Refined criteria toward boundedness in an attraction–repulsion chemotaxis system with nonlinear productions

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) ut=Δu−χ∇⋅ (u∇v) +ξ∇⋅ (u∇w) inΩ× (0, Tmax), τvt=Δv−φ (t, v) +f (u) inΩ× (0, Tmax), τw=Δw−ψ (t, w) +g (u) inΩ× (0, Tmax). (⋄) In this problem, Ω is a bounded and smooth domain of Rn, for n≥2, χ, ξ>0, f (u), g (u) reasonably regular functions generalizing, respectively, the prototypes f (u) =αuk and g (u) =γul, for some k, l, α, γ>0 and all u≥0. Moreover, φ (t, v) and ψ (t, w) have specific expressions, τ∈{0, 1 and Θ: =χα−ξγ. Once for any sufficiently smooth u (x, 0) =u0 (x) ≥0, τv (x, 0) =τv0 (x) ≥0 and τw (x, 0) =τw0 (x) ≥0, the local well-posedness of problem (◊) is ensured, and we establish for the classical solution (u, v, w) defined in Ω× (0, Tmax) that the life span is indeed Tmax=∞ and u, v and w are uniformly bounded in Ω× (0, ∞) in the following cases: For φ (t, v) =βv, β>0, ψ (t, w) =δw, δ>0 and τ=0, provided (I. 1) k0, ψ (t, w) =δw, δ>0 and τ=1, whenever (II. 1) l, k∈ (0, 1n]; (II. 2) l∈ (1n, 1n+2n2+4) and k∈ (0, 1n], or k∈ (1n, 1n+2n2+4) and l∈ (0, 1n]; (II. 3) l, k∈ (1n, 1n+2n2+4). For φ (t, v) =1|Ω|∫Ωf (u) and ψ (t, w) =1|Ω|∫Ωg (u) and τ=0, under the assumptions k<l or (I. 3) ). In particular, in this paper we partially improve what derived in Viglialoro Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system. Math Nachr. 2021;294 (12): 2441–2454 and solve an open question given in Liu and Li Finite-time blowup in attraction-repulsion systems with nonlinear signal production. Nonlinear Anal Real World Appl. 2021;61: Paper No. 103305, 21. Finally, the research is complemented with numerical simulations in bi-dimensional domains.