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

• DOI: 10.1016/j.nonrwa.2024.104135
• Publication Date: 2024-05-15T21:18:39Z
• Authors: Alessandro Columbu; Rafael Díaz Fuentes; Silvia Frassu
• Keywords: Mathematics - Analysis of PDEs; Chemotaxis; Attraction-repulsion; Nonlinear production; Boundedness; FOS: Mathematics; Analysis of PDEs (math.AP)

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