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

Abstract

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) max[m2+k,m3+l]<r, (ii) max[m2+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).

Topics & Concepts