Boundedness and blow up in the higher-dimensional attraction–repulsion chemotaxis system with nonlinear diffusion

This paper deals with the following quasilinear attraction–repulsion chemotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,vt=Δv+αu−βv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0 in a bounded domain Ω⊂Rn(n≥2)Ω⊂Rn(n≥2) under homogeneous Neumann boundary conditions. The parameters χ,ξ,α,β,γχ,ξ,α,β,γ and δ   are positive and the diffusion D(u)D(u) is supposed to generalize the prototype D(u)=D0u−θD(u)=D0u−θ with D0>0D0>0 and θ∈Rθ∈R. Under the assumption θ<2n−1, it is proved that the corresponding initial–boundary problem possesses a nonnegative globally bounded solution. Moreover, by constructing a Lyapunov functional and by assuming that θ=2n−1 and n=3n=3, it is proved that there exist radially symmetric solutions which blow up in finite time in Ω=BR(0)Ω=BR(0) with R>0R>0. Finally, it is also shown that blow up may occur if θ>2n−1 and χα−ξγ>0χα−ξγ>0 when Ω is a ball in RnRn with n≥3n≥3.