Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

In this paper, we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domain $??\mathbb{R}^n$ $(n ≥1)$, where $b ≥0$, $γ ≥1$, $a_i ≥1$, $µ$, $b_i >0$ $(i =1,2)$, $D$, $S∈ C^2(0,∞))$ fulfilling $D(s) ≥ a_0(s+1)^{?α}$, $0 ≤ S(s) ≤ b_0(s+1)^β$ for all $s ≥ 0,$ where $a_0,b_0 > 0$ and $α,β ∈ \mathbb{R}$ are constants. The purpose of this paper is to prove that if $b ≥ 0$ and $µ > 0$ sufficiently large, the globally bounded solution $(u,v,w)$ with nonnegative initial data $(u_0,v_0,w_0)$ satisfies $$\Big\| u(·,t)? \Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\|_{L^∞(?)}+\Big\| v(·,t)?\frac{b_1b_2}{a_1a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(?)} +\Big\| w(·,t)?\frac{b_2}{a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(?)}→0$$ as $t→∞$.