# 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
##### References:

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##### References:
 [1] Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 [2] Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 269-278. doi: 10.3934/dcdss.2020015 [3] Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034 [4] Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328 [5] Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019211 [6] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [7] Martin Burger, Marco Di Francesco. Large time behavior of nonlocal aggregation models with nonlinear diffusion. Networks & Heterogeneous Media, 2008, 3 (4) : 749-785. doi: 10.3934/nhm.2008.3.749 [8] Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581 [9] Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013 [10] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019229 [11] Youshan Tao, Lihe Wang, Zhi-An Wang. Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 821-845. doi: 10.3934/dcdsb.2013.18.821 [12] Youshan Tao, Michael Winkler. A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2047-2067. doi: 10.3934/cpaa.2019092 [13] Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087 [14] Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069 [15] Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737 [16] Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure & Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959 [17] Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 [18] Jiashan Zheng. Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 627-643. doi: 10.3934/dcds.2017026 [19] Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463 [20] Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481

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