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

Abstract

This paper deals with the quasilinear attraction-repulsion chemotaxis systemwith smooth boundary ∂Ω, where m, p, q ∈ R, χ, ξ, α, β, γ, δ > 0 are constants. Moreover, it is supposed that the function f satisfies f (u) ≡ 0 in the study of boundedness, whereas, when considering blow-up, it is assumed that m > 0 and f is a function of logistic type such as f (u) = λu − µu κ with λ ≥ 0, µ > 0 and κ > 1 sufficiently close to 1, in the radially symmetric setting. In the case that ξ = 0 and f (u) ≡ 0, global existence and boundedness have been proved under the condition p < m + 2 n . Also, in the case that m = 1, p = q = 2 and f is a function of logistic type, finite-time blow-up has 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.