Comptes Rendus
no. 2
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

Xuegang Hu1 ; Liangchen Wang1 ; Chunlai Mu2 ; Ling Li1
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, Volume 355 (2017) no. 2, pp. 181-186.

Nous considérons le système quasi-linéaire de chimio-haptotaxie

{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)ut=+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0,
sous des conditions aux limites de Neumann homogènes, dans un domaine borné et lisse Ω⊂R3. Ici χ>0, ξ>0, μ>0, D(u)≥cDum−1 pour tout u>0 et un cD>0, et D(u)>0 pour tout u≥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

{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)ut=+μu(1−u−w),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0,
is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂R3. Here χ>0, ξ>0 and μ>0, D(u)≥cDum−1 for all u>0 with some cD>0 and D(u)>0 for all u≥0. It is shown that if the ratio χμ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.12.005
Xuegang Hu 1 ; Liangchen Wang 1 ; Chunlai Mu 2 ; Ling Li 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 
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Xuegang Hu; Liangchen Wang; Chunlai Mu; Ling Li. Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 181-186. doi : 10.1016/j.crma.2016.12.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.12.005/

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