Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion

Chemotaxis refers to the effect of chemical substances in the environment on the movement of species. This can lead to strict directional movement or partial orientation and partial tumbling movement. The movement to higher chemical concentrations is called positive chemotaxis, and the movement to lower chemical concentrations is called negative chemotaxis. Chemotaxis is an important means of cellular communication. After the pioneering work of Keller and Segel [1], a number of works concerning on the classical KellerSegel model and its variations are investigated. This paper is devoted to making development for the following two species chemotaxis system with nonlinear diffusion and consumption of chemoattractant ⎧ ⎨ ⎪⎪

Chemotaxis refers to the effect of chemical substances in the environment on the movement of species. This can lead to strict directional movement or partial orientation and partial tumbling movement. The movement to higher chemical concentrations is called positive chemotaxis, and the movement to lower chemical concentrations is called negative chemotaxis. Chemotaxis is an important means of cellular communication. After the pioneering work of Keller and Segel [1], a number of works concerning on the classical Keller-Segel model and its variations are investigated. This paper is devoted to making development for the following two species chemotaxis system with nonlinear diffusion and consumption of chemoattractant where Ω 3 ⊂ is a bounded domain with smooth boundary Ω ∂ , and m n , 1 > , μ μ , 0 1 2 > , a a , 0 1 2 > , α β , 0 > are positive constants and υ is the outward normal vector to Ω ∂ . The functions u u x t , ( ) = and v v x t , ( ) = denote, respectively, the unknown population density of two species, and w w x t , ( ) = represents the concentration of the chemoattractant. χ w i 1, 2 i ( ) ( ) = is the sensitivity function of aggregation induced by the concentration changes of chemoattractant, μ u u av 1 1 1 are both the proliferation and death of bacteria according to a generalized logistic law and αu βv w ( ) − + denotes the consumption of chemoattractant.
In order to better understand model (1), we recall some previous contributions in this direction. Consider the following chemotaxis model with consumption of chemoattractant where Ω N ⊂ is a bounded domain with smooth boundary Ω ∂ , χ 0 > is a parameter referred to as chemosensitivity. When D u 1 ( ) ≡ , in [2], Tao and Winkler proved that problem (2) possesses global bounded smooth solutions in the spatially two-dimensional setting, whereas in the three-dimensional counterpart, at least global weak solutions can be constructed, which eventually become smooth and bounded. When the nonlinear nonnegative function D u D u D 1 0 , it has been shown in [3] that system (2) admits a unique global classical solution that is uniformly bounded when m ≥ . Later, Zheng and Wang [4] improved this result to m 2 N If the reproduction and death of species themselves are taken into account, some logistic type sources will be added to the first equation of (2). For instance, when D u 1 ( ) ≡ , in the three-dimensional case, Zheng et al. proved that the system (2) with a logistic type source μu admits a unique global classical solution if the initial data of w are small in [5]. In arbitrary N-dimensional bounded smooth domain, Lankeit and Wang obtained the global bounded classical solutions of (2) for any large initial data in [6] when μ is appropriately large, and they also proved the existence of global weak solutions for any large μ. In a bounded domain [7] studied the issue of boundedness to solutions of (2) without any restriction on the space dimension, if where λ 0 is a positive constant which is corresponding to the maximal Sobolev regularity, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. Also, multi-species chemotaxis systems have been extensively studied by many authors. When the two species have effect on each other, the system involved Lotka-Volterra competitive kinetics has been proposed to describe the evolution of two competing species that react on a single chemoattractant. Here u v , , and w are represented as model (1), the chemotactic function χ w i 1, 2 i ( ) ( ) = is smooth. In [8], it is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution. Moreover, the authors there also proved the asymptotic stabilization of solution. When αu βv w ( ) − + in (3) is replaced by w αu βv − + + , this model has been extensively studied by many authors. In the two-dimensional case, Bai and Winkler [9] obtained global existence of solution to the system if χ w χ i i ( ) = are positive constants. Moreover, they also considered asymptotic behavior of solutions to the system: when a a , 0,1 when a a 1 0 In the three-dimensional case, Lin and Mu [10] obtained similar results if μ 1 and μ 1 are sufficiently large. When the two species have no effect on each other, the competitive kinetics terms μ u u av 1 1 1 where Ω is a bounded domain in N with smooth boundary Ω ∂ . Negreanu and Tello [11,12] proved global existence and asymptotic behavior of solutions to the above system when d 0 1 ≤ < . This result was later improved by Mizukami and Yokota, and they removed the restriction of d 0 1 ≤ < in [13]. Assuming that the random movement of species is nonlinearly enhanced at large densities, several works addressed a porous medium-type diffusion chemotaxis model (see, e.g. [14][15][16][17]). Inspired by the aforementioned works, in this paper, we consider the two species chemotaxis system (1) with nonlinear diffusion. The purpose of this paper is to obtain global existence and uniform boundedness of weak solution in a three-dimensional setting.
Throughout this paper, we assume that The chemotactic sensitivity function χ i 1, 2 i ( ) = satisfies the following conditions: Now, we state the main results of this paper as follows.
be a bounded domain with smooth boundary. Assume that (5) and (6) hold, and m n , 1 > , a a , 0 where C depends only on μ μ u v w , , , , The rest of this paper is organized as follows. In Section 2, we introduce the conception of the weak solution and summarize some basic definitions and useful lemmas. In Section 3, we shall first establish the existence of global classical solutions to the regularized problems and second show the convergence of the solution of regularized problems and thus obtain the proof of Theorem 1.

Some preliminaries
We first give some notations, which will be used throughout this paper.
Next, we introduce the definition of weak solutions.
for any T 0 > , and Before going further, we give some lemmas, which will be used later. We firstly list the following lemma, a proof of which can be found in [18] (see also [19]).
∞ is absolutely continuous, and satisfies , where M is a constant independent of τ.

Global existence and boundedness of weak solution
The degeneracy at u 0 = of system (1) results in the failure of the classical parabolic regularity theory. To overcome this difficulty, we shall first consider the following regularized version: We begin with the local existence of classical solutions to system (18), the proof of which is similar (refer to, e.g., [17,[22][23][24][25], for the details).
are some constants independent of ε.
Next, we give some estimates of u v w , , It is easy to see that τ 1 ≤ .
Proof. By comparison principle, it is easy to see that (22) holds. Integrating the first equation of (18) and using Young's inequality, we obtain We get by (26) and Hölder's inequality that By the same way, there exists C 0 2 > such that (24) holds. From the third equation of (18), we have (23), (24), and Minkowski's inequality, we have The proof is complete. □ Lemma 6. Suppose that (5) and (6) hold. Let u v w , , ε ε ε ( ) be the classical solution of (18). Then for any k 1, 2, 3, = …, where C k , C k ′, C k  , C k  ′ depend on k and all of them are independent of T ε max, , τ and ε.
Proof. Multiplying the first equation of (18) by u ε r for any r 0 > and integrating over Ω by parts, where α 1 2 = , C 3 depends on m k , , Ω, and w L (6) and (22), we infer from Young's inequality that in T 0, then, taking k 0 = in (33) and using (25), we infer from Lemma 1 that By the same way, we have Due to u v , 0 ≥ , and the third equation of (18), using Lemma 3, (22), we derive Arguing similarly as above, we see that The proof is complete. □ Lemma 7. Suppose that (5) and (6) hold. Let u v w , , ε ε ε ( ) be the classical solution of (18). Then where C is independent of T ε max, and ε.
Proof of Proposition 1. By Lemmas 4 and 7, for any ε 0 > , T ε max, = +∞, the system (18) has a global classical solution which is, furthermore, bounded. Namely, there exists C that is independent of ε T , ε max, such that