxis model with nonlinear diffusion},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {181--186},
     publisher = {Elsevier},
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Hu, Xuegang; Wang, Liangchen; Mu, Chunlai; Li, Ling. Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. Comptes Rendus. Mathématique, Tome 355 (2017) no. 2, pp. 181-186. doi : 10.1016/j.crma.2016.12.005. http://www.numdam.org/articles/10.1016/j.crma.2016.12.005/

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