Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-species chemotaxis system  ut = ∇ · (D1(u)∇u)−∇ · (uχ1(w)∇w) + μ1u(1− u− a1v), x ∈ Ω, t > 0, vt = ∇ · (D2(v)∇v)−∇ · (vχ2(w)∇w) + μ2v(1− a2u− v), x ∈ Ω, t > 0, wt = ∆w− (αu + βv)w, x ∈ Ω, t > 0, where Ω ⊂ Rn (n ≥ 1) is a bounded domain with smooth boundary ∂Ω; χi(i = 1, 2) are chemotactic functions satisfying χ′ i ≥ 0; the parameters μ1, μ2 > 0, a1, a2 > 0 and α, β > 0, the initial data (u0, v0) ∈ (C0(Ω))2 and w0 ∈ W1,∞(Ω) are non-negative. Based on the maximal Sobolev regularity, it is shown that this system possesses a unique global bounded classical solution provided that the logistic growth coefficients μ1 and μ2 are sufficiently large.

(1.3) Furthermore, we assume that the diffusion function D i (s) ∈ C 2 ([0, ∞)) (i = 1, 2) as well as where c D i > 0 and m ∈ R. In model (1.1), u = u(x, t) and v = v(x, t) represent densities of two populations, respectively, and w = w(x, t) denotes the concentration of oxygen.System (1.1) is used in mathematical biology as a model to study the mechanism of twospecies chemotaxis.The model describes the nonlinear diffusion of competing species which move towards the gradient of a substance called chemoattractant.Chemotaxis system plays a crucial role in cellular communication, for instance, in the governing of immune cells migration, in wound healing, in tumours growth or in the organization of embryonic cell positioning (see e.g.[3,5,38,40]).
The classical Keller-Segel model was proposed by Keller and Segel [14], and the existence of traveling wave solutions was proved under some conditions.Based on the Keller-Segel model, various chemotaxis models have attracted many authors to explore their mathematical properties, such as the boundedness, the stabilization of solutions and the blow-up of solutions [4, 6, 8, 12, 17, 18, 21, 23, 24, 27, 34-37, 39, 41].
A typical chemotaxis process is considered where the signal is degraded, but not produced by the cells.More precisely, the following oxygen consumption model is studied where u and v represent the density of the bacteria and the concentration of oxygen, respectively.D(u) denotes the diffusion function and f (u) is the logistic source.The analysis of this model has attracted many interests and many results are presented.For instance, in the absence of the logistic source (i.e.f (u) ≡ 0), when D(u) = 1, the global bounded solutions have been shown by Tao [20] under the condition of v 0 L ∞ (Ω) ≤ 1 6(n+1)χ .For arbitrarily large initial data, in three-dimensional case, the global bounded weak solutions and smoothness in Ω × (T, +∞) are proved with some T > 0 by Tao and Winkler [22].Moreover, when D(u) satisfies (1.4), Wang et al. prove that system (1.5) possesses a unique global bounded classical solution if m > 1 2 in the case n = 1 or m > 2 − 2 n in the case n ≥ 2 [32], the domain can be extended to m > 2 − 6 n+4 in the case n ≥ 3, but the solutions maybe unbounded in [31].Furthermore, the global bounded solutions are proved [9,33] provided that m > 2 − n+2 2n , which improves the results in [31,32].Recently, the diffusivity D(u) exponential decay as u → ∞ is studied in [16,26].
If the logistic source f (u) = au − µu γ with γ > 1 and D(u) = δ in system (1.5), the global bounded solution is studied if in [2].Similarly, Lankeit and Wang [15] prove this system has global bounded solutions if , where c 1 (n) and c 2 (n) are constants about n.The chemotaxis-consumption model (1.5) with nonlinear diffusion function and nontrivial source terms has also already been considered in [28,30].
To better discuss model (1.1), we need to mention the following two species chemotaxis(-Navier)-Stokes system with Lotka-Volterra competitive kinetics [25] which describes the evolution of two competing species that reacts on a chemoattractant in the environment of fulling the fluid.Here u, v and w are represented as model (1.1), and V denotes the velocity field of the fluid belonging to an incompressible Navier-Stokes equation with pressure P.Moreover, φ is a potential function, and κ is a constant concerning the strength of nonlinear fluid convection.Boundedness and asymptotic behavior of model (1.6) are researched in the case two-dimension and three-dimension [7,11,13].When the fluid is stationary or the effect of fluid is absent, i.e.V ≡ 0, model (1.6) is ascribed to the fundamental chemotaxis model (1.1).Motivated by the arguments in [19,29,30,37,41], in this paper, we extend their method and then obtain global boundedness of solution of model (1.1).Our main results are as follows.
) has global bounded solutions in [29], but which is independent of µ 1 and µ 2 .Theorem 1.1 gives a qualitative result, namely, if µ i (i = 1, 2) are sufficiently large, model (1.1) has global bounded solutions, which improves above results in some sense.
The rest of this paper is organized as follows.In the next section, we show the local existence of a solution to model (1.1) and give some preliminary inequalities those are important for our proofs.In Section 3, we will give the complete proof of Theorem 1.1.

Preliminaries
In order to prove our result, we first give one result concerning local-in-time existence of a classical solution to system (1.1).
The following characteristic of the solution of the third equation in model (1.1) plays an essential role in the later proof.Lemma 2.2.Let (u, v, w) be the solution of model (1.1), then we have for all t ∈ (0, T max ).
Proof.According to the third equation of model (1.1), and the non-negative u, v, w and α, β > 0, we claim result (2.4) upon an application of the maximum principle.

Global boundedness
In this section, global boundedness of solutions is proved to model (1.1).Firstly, to prove Theorem 1.1, we make an estimate for (u, v, w, ∆w) when s 0 ∈ (0, T max ) and s Next, we prove boundedness in t ∈ (s 0 , T max ).