GLOBAL DYNAMICS IN A TWO-SPECIES CHEMOTAXIS-COMPETITION SYSTEM WITH TWO SIGNALS

. In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type 0 , with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain Ω ⊂ R n , n ≥ 2, where χ i , µ i and a i ( i = 1 , 2) are positive constants. It is shown that for any positive parameters χ i , µ i , a i ( i = 1 , 2) and any suitably regular initial data ( u 0 ,w 0 ), this system possesses a global bounded classical solution provided that χ i µ i are small. Moreover, when a 1 ,a 2 ∈ (0 , 1) and the parameters µ 1 and µ 2 are suﬃciently large, it is proved that the global solution ( u,v,w,z ) of this system exponentially approaches to the steady state (cid:16) 1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 2 , 1 − a 1 − a 1 2 1 − a 1 − a 1 a 2 (cid:17) in the norm of L ∞ (Ω) as t → ∞ ; If a 1 ≥ 1 > a 2 > 0 and µ 2 is suﬃciently large, the solution of the system con- verges to the constant stationary solution (0 , 1 , 1 , 0) as time tends to inﬁnity, and the convergence rates can be

Recently, the two-species and two-stimuli chemotaxis system was studied by Tao and Winkler [35], they studied the global boundedness and finitetime blow-up for (5) when χ 1 , χ 2 ∈ {−1, 1}. Especially, when n = 2, χ 1 = χ 2 = 1 (i.e. attraction-attraction case), and the initial data (u 0 , w 0 ∈ (C 0 (Ω)) 2 satisfies max Ω u 0 dx, where C GN > 0 is a constant, system (5) possesses a unique global bounded classical solution. Recently, Zheng [48] generalized the results of [35] to the quasilinear cases. Later, in the case of n = 2, Zheng and Mu [49] removed the condition of small initial masses under the Lotka-Volterra-type competition. Recently, Zheng et al. [50] studied the persistence property of global bounded solutions for a fully parabolic two-species chemotaxis system with two signals. Moreover, Zheng et.al. [51] studied the boundedness and stabilization of global classical solutions for (5) with the Lotka-Volterra-type weak competition (i.e. a 1 , a 2 ∈ (0, 1)), however, the large time behavior of global solutions is still open under the strong competition (i.e. a 1 ≥ 1 > a 2 > 0). In this paper, we consider the two-species and two-stimuli chemotaxis system x ∈ Ω, t > 0, where Ω ⊂ R n (n ≥ 2) is a bounded domain with smooth boundary ∂Ω, ∂ ∂ν represents differentiation with respect to the outward normal on ∂Ω, and the parameters χ i , µ i and a i (i=1,2) are positive. Here u and w denote the density of two species, v and z stand for the concentration of the chemical substance.
However, there are only few results for the multi-species and multi-stimuli chemotaxis model. In this paper, we will show that for any positive parameters χ i , µ i , a i (i = 1, 2) and any suitably regular initial data (u 0 , w 0 ), this system possesses a global bounded classical solution provided that χi µi are small. Moreover, when a 1 , a 2 ∈ (0, 1) and the parameters µ 1 and µ 2 are sufficiently large, it is proved that the global solution (u, v, w, z) of this system exponentially approaches to the steady state in the norm of L ∞ (Ω) as t → ∞; while a 1 ≥ 1 > a 2 > 0 and µ 2 is sufficiently large, the solution of the system converges to the constant stationary solution (0, 1, 1, 0) as time tends to infinity, and the convergence rates can be calculated accurately.

XINYU TU, CHUNLAI MU, PAN ZHENG AND KE LIN
Under these assumptions, our results in this paper are stated as follows. Firstly, we give the global boundedness of solutions to (7) for all positive parameters χ i , µ i and a i , i = 1, 2.
Theorem 1.1. Let Ω ⊂ R n (n ≥ 2) be a smoothly bounded domain, and let χ i > 0, µ i > 0 and a i > 0, i = 1, 2. Suppose that χ 1 , χ 2 , µ 1 and µ 2 satisfy Then for any (u 0 , w 0 ) fulfilling (8) with some q > n, and for any a 1 ≥ 0 and a 2 ≥ 0, problem (7) possesses a unique global classical solution (u, v, w, z), which is uniformly bounded in the sense that And the solutions u, v, w, z are the Hölder continuous functions, i.e. there exist σ ∈ (0, 1) and Next, we shall discuss the asymptotic behavior of global solutions to (7) in the weak competition case a 1 , a 2 ∈ (0, 1).

Remark 1.
In the previous reference [51], the boundedness and stabilization of global solutions to (7) were only derived under the case a 1 , a 2 ∈ (0, 1). In this paper, we further study the conditions which assert global existence of classical bounded solutions for all a 1 , a 2 ∈ (0, ∞), which covers the case that a 1 , a 2 ∈ [1, ∞). Moreover, the convergence rates can be calculated accurately under the cases a 1 , a 2 ∈ (0, 1) and a 1 ≥ 1 > a 2 > 0, respectively. Hence, the results of this paper improve the previous works.
Remark 2. In both Theorems 1.2 and 1.3, we only give the convergence rates of global solutions for (7) under the cases a 1 , a 2 ∈ (0, 1) and a 1 ≥ 1 > a 2 > 0, respectively. For the case a 2 ≥ 1 > a 1 > 0, it is easy to derive the convergence rates by the same method used in the proof of Theorem 1.3. Thus, we don't discuss this condition here. However, for the case a 1 , a 2 ∈ [1, ∞), there is still an open problem about stabilization of global bounded solutions.
The paper is organized as follows. In Section 2, we prove global existence and boundedness by extending a method in [23] by establishing the L p -estimate for u and w with p > n 2 . In Section 3, inspired by the method in [23,1], we construct some energy-type functionals to obtain the convergence properties of solutions to (7) in the cases a 1 , a 2 ∈ (0, 1) and a 1 ≥ 1 > a 2 > 0 respectively.

2.
Global existence and boundedness. In this section, we first prove the local existence of solutions to (7) by means of a straightforward fixed point argument and the strong maximum principle which are adapted in [49].
and Ω ⊂ R n (n ≥ 2) be a smoothly bounded domain. Then for any (u 0 , w 0 ) satisfying (8), there exist a maximal T max ∈ (0, ∞] and a unique nonnegative function (u, v, w, z) (7) classically. Moreover, In order to derive an estimate for (u, w) in L ∞ (Ω), we next estimate the bounds for u L p (Ω) and w L p (Ω) with some p > n 2 . Lemma 2.2. Suppose that (8) and (9) are satisfied. Then for any solution (u, v, w, z) of (7) and p ∈ I 1 I 2 , there exists C(p) > 0 such that where Proof. Noting from the condition (9) to obtain that I 1 I 2 = ∅, we fix p ∈ I 1 I 2 .
Testing the first equation in (7) by u p−1 and using the second equation in (7), we see that By the positivity of u and v, we obtain that From the condition p ∈ I 1 I 2 , we infer that By the Hölder inequality, we can estimate Thus there exists ρ 1 > 0 satisfying This yields Similarly, we can obtain the L p -estimate for w w(·, t) L p ≤ min where ρ 2 > 0, This readily yields (16).
Next by using Lemma 2.2 and the semigroup estimates, we can obtain the following lemma.  (8) and (9). Then for any solution (u, v, w, z) of (7), there exists C > 0 such that for all t ≥ 0. Furthermore, there exist some K > 0 and σ ∈ (0, 1) satisfying for all t ≥ 1.
As for the φ 3 , we note that thus by the maximal principle, we can obtain that there exists C 6 > 0 such that Since t ∈ (0, T ) is arbitrary, C 1 , ..., C 5 are independent of t, then we can infer from (29)-(32) that there exists C 7 > 0 such that since C 7 > 0 is independent of T , and by 1 − p rσ > 0, we can obtain that Λ(T ) ≤ C 8 , for all T ∈ (0, T max ) (34) with some C 8 > 0. Thus we obtain T max = ∞ in view of Lemma 2.1. Similarly, we can verify the L ∞ -estimate for w. Next by (21) and (22), we can obtain v W 1,q (Ω) ≤ C 9 , z W 1,q (Ω) ≤ C 9 f or all t ∈ (0, T max ) with some C 9 > 0. Furthermore, by the known regularity argument (see Proposition 2.3 in [3]), we can find some K > 0 and σ ∈ (0, 1) satisfying Proof of Theorem 1.1. The assert of Theorem 1.1 is an immediate consequence of Lemma 2.3 .
3. Asymptotic behavior. In order to derive the asymptotic behavior of the solutions to (7), it is essential to assume that the solutions of (7) satisfy the conditions in Theorem 1.1, and next we will recall the following important lemmas for the proofs of Theorems 1.2 and 1.3.
Assume that there exists some positive constant M * such that Proof. This lemma is a straightforward result from lemma 4.6 in [8].
Next, we generalize Lemma 3.3 in [23] to our model and obtain the following lemma.
Lemma 3.2. Let (u, v, w, z) be a solution to (7). Assume that g : [0, ∞) → R is a decreasing function satisfies that for all t > 0, then there exists > 0 such that for all t > 1, m ∈ N + .

GLOBAL DYNAMICS IN A CHEMOTAXIS SYSTEM 3627
Then applying the L p − L q estimate and (49) to establish that with ρ 8 > 0, ρ 9 > 0. Therefore a combination of (46), (47) and (50), we can obtain that Similarly, the estimate for w − w * L ∞ (Ω) can be obtained. Thus there exists C 3 such that and by using the maximal principle to which concludes the proof of this lemma.
3.1. The case of a 1 , a 2 ∈ (0, 1). In this section, we turn our attention to the asymptotic behaviour of solutions in (7) when a 1 , a 2 ∈ (0, 1) , the parameter χ i , µ i , i = 1, 2 satisfy (11) and (12), with (u * , v * , w * , z * ) as given by (13), the solution of (7) converges to the steady state exponentially. The cornerstone of our approach is based on the following energy inequality for all t > 0, where δ is a positive constant. Inspired by lemma 3.4 in [23], we define H 1 (u, v, w, z) by for all t > 0.
Lemma 3.4. Assume that (10)-(12) are satisfied. Let (u, v, w, z) be a global bounded classical solution of (7), which is under the same assumptions in Theorem 1.2. Then Proof. A combination of Lemma 3.1 and Lemma 3.3 implies this lemma.
Next, in order to establish the convergence rates for the solution of (7), in view of Lemma 3.3, we should obtain the L 2 -convergence rate for the solution.
Proof. According to L'Hôpital's rule, by Lemma 3.4 and a similar argument in the proof of Lemma 3.7 in [1], there exists some t 0 > 0 such that which implies with C 4 > 0. Finally, by (77) and (79), we derive Proof. A combination of Lemma 3.1 and Lemma 3.6 implies that Invoking the standard L p − L q estimates for the Neumann heat semigroup (see [15]) applied the second equation in (7), we establish v(·, t) − 1 L ∞ (Ω) → 0 as t → ∞, which completes the proof of this lemma.
Next we shall establish the the convergence rates for the solution of (7) when a 1 ≥ 1 > a 2 > 0 by using the similar method above, we should obtain the L 2convergence rate for the solution.
Proof of Theorem 1.3. A combination of Lemma 3.2, Lemma 3.8 and Lemma 3.9 yields the results in Theorem 1.3., Thus we complete our proof.