# American Institute of Mathematical Sciences

July  2018, 38(7): 3595-3616. doi: 10.3934/dcds.2018155

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

 Department of Mathematics, South China University of Technology, Guangzhou 510640, China

* Corresponding author: Hai-Yang Jin

Received  October 2017 Published  April 2018

Fund Project: The research of H.Y. Jin was supported by the NSF of China No. 11501218, and the Fundamental Research Funds for the Central Universities (No. 2017MS107).

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
 $K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$
.
Citation: Hai-Yang Jin. Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3595-3616. doi: 10.3934/dcds.2018155
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