Boundedness and asymptotic behavior in the higher dimensional fully parabolic attraction-repulsion chemotaxis system with nonlinear diffusion

Abstract

In this paper, we consider the initial-boundary value problem for the fully parabolic attraction-repulsion chemotaxis model with nonlinear diffusion{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,vt=D1Δv+αu−βv,x∈Ω,t>0,wt=D2Δw+γu−δw,x∈Ω,t>0, in a bounded domain Ω⊂Rn(n≥3) with smooth boundary subject to homogeneous Neumann boundary conditions, where χ,ξ,D1,D2,α,β,γ,δ are positive parameters. The function D satisfies D(u)≥CDuθ for all u>0 with constant CD>0. We study the higher dimensional case with D1≠D2 and obtain that the problem possesses a global bounded solution under ξγχα≥max⁡{D1D2,D2D1,βδ,δβ} and θ>1−4n+2. If the parameter further satisfies θ≤1, then the bounded solution converges to the constant steady state (u‾0,αβu‾0,γδu‾0) as t→∞, where u¯0=1|Ω|∫Ωu0dx and u0 is the initial data of u. The result extends some existing conclusions on two fronts: we obtain the boundedness with D1≠D2 in the higher dimensional case and expand the range of θ due to the repulsion effect.