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 ut=∇·(D(u)∇u)-∇·(S(u)∇v)+μ(u-uγ),x∈Ω,t>0,0=Δv-v+w,x∈Ω,t>0,0=Δw-w+u,x∈Ω,t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (S(u)\nabla v)+\mu (u-u^\gamma ) ,&\qquad \quad x\in \Omega ,\,t>0,\\&0=\Delta v- v+ w,&\qquad \quad x\in \Omega ,\,t>0,\\&0=\Delta w- w+ u,&\qquad \quad x\in \Omega ,\,t>0 \end{aligned} \right. \end{aligned}$$\end{document}under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n≥1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Omega \subset \mathbb {R}^n(n\ge 1)$$\end{document}, where μ>0,γ>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu>0, \gamma >1$$\end{document}, and D,S∈C2([0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D, S\in C^2\,([0,\infty ))$$\end{document} fulfilling D(s)≥a0(s+1)α,|S(s)|≤b0s(s+1)β-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D(s)\ge a_0(s+1)^{\alpha },\, |S(s)|\le b_0s(s+1)^{\beta -1}$$\end{document} for all s≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\ge 0$$\end{document} with a0,b0>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_0, b_0>0$$\end{document} and α,β∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta \in \mathbb {R} $$\end{document} are constants. The purpose of this paper is to prove that if β≤γ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \le \gamma -1$$\end{document}, the nonnegative classical solution (u, v, w) is global in time and bounded. In addition, if μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >0 $$\end{document} 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)→0ast→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u(\cdot ,t)-1\Vert _{L^\infty (\Omega )}+\Vert v(\cdot ,t)-1\Vert _{L^\infty (\Omega )}+\Vert w(\cdot ,t)-1\Vert _{L^\infty (\Omega )} \rightarrow 0 \quad as \quad t\rightarrow \infty . \end{aligned}$$\end{document}