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

Hai-Yang Jin; Tian Xiang

doi:10.1016/j.crma.2018.07.002

Partial differential equations

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

Presented by: the Editorial Board

Hai-Yang Jin ¹; Tian Xiang 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

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

Abstract
Résumé

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

{\begin{matrix} u_{t} = \nabla \cdot (\nabla u - χ u \nabla v) + u - μ u^{2}, & x \in Ω, t > 0, \\ v_{t} = Δ v - v + u, & x \in Ω, t > 0 \end{matrix}

in a smooth bounded domain

Ω \subset R^{2}

with

χ, μ > 0

, nonnegative initial data

u_{0}

,

v_{0}

, 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 (u_{0}, v_{0}, Ω) > 0

such that

{‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} \leq C [1 + \frac{1}{μ} + χ K (χ, μ) N (χ, μ)]

and

{‖ v (\cdot, t) ‖}_{W^{1, \infty} (Ω)} \leq C [1 + \frac{1}{μ} + \frac{χ^{\frac{8}{3}}}{μ} K^{\frac{8}{3}} (χ, μ)] = : C N (χ, μ)

uniformly on

[0, \infty)

, where

K (χ, μ) = M (χ, μ) E (χ, μ), M (χ, μ) = 1 + \frac{1}{μ} + \sqrt{χ} (1 + \frac{1}{μ^{2}})

and

E (χ, μ) = \exp [\frac{χ C_{G N}^{2}}{2 \min {1, \frac{2}{χ}}} (\frac{4}{μ} {‖ u_{0} ‖}_{L^{1} (Ω)} + \frac{13}{2 μ^{2}} | Ω | + {‖ \nabla v_{0} ‖}_{L^{2} (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 - μ u^{2}

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 :

{\begin{matrix} u_{t} = \nabla \cdot (\nabla u - χ u \nabla v) + u - μ u^{2}, & x \in Ω, t > 0, \\ v_{t} = Δ v - v + u, & x \in Ω, t > 0 \end{matrix}

dans un domaine borné, lisse

Ω \subset R^{2}

avec

χ, μ > 0

, des données initiales

u_{0}

,

v_{0}

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 (u_{0}, v_{0}, Ω) > 0

tel que

{‖ u (\cdot, t) ‖}_{L^{\infty} (Ω)} \leq C [1 + \frac{1}{μ} + χ K (χ, μ) N (χ, μ)]

et

{‖ v (\cdot, t) ‖}_{W^{1, \infty} (Ω)} \leq C [1 + \frac{1}{μ} + \frac{χ^{\frac{8}{3}}}{μ} K^{\frac{8}{3}} (χ, μ)] = : C N (χ, μ)

uniformément sur

[0, \infty [

, où

K (χ, μ) = M (χ, μ) E (χ, μ), M (χ, μ) = 1 + \frac{1}{μ} + \sqrt{χ} (1 + \frac{1}{μ^{2}})

et

E (χ, μ) = \exp [\frac{χ C_{G N}^{2}}{2 \min {1, \frac{2}{χ}}} (\frac{4}{μ} {‖ u_{0} ‖}_{L^{1} (Ω)} + \frac{13}{2 μ^{2}} | Ω | + {‖ \nabla v_{0} ‖}_{L^{2} (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 - μ u^{2}

dans la première équation) a des solutions qui explosent pour les grandes données initiales.

Received: 2018-03-27
Accepted: 2018-07-03
Published online: 2018-07-14

DOI: 10.1016/j.crma.2018.07.002

Author's affiliations:

Hai-Yang Jin ¹; Tian Xiang 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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     author = {Hai-Yang Jin and Tian Xiang},
     title = {Chemotaxis effect vs. logistic damping on boundedness in the {2-D} minimal {Keller{\textendash}Segel} model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {875--885},
     publisher = {Elsevier},
     volume = {356},
     number = {8},
     year = {2018},
     doi = {10.1016/j.crma.2018.07.002},
     language = {en},
}

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Hai-Yang Jin; Tian Xiang. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.07.002/

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