Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent. (December 2021)

• Record Type: Journal Article
• Authors: Frassu, Silvia; Viglialoro, Giuseppe
• Is Part Of: Nonlinear analysis. Volume 213(2021)
• Publication Date: 2021-12
• Subjects: primary 35A01 35K55 35Q92 -- secondary 92C17; Chemotaxis -- Global existence -- Boundedness -- Nonlinear production; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112505
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study this zero-flux attraction–repulsion chemotaxis model, with linear and superlinear production g for the chemorepellent and sublinear rate f for the chemoattractant: ( ♢ ) u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) in Ω × ( 0, T m a x ), v t = Δ v − f ( u ) v in Ω × ( 0, T m a x ), 0 = Δ w − δ w + g ( u ) in Ω × ( 0, T m a x ) . In this problem, Ω is a bounded and smooth domain of R n, for n ≥ 1, χ, ξ, δ > 0, f ( u ) and g ( u ) reasonably regular functions generalizing the prototypes f ( u ) = K u α and g ( u ) = γ u l, with K, γ > 0 and proper α, l > 0 . Once it is indicated that any sufficiently smooth u ( x, 0 ) = u 0 ( x ) ≥ 0 and v ( x, 0 ) = v 0 ( x ) ≥ 0 produce a unique classical and nonnegative solution ( u, v, w ) to (1), which is defined in Ω × ( 0, T m a x ), we establish that for any such ( u 0, v 0 ), the life span T m a x = ∞ and u, v and w are uniformly bounded in Ω × ( 0, ∞ ), (i) for l = 1, n ∈ { 1, 2 }, α ∈ ( 0, 1 2 + 1 n ) ∩ ( 0, 1 ) and any ξ > 0, (ii) for l = 1, n ≥ 3, α ∈ ( 0, 1 2 + 1 n ) and ξ larger than a quantity depending on χ ‖ v 0 ‖ L ∞ ( Ω ), (iii) for l > 1, α ∈ ( 0, 1 2 + 1 n ) ∩ ( 0, 1 ), any ξ > 0, and in any dimensional settings. Finally, an illustrative analysis about the effect by logistic and repulsive actions on chemotactic phenomena is proposed by comparing the results herein derived for the linear production case with those in Lankeit and Wang (2017).