Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

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:in Ω × (0, Tmax).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) = Ku α and g(u) = γu 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 (✸), which is defined in Ω × (0, Tmax), we establish that for any such (u 0 , v 0 ), the life span Tmax = ∞ 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 any ξ > 0, and in any dimensional settings. Finally, an indicative 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 [11].

Uniform in time $L^\infty$-estimates for an attraction-repulsion chemotaxis system with double saturation

“…in Ω × (0, T max ), k, µ > 0, the authors establish that the resulting Cauchy problem admits classical bounded solutions for arbitrarily large χ v 0 L ∞ (Ω) provided µ is also larger than a certain expression increasing with χ v 0 L ∞ (Ω) . But, going towards attraction-repulsion models, when in (1) one considers h(u, v) ≈ βv − u α v in the second equation (with τ = 1) and k(u, v) ≈ u l in the third (with τ = 0), a model with produced chemorepellent and saturated chemoattractant is obtained; in [3] boundedness is proved (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 any ξ > 0, and in any dimensional settings. We point out that papers [3,9] will be mentioned again in Remark 2 of §5.2, when technical discussions on models with only attraction and with attraction and repulsion are given.…”

“…But, going towards attraction-repulsion models, when in (1) one considers h(u, v) ≈ βv − u α v in the second equation (with τ = 1) and k(u, v) ≈ u l in the third (with τ = 0), a model with produced chemorepellent and saturated chemoattractant is obtained; in [3] boundedness is proved (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 any ξ > 0, and in any dimensional settings. We point out that papers [3,9] will be mentioned again in Remark 2 of §5.2, when technical discussions on models with only attraction and with attraction and repulsion are given. (Even though herein we are not strictly interested in the influence of logistics in chemotaxis models, we discussed its role in connection to problems of the form in (2).…”

“…Theorem 2.1. For any n ≥ 2 and some r > n, let Ω be a smooth and bounded domain of R n and (u 0 , v 0 , w 0 ) ∈ (W 1,r (Ω)) 3 any nontrivial initial data with u 0 , v 0 , w 0 ≥ 0 on Ω. Then, for 0 < χ <…”

“…With these preparations in our hands, we give the following Definition 3.1. For some r > n, let (u 0 , v 0 , w 0 ) ∈ (W 1,r (Ω)) 3 and T > 0. We say that ω ∈ (C([0, T ); W 1,r (Ω))) 3 is a weak W 1,r -solution of problem (6) in Ω × (0, T ), if for all test function ϕ ∈ C ∞ 0 (Ω × [0, T )) we have that for I, J, K = 1, 2, 3, with…”

“…More precisely Lemma 3.2 (Local existence). For any n ≥ 2 and some r > n, let Ω be a smooth and bounded domain of R n and (u 0 , v 0 , w 0 ) ∈ (W 1,r (Ω)) 3 any nontrivial initial data with u 0 , v 0 , w 0 ≥ 0 on Ω. Then there exists T max ∈ (0, ∞] such that problem (3) admits a unique nonnegative local-in-time classical solution…”