THE SCHNAKENBERG MODEL WITH PRECURSORS

. In this paper, we mainly consider the following Schnakenberg model with a precursor µ ( x ) on the interval ( − 1 , 1): 0 , where D 1 > 0, D 2 > 0, B > 0. We establish the existence and stability of N − peaked steady-states in terms of the precursor µ ( x ) and the diﬀusion coeﬃcients D 1 and D 2 . It is shown that µ ( x ) plays an essential role for both existence and stability of the above pattern. Similar result has been obtained for the Gierer-Meinhardt system by Wei and


1.
Introduction. Since the work of Turing [11] in 1952, a lot of models have been established and studied to explore the so-called Turing diffusion-driven instability.
One of the most interesting models in biological pattern formation is the schnakenberg model [10] on a one-dimension interval, which can be stated as follows:    u t = D 1 u xx − u + vu 2 x ∈ (−1, 1), t > 0, v t = D 2 v xx + B − vu 2 x ∈ (−1, 1), t > 0, u x (±1, t) = v x (±1, t) = 0, (1.1) where D 1 > 0, D 2 > 0 and B > 0 are positive constants. Substituting u =û/(2B), v = 2Bv, and dropping hats we obtain the following form of the system To find a scaling appropriate for spike solutions we assume that u diffuses more slowly than v, so that where ε 2 << D << 1. We then introduce the new variables Substituting (1.3) into (1.1), and dropping the variables, we obtain the following singularly perturbed reaction-diffusion system of interest: The stationary solution to (1.4) (1. 5) We note that the schnakenberg model has been widely studied by analytical and numerical methods. In the one-dimension case. For problem (1.2), the existence and stability of symmetric N −peaked solution were established by Wei and Winter [7] using asymptotic analysis. They mainly consider the stability of symmetric N −peaked solutions to problem (1.2). In this case, the parameter D effect the stability. For D small, the N − spike solution is stable, while for D large, the N − spike solution is unstable for N ≥ 2. For problem (1.5), using asymptotic expansions, Ward and Wei studied the existence and stability of asymmetric equilibrium spike patterns for the Schnakenberg model [13]. In this article, as ε → 0, they constructed an asymmetric k− spike equilibrium solution to problem (1.5) in the form of a sequence of spikes of different heights. Moreover, they considered the stability of the asymmetric k− spike equilibrium solution.
In two dimension case. We refer to [2] and references therein-in which the Schnakenberg model is posed in a two-dimensional square.
Our interest is to consider the existence and stability of N −peaked solutions to the Schnakenberg model (1.6). For the existence of N −peaked solutions, we must consider the effect of the precursor µ(x). Since the precursor µ(x) may be not symmetric in (−1, 1), we can't consider this problem in symmetric space. That is the solution of (1.6) may be not symmetric. Hence, we need to construct solution by a new method. In this paper, we will construct N −peaked solutions by using the method of Liapunov-Schmidt reduction which has been used for the one-dimension Schrödinger equation [3][8] [9] and then extended to the higher-dimensional Cahn-Hilliard equation [15] [16] and semilinear elliptic equations [1][5] [6]. This method has also been applied to the Schnakenberg model [7][13] [22]. For the stability of N −peaked solutions, we study it by using the asymptotic analysis, which has been used to study the Gierer-Meinhardt system [17] [18] [19] [20]. In [12], the authors have used the Lypunov-Schmidt reduction to consider the effect of precursor for the Gierer-Meinhardt system. In this paper, we will employ the same idea to deal with the Schnakenberg model. It turns out that unlike the homogeneous case for which the N −spike solution is stable for D small, the precursor may effect the stability of the spike solutions.
Before we state our main results in Section 2, we introduce some notation. Throughout this paper, we always assume Ω = (−1, 1), Ω ε = (− 1 ε , 1 ε ). With L 2 (Ω) and H 2 (Ω) denote the usual Sobolev spaces. The function ω we denote the solution of the following problem: (1.7) An explicit representation is ω(y) = 3 2 cosh −2 ( y 2 ). We list some properties of ω:    ω is a even function on R 1 ; ω (y) < 0, if y > 0; R ω 3 (y)dy = 7.2,´R(ω ) 2 (y)dy = 1.2,´R ω 2 (y)dy = 6. (1.8) We assume that the precursor µ(x) satisfies Let G D (x, z) be Green's function given by (1.10) We easy calculate The paper is organized as follows. In Section 2, we will state the main existence and stability result for the Schnakenberg model. Section 3-Section 6 concerns the existence part. In Section 7-Section 8, we prove the stability result. Section 9 contains some technical computations and the analysis of the Green's function is contained in the Appendix.
We use the notation e.s.t. to denote an exponentially small term of order O(e − d ε ) for some d > 0 in the corresponding norm. By C we denote a generic constant which may change from line to line.

2.
Main results: Existence and stability of N-peak solutions. In this paper, we always consider the following situations: 0 < ε << 1 and D > 0.
We define ω a (y) = aω(a 1 2 y), f or a > 0, (2.3) where ω satisfies (1.7), is the unique solution of the following problem: By some simple calculation, we have the following relations (ω ) 2 (y)dy. (2.5) We introduce several matrices for late use: Similarly, we define (2.7) Now the derivatives of G D (t i , t j ) are defined as follows: Next we state the first assumption: (H1) There exists a solution (ξ 0 1 , . . . , ξ 0 N ) of the equation where λ 1 = − 1 6DN . By Appendix (9.42), we know that Next we introduce some matrices: (2.10) Our second assumption is the following: (H2) It holds that is invertible.

(2.19)
Note that by Appendix (9.38) and (9.39), we have Therefore, we can obtain By (H1), moreover, we have To study the stability, we define where D and C are given by (2.10) (2.11) respectively .
Remark 2.4. By the same reasoning as for the matric M, the eigenvalues of B are real.
Our first result can be stated as follows: Theorem 2.5. Assume that assumptions (H1), (H2) and (H3) hold. Then for ε << 1, problem (1.6) has an N-peak solution centered at t ε 1 , . . . , t ε N , N ≥ 2. Moreover, it satisfies where ω i is given by The next theorem reduces the stability to the conditions on the matrices M and B.
We end this section with a few remarks.
Thus the height of different peaks may be different. This is strikingly different from the solutions constructed by Iron, Wei and Winter in [7]. Remark 2.8. For µ ≡ 1, the eigenvalues of matrices B and M have been computed explicitly in [7]. But for general precursor µ, the computation may be more complicated.
Remark 2.9. Generally speaking, the choice of the precursor µ and the parameter D will effect the stability of the spike solutions, since the eigenvalues of M and B are dependent on the precursor µ and the parameter D.
Let us consider the following case: Note that M is symmetric, H = ξI and µ = mI. So the eigenvalues of M are all real, The first matrix M 1 does not depend on A. Thus, if A is sufficiently large, the eigenvalues of (M) are all negative. Hence, by Theorem 2.6, we obtain (u ε , v ε ) is unstable. We conclude that precursors may give rise to instability. This new effect is not present in the homogeneous case.
3. Some preliminaries. In this section, we study a system of nonlocal linear operators. We first recall Theorem 3.1. [14]: Consider the following nonlocal eigenvalue problem (3.1) 1. If γ < 1, then there is a positive eigenvalue to (3.1).
Next, we consider the following system of linear operators where u ∈ H 2 (R).

WEIWEI AO AND CHAO LIU
Then the conjugate operator of L under the scalar product in L 2 (R) is As a consequence of Lemma 3.2, we have is an invertible operator if it is restricted as follows Moreover, L −1 is bounded.

4.
Study of approximate solutions. Let ξ 0 = (ξ 0 1 , . . . , ξ 0 N ) be the locally unique solution of (2.9). Recall that µ 0 i = µ(t 0 i ) and We now construct an approximate solution of (1.6) which concentrates near those Introduce a smooth cut-off function χ: R → [0, 1] such that We define approximate solutioñ Thenω i (x) satisfies the following equation: According to (H2), for t ∈ B ε (t 0 ) there exists a unique solution ξ(t) = (ξ 1 , . . . , ξ N ) such that For u ∈ H 2 (−1, 1), we define T [u] to be the solution of (4.7) and Green's function (1.10), we have Thus, by Appendix (9.42), (4.10) On the other hand, integrating (4.7), we obtain Hence, by (2.9) and(4.10), we obtain Substituting T [ω ε,t ] into (4.9), we have By assumption (H2) and the implicit function theorem, the equation (4.11) has a unique solution where T [ω ε,t ] is given by (4.9). We now compute S ε [ω ε,t ] as follows: and We first estimate E 1 : For E 2 , we have (Letting x = t i + εy and using (4.12)) This implies that The estimates derived in this section will enables us to carry out the existence proof in the next two sections.
5. The Liapunov-Schmidt reduction method. In this section, we use Liapunov-Schmidt method to solve the problem for real constants α i and some function φ ∈ H 2 (− 1 ε , 1 ε ) that is small in the corresponding normal, where ω ε,t is given by (4.6) andω by (4.3).
First, we need to study the following linear operator where Ω ε = (− 1 ε , 1 ε ), T [ω ε,t ] is given by (4.9), and for given φ ∈ L 2 (Ω) the function T [ω ε,t ]φ is defined as the uniqueness solution of We define the approximate kernel and co-kernel of the operatorL ε,t , respectively, as follows: Recall the definition of the following system of linear operators from (3.2): By lemma (3.3), we known that is invertible with a bounded inverse, where X 0 = span{ dω dy }. We will see that this system is a limit of the operator L ε,t . We also need to introduce the projection π ⊥ ε,t : L 2 (Ω) → C ⊥ ε,t and study the operator L ε,t := π ⊥ ε,t • L ε,t . By letting ε → 0, we will show that L ε,t : K ⊥ ε,t → C ⊥ ε,t is invertible with a bounded inverse provided ε is sufficiently small. This statement is contained in the following proposition, whose proof is given in proposition 5.1 of [20].
is surjective. Now we are in a position to solve the equation Since L ε,t | K ⊥ ε,t is invertible, we denote its inverse operator by L −1 ε,t , we can rewrite ) and the operator M ε,t is defined by (5.7) for φ ∈ H 2 N (Ω ε ). We are going to show that the operator M ε,t is a contraction on for ε and δ are small enough. According to (4.20) and proposition 5.1 If we choose δ and ε sufficiently small, then M ε,t is a contraction on B ε,δ . By standard contraction mapping principle, there exists a φ ε,t ∈ B ε,δ such that (5.7).
Let us now calculate W ε (t). By (4.16) and (4.18), we have where I 1 , I 2 and I 3 are defined by the last equality.
The computation of I 3 is as follows: note that by Taylor's expansion for (5.8), the first term in the expansion of N ε,t is quadratic in φ ε,t . So We now compute I 1 and I 2 .
For I 1 , we have where E 1 and E 2 have been given by (4.14) and (4.15), respectively. Using (4.16), we obtain Next, we calculate I 12 since P i (y) is an even function. For I 2 , by (4.12), (4.9) and (4.4), using the following results one has Combining the estimates for I 1 , I 2 and I 3 , we have . . , N . By assumption (H3), we have F (t 0 ) = 0 and det(D t 0 F (t 0 )) = 0.
Thus we have proved the following proposition.
7. Stability analysis I: Large eigenvalue. In this section, we consider the large eigenvalues of the associated linearized eigenvalue problem. Let (u ε , υ ε ) be the N peak solution constructed in previous section. We have We linear (1.6) at (u ε , υ ε ). The eigenvalue problem becomes Here λ ε is some complex number and We consider two case: The large eigenvalue case with λ ε → λ 0 = 0 and the small eigenvalue λ ε → 0. The second case will be considered in the next section.
We now analysis the large eigenvalues there exists some small c > 0 such that|λ ε | ≥ −c for ε sufficiently small. We are going looking for a condition under which Re(λ ε ) < 0 for all eigenvalues λ ε of (7.2) if ε is sufficiently small. If Re(λ ε ) < c, then λ ε is a stable large eigenvalue. Therefore for the rest of this section we assume that Re(λ ε ) ≥ −c and study the stability properties of such eigenvalues.
This proof is similar to Theorem 7.1 [21], more details can see Theorem 7.1 [21] or Theorem 8.1 [20]. Now, we study the stability of (7.2) for large eigenvalues explicitly and prove (2.25) and (2.27) of Theorem 2.6.
In conclusion, we have finished the study of large eigenvalues and derived results on their stability properties. It remains to study small eigenvalues which will be done in the next section.
8. Stability analysis II: Small eigenvalue. Now we study (7.2) for small eigenvalue. Namely, we assume that λ ε → 0 as ε → 0. Letω where t ε = (t ε 1 , . . . , t ε N ). Recall the eigenvalue problem (7.2): Our basic ideal is the following: the eigenfunction φ ε can be expended as . So when we differentiate ω ε,t with respect to t j , we also need to differentiate ξ j and µ(t j ) with respect to t j . Therefor, we need to expand φ ε up to O(ε 2 ).
Similarly, we can decompose where ψ ε,j satisfies and ψ ⊥ ε satisfies Throughout this section, we denote Substituting the decompositions of φ ε and ψ ε into (8.2) we have, using (8.5) a ε jω ε,j . (8.10) Let us first compute Let us also putL and a ε = (a ε 1 , . . . , a ε N ) . Multiplying both sides of (8.10) byω ε,l and integrating over (−1, 1), we have, using (2.5), (8.14) and, using(8.11), where J i,l , l = 1, . . . , 5 are defined by the last equality. For J 2,l , integrating by parts gives For J 3,l , we have We define the vectors We have the following lemma: where G D , Q and H are defined by(2.6), (2.20) and (2.14), respectively, a ε is given by (8.13) and The proof of lemma 8.1 is delayed to the next section.
Proof of Theorem 2.6. By the previous lemma, we obtain l.h.s.
Arguing as in Theorem 7.1, this shows that if all the eigenvalues of M(t 0 ) are positive, then the small eigenvalues are stable. On other hand, if M(t 0 ) has a negative eigenvalue, then we can construct eigenfunctions and eigenvalues to make the system unstable.

9.
Computation of the small eigenvalues II: Proof of Lemma 8.1. In this section, we prove lemma 8.1. First note that So we need to study the asymptotic behavior of ψ ε,j near t ε . Since ψ ε,j satisfies (8.7), we have that Hence we have 2) where I 1 , I 2 are defined by the last equality.