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

$\begin{cases}\tag{*}n_t+u·\nabla n = \nabla ·(d(c)\nabla n)-\nabla ·(χ (c) n\nabla c)+a n-bn^2, &x∈ Ω, ~~t>0, \\ c_t+u·\nabla c = Δ c+ n-c,&x∈ Ω, ~~t>0, \\ u_t+ u·\nabla u = Δ u-\nabla P+n\nabla φ,&x∈ Ω, ~~t>0, \\\nabla · u = 0& x∈ Ω, \ t>0, \end{cases}$

in a bounded smooth domain $Ω\subset \mathbb{R}^2$ with homogeneous Neumann boundary conditions, where $a≥0$ and $b>0$ are constants, and the functions $d(c)$ and $χ(c)$ satisfy the following assumptions:

● $(d(c), χ (c))∈ [C^2([0, ∞))]^2$ with $d(c), χ(c)>0$ for all $c≥0$, $d'(c)<0$ and $\lim\limits_{c\to∞}d(c) = 0$.

● $\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$ and $\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$ exist.

The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition $\lim\limits_{c\to∞}d(c) = 0$. In this paper, we will use function $d(c)$ 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 $(n, c, u)$ will converge to the constant state $(\frac{a}{b}, \frac{a}{b}, 0)$ if $b>\frac{K_0}{16}$ with $K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$.