Partial differential equations
Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model
Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 875-885.

We study the chemotaxis effect vs. logistic damping on boundedness for the well-known minimal Keller–Segel model with logistic source:

{ut=(uχuv)+uμu2,xΩ,t>0,vt=Δvv+u,xΩ,t>0
in a smooth bounded domain ΩR2 with χ,μ>0, nonnegative initial data u0, v0, and homogeneous Neumann boundary data. It is well known that this model allows only for global and uniform-in-time bounded solutions for any χ,μ>0. Here, we carefully employ a simple and new method to regain its boundedness, with particular attention to how upper bounds of solutions qualitatively depend on χ and μ. More, precisely, it is shown that there exists C=C(u0,v0,Ω)>0 such that
u(,t)L(Ω)C[1+1μ+χK(χ,μ)N(χ,μ)]
and
v(,t)W1,(Ω)C[1+1μ+χ83μK83(χ,μ)]=:CN(χ,μ)
uniformly on [0,), where
K(χ,μ)=M(χ,μ)E(χ,μ),M(χ,μ)=1+1μ+χ(1+1μ2)
and
E(χ,μ)=exp[χCGN22min{1,2χ}(4μu0L1(Ω)+132μ2|Ω|+v0L2(Ω)2)].
We notice that these upper bounds are increasing in χ, decreasing in μ, and have only one singularity at μ=0, where the corresponding minimal model (removing the term uμu2 in the first equation) is widely known to possess blow-ups for large initial data.

Nous étudions l'effet chimiotaxique versus l'amortissement logistique pour borner les solutions du modèle de Keller–Segel minimal bien connu avec source logistique :

{ut=(uχuv)+uμu2,xΩ,t>0,vt=Δvv+u,xΩ,t>0
dans un domaine borné, lisse ΩR2 avec χ,μ>0, des données initiales u0, v0 positives ou nulles et des données au bord de Neumann homogènes. Il est bien connu que ce modèle n'a que des solutions bornées globales et uniformes en temps, pour tout χ,μ>0. Nous utilisons ici une méthode nouvelle et simple pour retrouver ces bornes en portant une attention particulière à la dépendance en χ et μ des bornes supérieures des solutions. Plus précisément, nous montrons qu'il existe C=C(u0,v0,Ω)>0 tel que
u(,t)L(Ω)C[1+1μ+χK(χ,μ)N(χ,μ)]
et
v(,t)W1,(Ω)C[1+1μ+χ83μK83(χ,μ)]=:CN(χ,μ)
uniformément sur [0,[, où
K(χ,μ)=M(χ,μ)E(χ,μ),M(χ,μ)=1+1μ+χ(1+1μ2)
et
E(χ,μ)=exp[χCGN22min{1,2χ}(4μu0L1(Ω)+132μ2|Ω|+v0L2(Ω)2)].
Nous observons que ces bornes supérieures croissent avec χ, décroissent avec μ et n'ont qu'une singularité en μ=0. Il est bien connu que le modèle minimal correspondant (en ôtant le terme uμu2 dans la première équation) a des solutions qui explosent pour les grandes données initiales.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.07.002
Jin, Hai-Yang 1; Xiang, Tian 2

1 School of Mathematics, South China University of Technology, Guangzhou 510640, China
2 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China
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Jin, Hai-Yang; Xiang, Tian. Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model. Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 875-885. doi : 10.1016/j.crma.2018.07.002. http://www.numdam.org/articles/10.1016/j.crma.2018.07.002/

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