Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction–repulsion chemotaxis models with logistics

The following fully nonlinear attraction–repulsion and zero-flux chemotaxis model is studied: (♢) ut=∇⋅ (u+1) m1−1∇u−χu (u+1) m2−1∇v+ξu (u+1) m3−1∇w+λu−μurinΩ× (0, Tmax), τvt=Δv−ϕ (t, v) +f (u) inΩ× (0, Tmax), τwt=Δw−ψ (t, w) +g (u) inΩ× (0, Tmax). Herein, Ω is a bounded and smooth domain of Rn, for n∈N, χ, ξ, λ, μ, r proper positive numbers, m1, m2, m3∈R, and f (u) and g (u) regular functions that generalize the prototypes f (u) ≃uk and g (u) ≃ul, for some k, l>0 and all u≥0. Moreover, τ∈0, 1, and Tmax∈ (0, ∞] is the maximal interval of existence of solutions to the model. Once suitable initial data u0 (x), τv0 (x), τw0 (x) are fixed, we are interested in deriving sufficient conditions implying globality (i. e. , Tmax=∞) and boundedness (i. e. , ‖u (⋅, t) ‖L∞ (Ω) +‖v (⋅, t) ‖L∞ (Ω) +‖w (⋅, t) ‖L∞ (Ω) ≤C for all t∈ (0, ∞) ) of solutions to problem (1). This is achieved in the following scenarios: ⊳ For ϕ (t, v) proportional to v and ψ (t, w) to w, whenever τ=0 and provided one of the following conditions (I) m2+k<m3+l, (II) m2+k<r, (III) m2+k<m1+2n is accomplished or τ=1 in conjunction with one of these restrictions (i) maxm2+k, m3+l<r, (ii) maxm2+k, m3+l<m1+2n, (iii) m2+k<r and m3+l<m1+2n, (iv) m2+k<m1+2n and m3+l<r; ⊳ For ϕ (t, v) =1|Ω|∫Ωf (u) and ψ (t, w) =1|Ω|∫Ωg (u), whenever τ=0 if moreover one among (I), (II), (III) is fulfilled. Our research partially improves and extends some results derived in Jiao et al. (2024) ; Ren and Liu (2020) ; Chiyo and Yokota (2022) ; Columbu et al. (2023).