Global existence and decay for a chemotaxis-growth system with generalized volume-filling effect and sublinear secretion

This paper deals with a fully parabolic chemotaxis-growth system with generalized volume-filling effect and sublinear secretion ut=∇·(φ(u)∇u)-∇·(ψ(u)∇v)+ru-μu2,(x,t)∈Ω×(0,∞),vt=Δv-v+g(u),(x,t)∈Ω×(0,∞),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} u_t=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (\psi (u)\nabla v)+ru-\mu u^{2}, &{}\quad (x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta v-v+g(u), &{}\quad (x,t)\in \Omega \times (0,\infty ), \end{array} \right. \end{aligned}$$\end{document}under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^{2}$$\end{document}, where φ(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (u)$$\end{document} is a nonlinear diffusion function, ψ(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (u)$$\end{document} is a chemotactic sensitivity and g(u) is a production rate of the chemoattractant. Under some suitable assumptions on the nonlinearities φ(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (u)$$\end{document}, ψ(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (u)$$\end{document} and g(u), we study the global boundedness and decay of solutions for the system.