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Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 2
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Research Article

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

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Pages 415-431 | Received 11 Sep 2022, Accepted 18 Dec 2022, Published online: 15 Mar 2023
 

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=Δuχ(uv)+ξ(uw)inΩ×(0,Tmax),τvt=Δvφ(t,v)+f(u)inΩ×(0,Tmax),τw=Δwψ(t,w)+g(u)inΩ×(0,Tmax).(⋄) In this problem, Ω is a bounded and smooth domain of Rn, for n2, 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 u0. Moreover, φ(t,v) and ψ(t,w) have specific expressions, τ{0,1} and Θ:=χαξγ. 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:

  1. 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 Θ0.

  2. 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).

  3. 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.

Mathematics Subject Classification 2010:

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) ofthe Istituto Nazionale di Alta Matematica (INdAM).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are partially supported by the research projects Evolutive and Stationary Partial Differential Equations with a Focus on Biomathematics (2019, Grant Number: F72F20000200007), Analysis of PDEs in connection with real phenomena (2021, Grant Number: F73C22001130007), funded by https://www.fondazionedisardegna.it/, Fondazione di Sardegna. GV is also supported by MIUR (Italian Ministry of Education, University and Research) Prin 2017 Nonlinear Differential Problems via Variational, Topological and Set-valued Methods (Grant Number: 2017AYM8XW).

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