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

Abstract

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) k<l; (I.2) k,l∈(0,2n); (I.3) k = l and Θ<0, or l=k∈(0,2n) and Θ≥0.For φ(t,v)=βv, β>0, 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.

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