Boundedness and finite-time blow-up in a quasilinear parabolic–elliptic–elliptic attraction–repulsion chemotaxis system

This paper deals with the quasilinear attraction–repulsion chemotaxis system {ut=∇·((u+1)m-1∇u-χu(u+1)p-2∇v+ξu(u+1)q-2∇w)+f(u),0=Δv+αu-βv,0=Δw+γu-δwin a bounded domain Ω ⊂ Rⁿ (n∈ N) with smooth boundary ∂Ω , where m, p, q∈ R, χ, ξ, α, β, γ, δ> 0 are constants, and f is a function of logistic type such as f(u) = λu- μu^κ with λ, μ> 0 and κ≥ 1 , provided that the case f(u) ≡ 0 is included in the study of boundedness, whereas κ is sufficiently close to 1 in considering blow-up in the radially symmetric setting. In the case that ξ= 0 and f(u) ≡ 0 , global existence and boundedness have already been proved under the condition p<m+2n. Also, in the case that m= 1 , p= q= 2 and f is a function of logistic type, finite-time blow-up has already been established by assuming χα- ξγ> 0. This paper classifies boundedness and blow-up into the cases p< q and p> q without any condition for the sign of χα- ξγ and the case p= q with χα- ξγ< 0 or χα- ξγ> 0.