Global boundedness and large time behavior in a signal-dependent motility system with nonlinear signal consumption

In this paper, we deal with the following system with nonlinear signal consumption ut=Δγvu+ru-μuα,x∈Ω,t>0,vt=Δv-uβv,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}$$\begin{aligned} \left\{ \begin{array}{llll} {u_t} = \Delta \left( {\gamma \left( v \right) u} \right) + ru - \mu {u^\alpha },\quad &{}x\in \Omega ,\quad &{}t>0,\\ {v_t} = \Delta v - {u^\beta }v,\quad &{}x\in \Omega ,\quad &{}t>0,\\ \end{array} \right. \end{aligned}$$\end{document}under homogeneous Neumann boundary conditions in a smooth bounded domain Ω∈Rnn≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \in {\mathbb {R}^n} \left( {n \ge 2} \right) $$\end{document}. It shown that whenever r>0,μ>0,α>2,β>0andαβ>n+22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r> 0,\mu> 0,\alpha> 2, \beta> 0 \text { and } \frac{\alpha }{\beta } > \frac{{n + 2}}{2}$$\end{document}, then the original system will produce a global classical solution and the solution converges to equilibrium rμ1α-1,0ast→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( {{{\left( {\frac{r}{\mu }} \right) }^{\frac{1}{{\alpha - 1}}}},0} \right) \quad \text { as } t \rightarrow \infty . \end{aligned}$$\end{document}