Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity \begin{document}$\begin{cases}\tag{*}n_t+u·\nabla n = \nabla ·(d(c)\nabla n)-\nabla ·(χ (c) n\nabla c)+a n-bn^2, x'>in a bounded smooth domain \begin{document}$Ω\subset \mathbb{R}^2$\end{document} with homogeneous Neumann boundary conditions, where \begin{document}$a≥0$\end{document} and \begin{document}$b>0$\end{document} are constants, and the functions \begin{document}$d(c)$\end{document} and \begin{document}$χ(c)$\end{document} satisfy the following assumptions: ● \begin{document}$(d(c), χ (c))∈ [C^2([0, ∞))]^2$\end{document} with \begin{document}$d(c), χ(c)>0$\end{document} for all \begin{document}$c≥0$\end{document} , \begin{document}$d'(c) and \begin{document}$\lim\limits_{c\to∞}d(c) = 0$\end{document} . ● \begin{document}$\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$\end{document} and \begin{document}$\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$\end{document} exist. The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition \begin{document}$\lim\limits_{c\to∞}d(c) = 0$\end{document} . In this paper, we will use function \begin{document}$d(c)$\end{document} as weight function and employ the method of energy estimate to establish the global existence of classical solutions of (*) with uniform-in-time bound. Furthermore, by constructing a Lyapunov functional, we show that the global classical solution \begin{document}$(n, c, u)$\end{document} will converge to the constant state \begin{document}$(\frac{a}{b}, \frac{a}{b}, 0)$\end{document} if \begin{document}$b>\frac{K_0}{16}$\end{document} with \begin{document}$K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$\end{document} .