Global boundedness in quasilinear attraction–repulsion chemotaxis system with logistic source

Abstract

This paper studies the quasilinear attraction–repulsion chemotaxis system with logistic source ut=∇⋅(D(u)∇u)−χ∇⋅(Φ(u)∇v)+ξ∇⋅(Ψ(u)∇w)+f(u), τvt=Δv+αu−βv, τ∈{0,1}, 0=Δw+γu−δw, in bounded domain Ω⊂RN, N≥1, subject to the homogeneous Neumann boundary conditions, D,Φ,Ψ∈C2[0,+∞) nonnegative, with D(s)≥(s+1)p for s≥0, Φ(s)≤χsq, ξsr≤Ψ(s)≤ζsr for s>1, and f smooth satisfying f(s)≤μs(1−sk) for s>0, f(0)≥0. It is proved that if the attraction is dominated by one of the other three mechanisms with max{r,k,p+2N}>q, then the solutions are globally bounded. Under more interesting balance situations, the behavior of solutions depends on the coefficients involved, i.e., the upper bound coefficient χ for the attraction, the lower bound coefficient ξ for the repulsion, the logistic source coefficient μ, as well as the constants α and γ describing the capacity of the cells u to produce chemoattractant and chemorepellent respectively. Three balance situations (attraction–repulsion balance, attraction–logistic source balance, and attraction–repulsion–logistic source balance) are considered to establish the boundedness of solutions for the parabolic–elliptic–elliptic case (with τ=0) and the parabolic–parabolic–elliptic case (with τ=1) respectively.