Asymptotic behavior of a quasilinear parabolic–elliptic–elliptic chemotaxis system with logistic source

In this paper, we study the following quasilinear parabolic–elliptic–elliptic chemotaxis system with indirect signal production and logistic source { u_t=∇· (D(u)∇ u) -∇· (S(u)∇ v)+μ (u-u^γ ) , x∈Ω , t>0, 0=Δ v- v+ w, x∈Ω , t>0, 0=Δ w- w+ u, x∈Ω , t>0 . under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ℝ^n(n≥ 1) , where μ>0, γ >1 , and D, S∈ C^2 ([0,∞ )) fulfilling D(s)≥ a_0(s+1)^α, |S(s)|≤ b_0s(s+1)^β -1 for all s≥ 0 with a_0, b_0>0 and α ,β∈ℝ are constants. The purpose of this paper is to prove that if β≤γ -1 , the nonnegative classical solution ( u , v , w ) is global in time and bounded. In addition, if μ >0 is sufficiently large, the globally bounded solution ( u , v , w ) satisfies ‖ u(· ,t)-1‖ _L^∞ (Ω )+‖ v(· ,t)-1‖ _L^∞ (Ω )+‖ w(· ,t)-1‖ _L^∞ (Ω )→ 0 as t→∞ .