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Comptes Rendus Mathématique
Volume 355, n° 2
pages 181-186 (février 2017)
Doi : 10.1016/j.crma.2016.12.005
Received : 10 July 2016 ;  accepted : 9 December 2016
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

Xuegang Hu a, Liangchen Wang a , Chunlai Mu b, Ling Li a
a Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China 
b College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR China 


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  . Here  ,   and  ,   for all   with some   and   for all  . It is shown that if the ratio   is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m .

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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  . Ici  ,  ,  ,   pour tout   et un  , et   pour tout  . Lorsque le quotient   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 full text of this article is available in PDF format.

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