Boundedness in a nonlinear attraction-repulsion Keller–Segel system with production and consumption

Abstract

This paper is focused on the zero-flux attraction-repulsion chemotaxis model(◇){ut=∇⋅((u+1)m1−1∇u−χu(u+1)m2−1∇v in Ω×(0,Tmax),+ξu(u+1)m3−1∇w)vt=Δv−f(u)v in Ω×(0,Tmax),0=Δw−δw+g(u) in Ω×(0,Tmax), defined in Ω, which is a bounded and smooth domain of Rn, for n≥2, with χ,ξ,δ>0, m1,m2,m3∈R, and f(u) and g(u) reasonably regular functions generalizing the prototypes f(u)=Kuα and g(u)=γul, with K,γ>0 and appropriate α,l>0. Moreover Tmax is finite or infinite and (0,Tmax) stands for the maximal temporal interval where solutions to the related initial problem exist. Our main interest is to identify constellations of the impacts m1,m2 and m3 of the diffusion and drift terms, as well as of the growth l of the production g for the chemorepellent (i.e., w) and the rate α of the consumption f for the chemoattractant (i.e., v), which ensure boundedness of cell densities (i.e., u). Precisely, for any fixed α∈(0,12+1n) and l≥1, we prove that wheneverm1>min⁡{2m2+1−(m3+l),max⁡{2m2,n−2n}}, any sufficiently smooth initial data u(x,0)=u0(x)≥0 and v(x,0)=v0(x)≥0 produce a unique classical solution (u,v,w) to problem (◇) such that its life span Tmax=∞ and, moreover, u,v and w are uniformly bounded in Ω×(0,∞).