GLOBAL BOUNDEDNESS FOR A CHEMOTAXIS-COMPETITION SYSTEM WITH SIGNAL DEPENDENT SENSITIVITY AND LOOP

. In this work, the fully parabolic chemotaxis-competition system with loop 2 ) is considered under the homogeneous Neumann boundary condition, where x ∈ Ω ,t > 0, Ω ⊂ R n ( n ≤ 3) is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters µ 1 ,µ 2 are suﬃciently large, then the system possesses a unique and global classical solution for n ≤ 3. Speciﬁcally, when n = 2, the global boundedness can be attained without any constraints on µ 1 ,µ

When χ ij (i, j = 1, 2) are constants, h 1 = α 1 u 1 + β 1 u 2 , h 2 = α 2 u 1 + β 2 u 2 . Without respect to the kinetic terms, Espejo et al. derived the simultaneous blow-up phenomenon in [4] for the parabolic-elliptic case of (1) in the whole space R 2 . Considering the Lotka-Volterra-type competition, whether the parabolic-elliptic case or the fully parabolic case of (1), the global dynamics of solutions were detected, it was found that the solution of (1) is globally bounded without any requirement on the size of the parameters for the fully parabolic case in the lower dimensions n ≤ 2 [14], while the largeness of parameters µ 1 , µ 2 is needed to guarantee the global solvability of (1) for n = 3 [15], and the global solution of this system exponentially approaches to a steady state for all n ≥ 1 [14], specifically, the system was shown to exhibit the large population densities phenomenon in [16], that is, the solution exhibits unbounded peculiarity for the proper choice of initial data. As for the parabolic elliptic case, in [13], the global boundedness result were established for n ≥ 2 under the condition that χ11 µ1 , χ12 µ1 , χ21 µ2 and χ22 µ2 are suitably small, moreover, the large time behavior of solution was derived.
In summary, for the two-species and two-stimuli chemotaxis system, most of the results are focusing on the case that the chemotactic sensitivity functions are constants and the signal production is linear. Therefore, the objective in the present study is to investigate the global boundedness of solutions for (1) when χ ij , (i, j = 1, 2), h 1 , h 2 are general functions. Our work is motivated by the method in [17], but in contrast, the existence of the chemical signalling loop in our model makes the computations and analysis fairly subtle.
Remark 1. It is obvious that there exist functions χ ij , h i (i, j = 1, 2) which satisfy (H1)-(H4), such as, we can choose the standard chemotactic sensitivity functions χ ij (s) = c0 (1+cs) 2 with c 0 , c > 0, and choose h i = c 1 u 1 + c 2 u 2 with c 1 , c 2 > 0. In this paper, we deal with the quasilinear chemotaxis-competition system with loop. First, we give the local existence and some properties to prepare for the later work. Next, under the condition that µ 1 , µ 2 are sufficiently large, we establish the global boundedness result when n ≤ 3. At last, for the case n = 2, we obtain the boundedness result without any requirement on the size of µ 1 , µ 2 .
2. Preliminary. As a preliminary, we first give the local existence and some important estimates of solutions for (1).
Proof. The local existence of classical solution to (1) can be shown by using wellestablished methods for chemotaxis problems in [19]. And the relation (8)-(11) can be directly derived by a similar method in [15].
Next, we recall the following lemma (see Lemma 3.4 in [9] or Lemma 2.3 in [1]), which is significant for our latter proof.
loc ([0, T )), y(t) be a nonnegative absolutely continuous function on [0, T ). Assume that there exist a > 0, b > 0 such that and y (t) + ay(t) ≤ f (t) for almost all t ∈ (0, T ), then where τ = min 1, T 2 . Based on Lemma 2.1 and Lemma 2.2, we can now derive some basic properties of v 1 , v 2 . 3,4). Assume that h 1 , h 2 satisfy (H3)-(H4). Then there exist M 1 , M 2 , N 1 , N 2 > 0 such that the solution of (1) satisfies Proof. Multiplying the third equation in (1) by −∆v 1 and integrating the result equation over Ω, we have 1 2 for all t ∈ (0, T max ). By denoting Lemma 2.1 and the definition of f 1 entail that with some c 1 > 0, then it follows from Lemma 2.2 that with some c 2 > 0. Whence by an integration of (15) over (t, t + τ ) we find along with the nonnegativity of y 1 , we conclude (13). The other inequalities in (12) and (13) can be obtained by the similar method.
To improve the condition that warrants the global boundedness of solution for (1) when n = 2, we recall the following generalization of Gagliardo-Nirenberg inequality which is given in Lemma A.5 of [10].

Lemma 2.4.
Let Ω ⊂ R 2 be a smooth and bounded domain. Then for all ϕ ∈ W 1,2 (Ω), one can find C > 0 such that for any > 0 there exists C > 0 with the property that The following lemma plays an important role in the proof of Corollary 1, and the proof is similar to Lemma 2.5 in [14].
be the classical solution of (1). Assume that χ ij , h i (i = 1, 2, j = 1, 2, 3, 4) satisfy (H1)-(H4). Let p ≥ 1 be such that n 2 < p ≤ n and sup Then T max = ∞, and 3. The global boundedness of solutions. In this section, we first prove the global boundedness of solutions for n ≤ 3 under the condition that µ 1 , µ 2 are sufficiently large; next, we remove the requirement on the largeness of parameters µ 1 , µ 2 when n = 2.
3.1. Proof of Theorem 1.1. To prepare our analysis, we establish several differential inequalities in the following two lemmas..
for all t ∈ (0, T max ). And Proof. Multiplying the first equation in (1) by u 1 and integrating by parts over Ω, in light of (H1)-(H2) and the Young inequality, we can see that for all t ∈ (0, T max ), which directly yields (21). Similarly, we can derive (22). To derive (23), in light of the third equation in (1) and the identity 2∇v for all t ∈ (0, T max ), here, in view of (H4) and the relation |∆v and Consequently, plugging (26) and (27) into (25), we arrive at (23). In addition, (24) can be established in a same manner.
On the basis of the L 2 bound for u 1 , u 2 , we can now establish the L p estimates for u 1 , u 2 .
In the above section, the global boundedness of solution is derived under the condition that µ 1 , µ 2 are sufficiently large. In this section, motivated by the method in [2,3,17], we remove the restriction on µ 1 , µ 2 when n = 2. The key point of the proof is to establish the boundedness of Ω |∇v 1 | 4 dx and Ω |∇v 1 | 4 dx, to this end, we first establish the boundedness of t+τ t Ω |∇v i | 4 dx, i = 1, 2 in the following lemma.
In the second step, we derive the boundedness of Ω |u i ln u i |dx, i = 1, 2.
Proof of Corollary 1. A combination of Lemma 2.5 and Lemma 3.7 directly yields Corollary 1.