Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

This paper deals with the two-species chemotaxis-competition system \begin{document}$\left\{ {\begin{array}{*{20}{l}}{{u_t} = {d_1}\Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u - {a_1}v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{v_t} = {d_2}\Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - {a_2}u - v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{w_t} = {d_3}\Delta w + h(u,v,w)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\end{array}} \right.$ \end{document} where \begin{document}$\Omega$\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with smooth boundary \begin{document}$\partial \Omega$\end{document} , \begin{document}$n\in \mathbb{N}$\end{document} ; \begin{document}$h$\end{document} , \begin{document}$\chi_i$\end{document} are functions satisfying some conditions. In the case that \begin{document}$\chi_i(w)=\chi_i$\end{document} , Bai–Winkler [ 1 ] proved asymptotic behavior of solutions to the above system under some conditions which roughly mean largeness of \begin{document}$\mu_1, \mu_2$\end{document} . The main purpose of this paper is to extend the previous method for obtaining asymptotic stability. As a result, the present paper improves the conditions assumed in [ 1 ] , i.e., the ranges of \begin{document}$\mu_1, \mu_2$\end{document} are extended.

1. Introduction. Nowadays, mathematics is useful in many things, for example, physics, chemistry, biology, computer, medical, architecture, and so on. Here we focus on biology. One of the famous and basic models in biology is the Lotka-Volterra competition system. On the other hand, many mathematicians study a chemotaxis system lately, which describes a part of the life cycle of cellular slime molds with chemotaxis. After the pioneering work of Keller-Segel [8], a number of variations of the chemotaxis system are proposed and investigated (see e.g., [2], [4] and [5]). Also, multi-species chemotaxis systems have been studied by e.g., [7] and [13]. In this paper we focus on a two-species chemotaxis system which describes a situation in which multi populations react on a single chemoattractant. Moreover, we assume that both populations reproduce themselves, and mutually compete with the other, according to the classical Lotka-Volterra kinetics, i.e., with coupling coefficients a 1 , a 2 > 0 in

MASAAKI MIZUKAMI
We consider the two-species chemotaxis system x ∈ Ω, t > 0, ∇u · ν = ∇v · ν = ∇w · ν = 0, x ∈ ∂Ω, t > 0, where Ω is a bounded domain in R n (n ∈ N) with smooth boundary ∂Ω and ν is the outward normal vector to ∂Ω. The initial data u 0 , v 0 and w 0 are assumed to be nonnegative functions. The unknown functions u(x, t) and v(x, t) represent the population densities of two species and w(x, t) shows the concentration of the substance at place x and time t.
These conditions are not natural because they are not symmetric. When a 1 ≥ 1 ≥ a 2 > 0, they obtained that u(t) → 0, v(t) → 1, w(t) → 1 γ in L ∞ (Ω) as t → ∞ under the condition that there exists a 1 ∈ [1, a 1 ] such that a 1 a 2 < 1 and In the non-competitive case (a 1 = a 2 = 0), global existence and asymptotic stability were established under some conditions for χ i (w) ( [10]). In the case that d 1 = d 2 = d 3 = 1 and h(u, v, w) = u+v −w, Zhang-Li [14] proved global existence of solutions to (1) under the assumption that µ i > 0 is small and The purpose of the present paper is to improve the methods in [1], [10] for obtaining global existence and asymptotic stability of solutions to (1) under a more general and sharp condition for the sensitivity function χ i (w). We shall suppose throughout this paper that h, χ i (i = 1, 2) satisfy the following conditions: We also assume that there exists p > n such that The above conditions cover the prototypical example Now the main results read as follows. The first one is concerned with global existence and boundedness in (1). (9). Then for any u 0 , v 0 , w 0 satisfying (10) for some q > n, there exists an exactly one pair (u, v, w) of nonnegative functions which satisfy (1). Moreover, the solutions u, v, w are uniformly bounded, i.e., there exists a constant C 1 > 0 such that for all t ≥ 0, and the solutions u, v, w are the Hölder continuous functions, i.e., there exist α ∈ (0, 1) and C 2 > 0 such that Remark 1. Theorem 1.1 improves the result in [14] when χ i (w) = Ki (1+w) σ i . Indeed, we do not need any condition for µ i . Moreover, the condition in [14] is with some r > 0, while the condition (9) becomes K i ≤ σi √ p when χ i (w) = Ki (1+w) σ i . Since Theorem 1.1 guarantees that u, v and w exist globally and are bounded and nonnegative, it is possible to define nonnegative numbers α 1 , α 2 , β 1 , β 2 by where I = (0, C 1 ) 3 and C 1 is defined in Theorem 1.1.
In the case a 1 , a 2 ∈ (0, 1) asymptotic behavior of solutions to (1) will be discussed under the following additional conditions: there exists δ 1 > 0 such that and .
This remark implies the following result which improves the previous work in [1]. Theorem 1.3. Let d 1 , d 2 , d 3 > 0, µ 1 , µ 2 > 0 and a 1 , a 2 ∈ (0, 1). Assume that there exists a unique global solution (u, v, w) of (1) satisfying that there exists C > 0 such that ≤ C for all t ≥ 1 and χ 1 , χ 2 satisfies that there exist the positive constants M 1 , M 2 > 0 and δ 1 > 0 such that Then the same conclusion as in Theorem 1.2 holds.
In the case a 1 ≥ 1 > a 2 > 0 asymptotic behavior of solutions to (1) will be discussed under the following additional conditions: there exist δ 1 > 0 and a 1 ∈ [1, a 1 ] such that .
The third one is concerned with asymptotic behavior of solutions to (1) in the case a 1 ≥ 1 > a 2 > 0.
Then the same conclusion as in Theorem 1.4 holds.
Remark 4. In Theorems 1.2 and 1.4 we can find w * ≥ 0 satisfying h(u * , v * , w * ) = 0 and w ≥ 0 satisfying h(0, On the other hand, (5) and (6) imply The strategy for the proof of Theorem 1.1 is to construct estimates for Ω u p and Ω v p by modifying a method in [10]. One of the keys for this strategy is to derive inequality Applying the new method in [10] for obtaining the above inequality, we can improve the result in [14]. On the other hand, the strategy for the proof of Theorems 1.2 and 1.4 is to modify an argument in [1]. The key for this strategy is to construct the following energy estimate: with some function E(t) ≥ 0 and some ε > 0, where (u, v, w) ∈ R 3 is a solution of (1). For finding the above inequality we apply more "suitable" estimates for These enable us to improve the conditions (2) and (3). This paper is organized as follows. In Section 2 we prove global existence and boundedness (Theorem 1.1). Sections 3 and 4 are devoted to the proof of asymptotic stability (Theorems 1.2, 1.4).

Global existence and boundedness.
In this section we shall show global existence and boundedness in (1). Firstly we will recall the well-known result about local existence of solutions to (1). (7). Then for any u 0 , v 0 , w 0 satisfying (10) for some q > n, there exist T max ∈ (0, ∞] and an exactly one pair (u, v, w) of nonnegative functions which satisfy (1). Moreover, Proof. The proof of local existence of classical solutions to (1) is based on a standard contraction mapping argument, which can be found in [11,12]. Finally the maximum principle is applied to yield u > 0, v > 0, w ≥ 0 in Ω × (0, T max ).
Let (u, v, w) be the solution of (1) on [0, T max ) as in Lemma 2.1. We introduce the functions to prove the following lemma.
Assume that χ 1 , χ 2 satisfy (4) and (9) with some p > n. Then there exist positive constants and Proof. We let p ≥ 1 and r > 0 be fixed later. From the first and third equations in Denoting by I 1 and I 2 the first and third terms on the right-hand side as we can write as We shall show the following inequality: ∃ p > n, ∃ r > 0 ; I 1 + I 2 ≤ 0.
Noting that we obtain Similarly, we see that Therefore it follows that where a 1 , a 2 , a 3 are given by Then there exists p > n such that the discriminant of a 1 r 2 + 2a 2 r + a 3 is nonnegative in view of (9). Therefore we have that there exists r > 0 such that a 1 r 2 + a 2 r + a 3 ≤ 0 and hence On the other hand, we can see from the positivity of u and v that This means that (17) holds. In the same way, we obtain (18). Lemma 2.3. Let d 1 , d 2 , d 3 > 0, µ 1 , µ 2 > 0 and a 1 , a 2 > 0. Assume that h, χ i satisfy (4)-(6), (8), and (9) with some positive constants k i (i = 1, 2) and p > n, then v(t) L p (Ω) ≤ e χ2 L 1 (0,∞) Proof. The proof is same as in [10,Lemma 3.2].

From (21), (22) and
Noting that we deduce that Since (9) implies χ 1 < 0, it follows from (20) and (23) that for all t ∈ (τ /2, T max ), Employing the variation of constants formula for u yields Let η ∈ n 2p , 1 2 and ε ∈ 0, 1 2 − η . Then we see that 0 < 2η − n p and η + ε + 1 2 < 1. By (21), (22) and Lemma 2.3 we observe that for all t ∈ (τ, T max ), Now we recall the well-known fact ( [6]). Let p ∈ (1, ∞), then there exists λ > 0 such that for every ε > 0 we can find c 5 > 0 satisfying for all R n -valued w ∈ L p (Ω). Using (21), (24) and (25), we obtain Since the Neumann heat semigroup (e t∆ ) t≥0 has the order preserving property, we infer and moreover, by the maximum principle we have Therefore we obtain that there exists C u (τ ) > 0 such that The positivity of u yields that The same argument as for u gives the L ∞ (Ω) bound for v. Finally, the Hölder continuity of the solution (u, v, w) comes from standard parabolic regularity theory ( [9]). This completes the proof of Theorem 1.1.
3. Asymptotic behavior. Case 1: a 1 , a 2 ∈ (0, 1). In this section we will establish asymptotic stability of solutions to (1) in the case a 1 , a 2 ∈ (0, 1). For the proof of Theorem 1.2, we shall prepare some elementary results. Then holds for all x, y, z ∈ R.
Proof. From straightforward calculations we obtain In view of the above equation, (26) leads to (27). Now we will prove the key estimate for the proof of Theorem 1.2.
We denote by A 1 (t), B 1 (t), C 1 (t) the functions defined as and we write as The Taylor formula applied to ) is a nonnegative function for t > 0 (more detail, see [1,Lemma 3.2]). Similarly, we have that B 1 (t) is a positive function. By the straightforward calculations we infer with some derivatives h u , h v and h w . Hence we have where At first, we shall show from Lemma 3.2 that there exists ε 1 > 0 such that To see this, we put Since µ 1 > 0, we have g 1 (0) = µ 1 > 0. Due to (12), we infer In light of (6) and the definitions of δ 2 > 0, α i , β i ≥ 0 (defined in (11)) we obtain Combination of the above inequalities and the continuity argument yields that there exists ε 1 > 0 such that g i (ε 1 ) > 0 hold for i = 1, 2, 3. Thanks to Lemma 3.2 with we obtain (31) with ε 1 > 0. Lastly we will find ε 2 > 0 satisfying By virtue of the definition of δ 2 > 0, we can find δ 3 ∈ χi(0) 2 u * (1+δ1) 4d1d3δ2 , 1 . Noting that χ i < 0 (from (9)) and then using the Young inequality, we have Plugging these into (30) we infer We note from the definitions of δ 2 > 0 and δ 3 > 0 that Therefore we obtain that there exists ε 2 > 0 such that (32) holds. Combination of (29), (31) and (32) implies the end of the proof.
In order to complete the proof of Theorem 1.2, we will prepare the following lemma.
Lemma 3.5. Let (u, v, w) ∈ R 3 be any solution of (1) and (u, v, w) a global bounded classical solution to (1). Suppose that there exist two decreasing functions h 1 , h 2 on (0, ∞) and t 0 > 0 such that for all t > t 0 .
Then there exist C > 0 and t 1 > 0 such that for all t > t 1 .
Proof. The arguments in [1, Lemma 3.6] and Theorem 1.1 lead to the proof of this lemma.
Proof of Theorem 1.2. From the L'Hôpital theorem applied to H 1 (s) := s − u * log s we can see In view of the combination of (33) and u − u * L ∞ (Ω) → 0 from Lemma 3.4 we obtain that there exists t 0 > 0 such that A similar argument yields that there exists t 1 > t 0 such that 1 4v * We infer from (34) and the definitions of E 1 (t), F 1 (t) that E 1 (t) ≤ c 6 F 1 (t) for all t > t 1 with some c 6 > 0. Plugging this into (28), we have which implies that there exist c 7 > 0 and > 0 such that Thus we obtain from (34) and (35) that for all t > t 1 with some c 8 > 0. Moreover, there exists c 9 > 0 such that Thanks to Lemma 3.5, we achieve that there exist C > 0 and λ > 0 such that . This completes the proof of Theorem 1.2.
We denote by A 2 (t), B 2 (t), C 2 (t) the nonnegative functions defined as and we write as Then by the straightforward calculations we infer with some derivatives h u , h v and h w . Hence we have where From the same argument as in the proof of Lemma 3.3 we obtain that there exists ε 1 > 0 such that On the other hand, thanks to χ 2 < 0 and the Young inequality, we infer that Plugging this into (38), we have Noting by the definition of δ 2 > 0 that d 3 δ 2 − a 1 µ 1 χ 2 (0) 2 δ 1 4a 2 µ 2 d 2 > 0, we obtain that there exists ε 2 > 0 such that Combination of (37), (39) and (40) implies the end of the proof.
In light of a similar argument to seeing (34) we obtain that there exists t 1 > t 0 such that The definitions of E 2 (t), F 2 (t) and (42), (43) yield that with some c 10 > 0. Plugging this into (36), we have which implies that there exist c 11 and > 0 such that Therefore from (42) and (43) we can find c 12 > 0 satisfying Moreover, there exists c 13 > 0 such that Thanks to Lemma 3.5, we achieve that there exist C > 0 and λ > 0 such that u(t) L ∞ (Ω) + v(t) − 1 L ∞ (Ω) + w(t) − w L ∞ (Ω) ≤ Ce −λt (t > 0), which implies the end of the proof.

Proof.
We have already known that there exists t 0 > 0 such that (43) holds for all t > t 0 . Hence the Cauchy-Schwarz inequality and the boundedness of (u, v, w) imply that there exists c 14 > 0 satisfying for all t > t 0 . Thus from (36) we can find c 15 > 0 such that which implies that there exists c 16 > 0 satisfying E 2 (t) ≤ c 16 t + 1 (t > t 0 ).