PATTERN FORMATION OF A COUPLED TWO-CELL SCHNAKENBERG MODEL

. In this paper, we study a coupled two-cell Schnakenberg model with homogenous Neumann boundary condition, i.e., Ω . We give a priori estimate to the positive solution. Meanwhile, we obtain the non-existence and existence of positive non-constant solution as parameters d 1 ,d 2 ,a and

in Ω, in Ω, We give a priori estimate to the positive solution. Meanwhile, we obtain the non-existence and existence of positive non-constant solution as parameters d 1 , d 2 , a and b changes.

1.
Introduction. The Schnakenberg model [12] is a well-known autocatalytic chemical reaction model with limit cycle behavior, which was introduced by Schnakenberg in 1979. The trimolecular reactions between two chemical products X, Y and two chemical sources A, B are described by the following equations: Using the law of mass action, one can obtain a system of reaction-diffusion equations for the concentrations u, v of the chemical products X, Y which describes the reactions in (1). The non-dimensional form of the equations is In the above equations, d 1 , d 2 are the diffusion coefficients of the chemicals X, Y , and a, b are the concentrations of A, B, respectively. It is also assumed that A, B are in abundance so a and b are kept constant. Model (2) has been studied by various researchers from both analytical and numerical points of view (see [2,9,11]). Nonexistence, existence of positive solutions and Hopf bifurcation of the steady state equation have been considered recently in [6,16]. The Turing patterns and spike solutions are shown in [4,15]. The study of two-cell model of two coupled components is a substantial progress from one-cell model of two-component reaction-diffusion system [3,5]. Coupled cells with diffusion-reaction and mutual mass exchange are often used to describe the processes in living cells and tissues, or in distributed chemical reactions [13,14] . The incentive for studying these problems is to investigate the self-oscillation produced by the reactions in systems of finite sizes. In fact, the study of two coupled components are much more complicated than that of two-component reaction-diffusion system [18]. The spatial patterns of a coupled two-cell reaction diffusion model with autocatalytic have been considered by utilizing the bifurcation theory and degree theory in [1,19,20].
In this paper, we mainly consider a coupled Schnakenberg model with homogeneous Neumann boundary condition, i.e., where a, b, c, d 1 and d 2 are positive constants, Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary ∂Ω and ν is the outward unit normal vector on ∂Ω.
The homogeneous Neumann boundary condition indicates that the system (3) is self-contained with zero-flux. The rest of the paper is organized as follows. In Section 2, we establish a priori estimates for the positive solutions of the system (4). In Section 3 and 4, we consider the non-existence and existence to the positive non-constant solutions of the system (4). 2. A priori estimates. In this section, we give a priori estimates for the solutions of a steady state system (4) which is consistent with system (3): We now recall the following result, due to Lou and Ni (see [7,8]).
Proof. By adding the first equation of (4) to the third one, we have Let x 0 be the minimum of u + w. Lemma 2.1 implies that 2a − (u + w)(x 0 ) ≤ 0, i.e., (u + w)(x 0 ) ≥ 2a, which yields u + w ≥ 2a, ∀x ∈ Ω. With the first equation of (4) and w ≥ 2a − u, we get We define x 1 by the minimum of u. Applying Lemma 2.1 again, it follows that have w ≥ a and z ≤ b a 2 . Adding the equations in (4) all together, we obtain , which yields, Similarly, by the fourth equation of (4), we have w( This concludes our proof. any solution (u, v, w, z) of (4) satisfies Furthermore, by the standard elliptic arguments and Theorem 2.2, we can obtain: Proposition 2. Let a, B, D 1 , D 2 > 0 be fixed. Then, for any positive integer k ≥ 1, there exists a positive constant C > 0 depending on a, B, D 1 , D 2 , k, N, Ω, such that for all where | · | k denotes the norm of C k (Ω).
3. Non-existence of non-constant solutions. In this section, we show some results for non-existence of positive non-constant solution of (4) as d 1 is sufficiently large or b is sufficiently small.
First, we give a lemma, which is essential in the proof of Theorem 3.1.
Proof. (i) By Proposition 1 and 2, for any positive integer k, there exist two positive constants C 1 and C 2 depending on a, b, d 2 and k, such that Choosing k > 2, since the embedding (C k (Ω)) 4 → (C 2 (Ω)) 4 is compact, there exists a subsequence of {u n , v n , w n , z n }, still denoted by itself, and positive functions u, v, w, z ∈ C 2 (Ω), such that (u n , v n , w n , z n ) → (u, v, w, z) in C 2 (Ω) as n → ∞.
(ii) The proof is similar to that of (i), we construct the operator F as follows then F (b, u, v, w, z) : R × (W 2,2 ν ) 4 → (L 2 (Ω)) 4 , it is also easy to verify that D (u,v,w,z) F (0, a, 0, a, 0) is bijection. Thus, by Lemma 3.2(ii) and Implicit Function Theorem, we complete the proof. 4. Existence of non-constant solution. In this section, we consider the existence of non-constant positive solutions to the system (4). Now, we study the influence of the size of the region Ω on the pattern formation of (4). LetΩ = Ω l , x ∈Ω, then system (4) can be written as follows Let λ = l 2 d 2 and θ = d 1 d 2 , writingû,v,ŵ,ẑ andΩ instead of u, v, w, z and Ω, then in Ω, in Ω, Clearly, λ is the measure of the size of the domain. Noting the relations λ = l 2 d 2 and θ = d 1 d 2 between λ, θ, d 1 and d 2 , we obtain the following corollary from (i) of Theorem 3.1.
Then system (13) can be written as and It is easy to calculate that So if 2b a + b > [(2c + 1)