GLOBAL CLASSICAL SOLUTION AND STABILITY TO A COUPLED CHEMOTAXIS-FLUID MODEL WITH LOGISTIC SOURCE

. In this paper, we deal with a coupled chemotaxis-ﬂuid model with logistic source γn − µn 2 . We prove the existence of global classical solution for the chemotaxis-Stokes system in a bounded domain Ω ⊂ R 3 for any large initial data. On the basis of this, we further prove that if γ > 0, the zero solution is not stable; if γ = 0, the zero solution is globally asymptotically stable; and if 0 < γ < 16 µ 2 , the nontrivial steady state (cid:16) γµ , γµ , 0 (cid:17) is globally asymptotically stable.

1. Introduction. In this paper, we consider a chemotaxis-Stokes model of characterizing a signal production mechanism, n t + u · ∇n = ∆n − ∇ · (n(1 + n) −α ∇c) + γn − µn 2 , in Q, c t + u · ∇c = ∆c − c + n, in Q, u t = ∆u − ∇π + n∇ϕ, in Q, ∇ · u = 0, in Q, ∂n ∂ν ∂Ω = ∂c ∂ν ∂Ω = 0, u| ∂Ω = 0, n(x, 0) = n 0 (x), c(x, 0) = c 0 (x), u(x, 0) = u 0 (x), x ∈ Ω, (1.1) where Q = Ω×R + , Ω ⊂ R 3 is a bounded domain with smooth boundary, n, c denote the bacterial or cell density, the chemical concentration respectively, J = n(1 + n) −α · ∇c with α > 0 is the chemotactic flux, γ ≥ 0, µ > 0 are parameters reflecting proliferation and death of cells in a logistic law, −c + n is the kinetics/source term, n represents the spontaneous production of the attractant and is proportional to the number of amoebae n, while −c represents decay of attractant activity, u = (u 1 , u 2 , u 3 ), π are the fluid velocity and the associated pressure. We see that this model is a coupled system of the chemotaxis model and the incompressible fluid model. The derivation of the chemotaxis model begins with a cell equation. Applying conservation law to the cell density n leads to the following equation where R n is the proliferation or death of cells, which obeys a logistic growth law. As for the flux J f lux , in addition to random motion, cells migrate toward higher concentrations of diffusible chemoattractant, so we take J f lux = J random + J chemotaxis . Taking into account the above factors, one obtain the following equation for the cell motion, ∂n ∂t = ∇ · (D n ∇n) dispersion − α∇ · (g(n)τ (c)∇c) where D n is the intrinsic dispersion coefficient, α is the signal detection coefficient, µ is the proliferation rate of the cells, γ is the maximum sustainable cell. c is the signal concentration, and τ is the mechanism of tactic responses in cell populations, such as chemotaxis. χ(c) = ατ (c) is the chemotactic sensitivity. Thus the taxis is positive or negative according to whether ατ (c) is positive or negative. (Here the word 'taxis' stems from the Greek 'taxis', meaning to arrange. In a living system, the individuals will always sense the environment where they reside and respond to the external stimulus, the response will lead to opposite behavior: movement toward or away from the external stimulus, and such a response is called taxis behavior [18].) For the positive taxis case, the equation includes two opposite phenomena, one is diffusion of cells due to their random walks, another is their aggregations toward higher concentrations of the chemical, which may result in blow up. In the second equation, the coupling between the cell equation and the chemical equation is a positive feedback, in which, the term +n implies that the more cells are aggregated, the more signals are produced to attract other cells, and −c represents the decay of attractant activity. While if one considers the chemotaxis phenomena in the fluid, then the velocity of the fluid must be considered, and the chemotaxis model and the fluid coupled together through transports and gravitational force ∇ϕ. For the derivation of a chemotaxis-fluid model, we also refer to a recent work [1] by Bellomo et al.
If we omit the fluid velocity u, then the system is reduced to the classical chemotaxis model, which was first introduced by Keller and Segel in 1970 [10]. In the past several decades, the study on the Keller-Segel models have attracted a great deal of attention, and the study on the chemotaxis models mainly concentrated into two types, one is the parabolic-elliptic Keller-Segel model, and another is the parabolic-parabolic model. In particular, a large amount of work has been devoted to determining whether the solutions are global in time or blow up in finite time. For example, for the Keller-Segel model of the form It is easy to see that this is a mass conservation model.  [2,7,16]. Similar conclusions are true for the parabolic-parabolic Keller-Segel model (τ = 1), but the available results were derived much latter and are less complete. While, the logistic-type growth restriction has been detected to prevent chemotactic collapse. In fact, it has been shown that for the following system, all solutions exist globally for the two dimensional space [17], and all solutions exist globally for the three dimensional space if γ is large [27].
The coupled chemotaxis-fluid model was first introduced by Tuval et. al [23] in 2005, which describes the dynamics of bacterial swimming and oxygen transport near contact lines, and the model is a coupled system of the chemotaxis model and the viscous incompressible Navier-Stokes equations. While if the fluid motion is slow, the Navier-Stokes equations can be replaced by the Stokes equations [15], namely, the nonlinear convective term u · ∇u can be abandoned. Recently, there has been an increasing interest in the study of this subject. Specially, the global existence of solutions of chemotaxis-fluid models have attracted considerable attention. Firstly, for the following problem with the chemical consumption In 2011, Liu and Lorz [13] considered the global solvability of weak solutions for chemotaxis(-Navier)-Stokes system, obtained global existence of weak solutions for the chemotaxis-Navier-Stokes system in two space dimensions and global existence of weak solutions of the chemotaxis-Stokes system with nonlinear diffusion in three space dimensions under some conditions. In 2012, Winkler [26] considered this problem in a bounded domain with zero-flux boundary for n, c and no-slip boundary for u, and obtained global existence of classical solution (with any τ ∈ R) in two space dimensions and global existence of weak solution (with τ = 0) in three space dimensions. Recently, Lankeit [12] studied the weak solution (with τ = 1) in three space dimensions for the system (1.3) with a logistic growth term, and the author also prove that after some waiting time the weak solution becomes smooth and finally converges to a steady state. When a signal production mechanism and a logistic reaction term appear in the chemotaxis-fluid model, it leads to the system (1.1). For the case α = 0, Espejo and Suzuki [4], Tao and Winkler [22] showed the global existence of weak solution for the two-dimensional case, Tao and Winkler [21] obtained the global classical solution under the condition µ > 23 for the three-dimensional case, however, the global existence or blow-up property for small µ > 0 remains open. While it is known that the cells will stop aggregating when a certain size of the aggregation is reached, so a modified chemotaxis model with volume-filling effect is proposed, that is the chemical term −∇ · (n∇c) can be replaced by −∇ · (n(1 + n) −α ∇c). For the problem (1.1) without the logistic growth term, that is the case γ = µ = 0, Wang and Xiang [25] studied the global existence of solutions in two dimensional space for any α > 0. When the logistic growth term is considered, Liu, Wang [14] and Zheng [29] studied the system (1.1) with the diffusion term ∆n being replaced by ∆n m , and proved the global existence of weak solutions for the case α > 6 5 − m. Clearly, for the equations (1.1), the global existence is only proved when α > 1 5 , and for the case α < 1 5 , it is still open. In this paper, we study the problem (1.1), we aim to prove the existence of global bounded solution for any α > 0. The main difficulty lies in improving regularity of the cell density n. For this purpose, we use an iterative technique to improve the regularity of ∆c and n alternately. On the basis of this, we use the Neumann heat semigroup to further raise the regularity of the solution, and the global bounded solution finally is obtained. On the other hand, we are also concerned with the stability of steady states. More precisely, we prove that if γ = 0, the zero solution is globally asymptotically stable; if γ > 0, the zero solution is not stable; and specially, if 0 < γ < 16µ 2 , the nontrivial steady state ( γ µ , γ µ , 0) is globally asymptotically stable.
Throughout this paper, we assume that the boundary conditions, the initial data and ϕ satisfy We give the main theorems of this paper as follows.

2.
Preliminaries. We first give some notations, which will be used throughout this paper.
Before going further, we list some inequalities, which will be used throughout this paper.
By Gagliardo-Nirenberg interpolation inequality, we have [20], we have the following lemma.
is absolutely continuous, and satisfies Then for any t > t 0 , we have Proof. Firstly, omitting the term f 1+σ , and by a direct calculation, we see that Next, for any t ∈ (τ, T ), integrating the inequality (2.2) from t − τ to t, and using mean value theorem of integrals, we see that there exists t 0 ∈ (t − τ, t), such that going back to (2.2), we see that which implies the first inequality in (2.4).
and the proof is complete.
By using a fixed point method, the local solvability of system (1.1) can be proved by using a similar process of Lemma 2.1 in [26].
Lemma 2.4. Assume α > 0 and (1.4) holds. Then there exists T max ∈ (0, ∞] such that the problem (1.1) admits a classical solution (n, c, u, π) with n, c ≥ 0, and There is exact one of the following alternatives: either T max = ∞, or for all β ∈ 3 4 , 1 . 3. Energy estimates. To prove the global existence of classical solution, we give some energy estimates for the solution (n, c, u, π) of (1.1) under the assumptions (1.4). Throughout this section, we fix τ = min{1, Tmax 2 } ≤ 1. So, in what follows, all these constants C, M, C i are independent of τ . For reader's convenience, here, we give an explanation of why these constants are independent of τ . In fact, if τ = 1, by Lemma 2.2 and Lemma 2.3, we see that the constants can be fixed. While if τ < 1, it implies that T max < 2, so the following estimates of t t−τ · ds can be replaced by Tmax 0 · ds. We first have the following lemma. where M is independent of T max and τ .
Proof. In fact, by integrating the first equation of (1.1) over Ω, it is easy to see that By Hölder's inequality and Cauchy's inequality with ε, we get Then we have By a direct calculation, we obtain sup t∈(0,Tmax) Ω ndx ≤ n 0 L 1 + γ|Ω| µ .
Recalling that τ ≤ 1, integrating the equality (3.2) from t − τ to t, we obtain where C is a constant, which is independent of T max and τ .
Proof. Multiplying the third equation of (1.1) by u, and integrating it over Ω, we see that by Poincaré inequality and Young's inequality, we get Applying the Helmholtz projection operator P to both sides of the third equation of (1.1), we have u t − ∆u = P (n∇ϕ). (3.5) Multiplying this equation by −∆u, then integrating it over Ω, we get If T max ≥ 2, then τ = 1, and by virtue of Lemma 2.2 and Lemma 3.1, we obtain sup t∈(0,Tmax) If T max < 2, then τ < 1, and by a direct calculation, the above inequality still holds. Next, we estimate the term u t . Multiplying the third equation of (1.1) by u t , and integrating it over Ω, we obtain for any t ∈ (τ, T max ), integrating it from t − τ to t yields where C is a constant independent of T max , and τ .
Proof. Multiplying the second equation of (1.1) by c, and integrating it over Ω, we get and thus by Lemma 2.2, we obtain sup t∈(0,Tmax) Multiplying the second equation of (1.1) by ∆c, integrating it over Ω, and using Lemma 2.1 and Young's inequality, we get Combining ( Multiplying the second equation of (1.1) by c t , and integrating it over Ω, we conclude The proof is complete.
where C depends on r, α, Ω, c 0 , n 0 , and it is independent of T max and τ .
Proof. Multiplying the first equation of (1.1) by ∆n, and integrating it over Ω, we get For I 1 , by virtue of Lemma 2.1 and Young's inequality, we see that Moreover for I 2 , by Lemma 3.4, we have Summing up, we obtain ≤C( ∇n 2 L 2 + ∆c 2 H 1 ). By Lemma 3.4, and similarly to the proof of (3.19) and (3.20), we obtain sup t∈(0,Tmax)

CHUNHUA JIN
Multiplying the first equation of (1.1) by n t , integrating it over Ω, and using a similar method as above, we also obtain Summing up, and we finally obtain the desired estimate.
4. Global existence and stability. By Lemma 2.4, we see that to prove the global existence of classical solutions, we only need to prove the boundedness of the following norm with β ∈ ( 3 4 , 1). Proof of Theorem 1.1. We see that the first equation of (1.1) is equivalent to By Duhamel's principle, we see that the solution of (4.1) can be expressed as follows where {e t∆ } t≥0 is the heat semigroup on the domain Ω under Neumann boundary condition, for more properties of Neumann heat semigroup, please refer to [28]. Then for any 3 < p < 6, by using the regularity estimates in Lemma 3.2-Lemma 3.5, we have Noticing that 1 2 + 3 2 ( 1 2 − 1 p ) < 1 since p < 6, then we further obtain sup t∈(0,Tmax) which implies that there exists a constant M > 0 independent of T max , such that sup t∈(0,Tmax) ∇n L p ≤ M, ∀ 3 < p < 6 (4.2) since p 3(p−2) < 1. By Sobolev embedding theorem, n ∈ L ∞ (Ω × (0, T max )). For u, we see that By embedding theorem, u ∈ L ∞ (Ω × (0, T max )) since β > 3 4 . For c, clearly, we have c ∈ L ∞ (Ω × (0, T max )) since c ∈ L ∞ ((0, T max ); H 2 (Ω)). Similarly, for ∇c, we have ≤C.
where M is independent of T max . Recalling Lemma 2.4, we see that the solution of problem (1.1) exists globally.
Next, we show that if γ > 0, the zero solution is unstable; and if γ = 0, the zero solution is globally asymptotically stable.
Proof. Suppose to the contrary, then there exists t 0 > 0 such that n(·, t) L ∞ < γ 2µ for t > t 0 , and n(·, t 0 ) L ∞ > 0. Then integrating the first equation of (1.1) over Ω yields which implies that It is a contradiction.
where M 1 , M 2 are constants. It implies that Proof. Integrating the first equation of (1.1) over Ω yields Integrating the above equality from 0 to t, then (4.5) is proved. Note that which implies (4.6). Integrating the second equation of (1.1) over Ω yields then we further have Recalling (3.4), we see that Similarly to the proof above, we obtain (4.8).
Next, we use an idea of [9] to show the stability of the nontrivial steady state  In fact, multiplying the first equation of (4.9) by ω, and noting that ∇ · ω = 0 then we have as t → ∞.

Proof. Let
By the mean value theorem, we see that where ξ is between n and γ µ . By a direct calculation, we see that  Multiplying equation (4.10) by v and ∆v respectively, and integrating them over Ω, we obtain Note that v 2 H 2 is uniformly continuous, then we have lim