A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source

Abstract

In this paper we study the quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source: ut=∇⋅(D(u)∇u−S(u)∇v)+f(u), vt=Δv−a1v+b1w, wt=Δw−a2w+b2u, under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥1), where ai,bi>0 (i=1,2), D,S∈C2([0,∞)) and f:R→R is a smooth function generalizing the logistic source f(s)=b−μsr for all s≥0 with b≥0, μ>0 and r≥1. We obtain the global boundedness of solutions in four cases: (i) the self-diffusion dominates the cross-diffusion; (ii) the logistic source suppresses the cross-diffusion; (iii) the logistic dampening balances the cross-diffusion with μ>0 suitably large; (iv) the self-diffusion and the logistic source both balance the cross-diffusion to some extent with μ>0 arbitrary. As corollaries, we also consider the global boundedness of solutions for the quasilinear attraction-repulsion chemotaxis model with logistic source: u˜t=∇⋅(D(u˜)∇u˜)−χ∇⋅(u˜∇z)+ξ∇⋅(u˜∇w˜)+f(u˜), zt=Δz−ρz+ηu˜, w˜t=Δw˜−δw˜+γu˜, where χ,η,ξ,γ,ρ,δ>0.