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

Abstract

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system ut=∇⋅(D(u)∇u)−∇⋅(G(u)χ(v)∇v)+∇⋅(H(u)ξ(w)∇w),x∈Ω,t>0,vt=d1Δv+αu−βv,x∈Ω,t>0,wt=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 0≤χ(s)≤χ0sk1 with χ0>0 and k1>1; 0≤ξ(s)≤ξ0sk2 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.