Existence of global bounded classical solution to a quasilinear attraction-repulsion chemotaxis system with logistic source

This paper concerns the global existence of bounded classical solution to a quasilinear attraction-repulsion chemotaxis system with logistic source ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+κu−μu2in Ω,vt=Δv+αu−βvin Ω,wt=Δw+γu−δwin Ω,under homogeneous Neumann boundary condition, with positive parameters χ,ξ,κ,μ,α,β,γ,δ and D(s)≥c0sm−1 for s>0, D(0)>0. We prove that in dimension three there exists a unique global bounded classical solution provided that m>65 and μ>0. With an additional assumption D(s)≤C0(sm−1+1) for s>0, we prove that for any m∈1,65, there exists a constant μ0=μ0(m) such that for all μ>μ0 the above problem admits a unique global bounded classical solution.