Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source

Abstract

This paper studies the quasilinear attraction–repulsion chemotaxis system with a logistic source ut=∇⋅(D(u)∇u)−∇⋅(Φ(u)∇v)+∇⋅(Ψ(u)∇w)+f(u), τ1vt=Δv+αu−βv, τ2wt=Δw+γu−δw, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂RN (N≥1), where τ1,τ2∈{0,1}, D,Φ,Ψ∈C2([0,+∞)) nonnegative with D(s)≥(s+1)p for s≥0, Φ(s)≤χsq, ξsr≤Ψ(s)≤ζsr for s≥s0>0, f(s)≤μs(1−sk) for s>0, f(0)≥0. In a previous paper of the authors (Tian et al., 2016), the criteria for global boundedness of solutions were established for the case of τ2=0, depending on the interaction among the multi-nonlinear mechanisms (diffusion, attraction, repulsion and source) in the model. This paper continuously determines the global boundedness conditions for the case of τ2=1. In particular, we obtain the large time behavior of the globally bounded solutions for the situation of D(s)=(s+1)p, Φ(s)=χsq, Ψ(s)=ξsr, f(s)=μs(1−s), s≥0 with p=2(q−1)=2(r−1)≥0, τ1,τ2∈{0,1}.