Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller–Segel model

Jin, Hai-Yang; Xiang, Tian

Comptes Rendus. Mathématique, Tome 356 (2018) no. 8, pp. 875-885.

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=∇⋅(∇u−χu∇v)+u−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0

dans un domaine borné, lisse Ω⊂R2 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}(4μ‖u0‖L1(Ω)+132μ2|Ω|+‖∇v0‖L2(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 u−μu2 dans la première équation) a des solutions qui explosent pour les grandes données initiales.

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

{ut=∇⋅(∇u−χu∇v)+u−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0

in a smooth bounded domain Ω⊂R2 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}(4μ‖u0‖L1(Ω)+132μ2|Ω|+‖∇v0‖L2(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 u−μu2 in the first equation) is widely known to possess blow-ups for large initial data.

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