Nonlocal logistics and nonlinear productions in an attraction-repulsion chemotaxis model: analysis of the global well-posedness

Abstract: This paper investigates a {three-component} chemotaxis system involving both attraction and repulsion effects, as well as a nonlocal logistic-type source term. Mathematically, if $u=u(x,t)$, $v = v(x,t)$ and $w = w(x,t)$ denote the cell distribution, and the attractive and the repulsive chemical signals, the model is then described by \begin{equation*} \begin{cases} u_t = Δu - χ\nabla \cdot (u \nabla v) + ξ\nabla \cdot (u \nabla w) + a u^α- b u^α\int_Ωu^β, & x \in Ω, \ t > 0, τv_t = Δv - v + f(u), & x \in Ω, \ t > 0, τw_t = Δw - w + g(u), & x \in Ω, \ t > 0. \end{cases} \end{equation*} Here, $Ω\subset \mathbb{R}<sup>n$ ($n \geq 1$) is a bounded smooth domain, $τ\in{0,1}$, $a,b,α,β,χ,ξ&gt;0$, the production functions $f(u)$ and $g(u)$ are assumed to satisfy algebraic growth conditions of order $\ell$ and $ρ$, generalizing prototypes of the form $u<sup>\ell$ and $u<sup>ρ$, $\ell,ρ&gt;0$. The work is devoted to proving the global existence and boundedness of classical solutions under a suitable balance between the signal production exponents $\ell, ρ$ and the nonlocal damping exponents $α, β$, for regular enough initial data and zero-flux boundary restrictions. In this regard, two main theorems are established for the cases where the chemical signals satisfy either elliptic ($τ=0$) or parabolic ($τ=1$) partial differential equations, highlighting how sufficiently strong nonlocal damping prevents the formation of singularities in time. We extend the results obtained in [Chiyo et al., Appl. Math. Optim. 89:9 (2024)], where the fully parabolic ($τ=1$) and only attraction version is studied. In our context, we establish well-posedness of the system and the long-time behavior of solutions.