Blow-up phenomena in a parabolic–elliptic–elliptic attraction–repulsion chemotaxis system with superlinear logistic degradation

This paper is concerned with the attraction–repulsion chemotaxis system with superlinear logistic degradation, ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+λu−μuk,x∈Ω,t>0,0=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,under homogeneous Neumann boundary conditions, in a ball Ω⊂Rn (n≥3), with constant parameters λ∈R, k>1, μ,χ,ξ,α,β,γ,δ>0. Blow-up phenomena in the system have been well investigated in the case λ=μ=0, whereas the attraction–repulsion chemotaxis system with logistic degradation has been not studied. Under the condition that k>1 is close to 1, this paper ensures a solution which blows up in L∞-norm and Lσ-norm with some σ>1 for some nonnegative initial data. Moreover, a lower bound of blow-up time is derived.