Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system u_t=∇⋅(D(u)∇u)−∇⋅(G(u)χ(v)∇v)+∇⋅(H(u)ξ(w)∇w),x∈Ω,t>0,v_t=d₁Δv+αu−βv,x∈Ω,t>0,w_t=d₂Δw+γu−δw,x∈Ω,t>0,under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂Rⁿ(n≥1) is a bounded domain with smooth boundary, d₁,d₂,α,β,γ,δ>0 are constants. Also, D,G,H∈C²([0,∞)) fulfill that a₀(s+1)^m−1≤D(s)≤a₁(s+1)^m−1 with a₀,a₁>0 and m∈R; G(0)=0, 0≤G(s)≤b₀(s+1)^q−1 with b₀>0 and q<min{2,m+1}; H(0)=0, 0≤H(s)≤c₀(s+1)^r−1 with c₀>0 and r<min{2,m+1}, and χ,ξ satisfy that [Formula presented] with χ₀>0 and k₁>1; [Formula presented] with ξ₀>0 and k₂>1. Global existence and boundedness in the case that w=0 were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.