EXACT MULTIPLICITY OF STATIONARY LIMITING PROBLEMS OF A CELL POLARIZATION MODEL

. We show existence, nonexistence, and exact multiplicity for stationary limiting problems of a cell polarization model proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. It is a nonlinear boundary value problem with total mass constraint. We obtain exact multiplicity results by investigat- ing a global bifurcation sheet which we constructed by using complete elliptic integrals in a previous paper.


Introduction.
We are interested in wave-pinning in a reaction-diffusion model for cell polarization proposed by Y.Mori, A.Jilkine and L.Edelstein-Keshet [8] and [9].
is proposed in [9], where W = W (x, t) denotes the density of an active protein, V = V (x, t) denotes the density of an inactive protein, ε, D are diffusion coefficients, W 0 (x) denotes an initial density of the active protein, and V 0 (x) denotes an initial density of the inactive protein. The mass conservation holds, where m is the total mass determined by the mass of the initial densities W 0 (x) and V 0 (x). The stationary problem of (TP) is where W = W (x), V = V (x), and m is a given initial total mass determined by initial densities. Letting D → ∞ in (SP), we obtain a limiting equation. In addition, we concentrate on monotone increasing solutions for simplicity, since we can obtain other solutions by reflecting this kind of solutions. Thus we obtain the following stationary limiting problem (SLP) (1.4) where m, ε are given positive constants, W = W (x) is an unknown function, andṼ is an unknown nonnegative constant. Here, we note that we may omit a condition W (0) > 0 since we can derive it from other conditions. Interesting bifurcation diagrams are obtained in [9] by numerical computations. Kuto and Tsujikawa [5] obtained several mathematical results for (SLP) with suitable change of variables.(see, also [3] and [4]) We have obtained the exact expressions of all the solutions of (SLP) by using the Jacobi elliptic functions and complete elliptic integrals in Mori, Kuto, Nagayama, Tsujikawa and Yotsutani [7]. The method to obtain all the exact solutions is essentially based on the method which started in Lou, Ni and Yotsutani [6]. It is developed by Kosugi, Morita and Yotsutani [2] to investigate the Cahn-Hilliard equation treated in Carr, Gurtin and Semrod [1]. Now, let us introduce an auxiliary problem to investigate (SLP). LetṼ > 0 be given, let us consider the problem (AP;Ṽ) The following fact is fundamental (see, e.g. Smoller and Wasserman [11], Smoller [10], and Theorem 2.1 in [7]).
We see from Theorem 1.4 and Figure 1.2 that the symmetric breaking occurs at secondary bifurcation point for the case m = 2 and there is an imperfect transcritical bifurcation phenomena in the neighborhood of m = 2. Moreover, this case is very delicate, since the shape of bifurcation curves drastically change.
We will discuss continuity, smoothness and limiting values of end points of ε 2 (Ṽ ) including the unique secondary bifurcation point in a forthcoming paper.
This paper is organized as follows. In Section 2 we give proofs of Theorems 1.2 -1.6 by using Proposition 1.1 and Theorem 1.1, which we prove in subsequent sections. In Section 3 we prepare some fundamental facts to give proofs of Proposition 1.1 and Theorem 1.1. In Section 4 we give a proof of Proposition 1.1. In Section 5 we give a proof of is concave in s for each fixed h. In the proof of Proposition 5.6 we use Lemmas 9.1 -9.6. In particular, Lemma 9.2 is main calculation. We prove Lemma 9.2 using by Lemmas A.1 -A.5, Lemmas B.1 and B.2 in Appendix A and B respectively. In Section 6 we give a proofs of Propositions 5.1 and 5.2. In Section 7 we give a proofs of Propositions 5.3 and 5.4. In Section 8 we give a proof of Proposition 5.5. In Section 9 we give a proof of Proposition 5.6. In Appendix A we give proofs of several inequalities including ratios of complete elliptic integrals. In Appendix B we treat more complicated inequalities.
2. Proofs of Theorem 1.2 -1.6. In this section we give proofs of Theorems 1.2 -1.6 by using Proposition 1.1 and Theorem 1.1 which we prove later. It is easy to see that the following lemma holds. (i) It holds that

Lemma 2.2. Let D m be defined by
Then it holds that We obtain conclusions of Theorems 1.2 -1.6 by using Proposition 1.1, Theorem 1.1, Lemma 2.1, Lemma 2.2 and (1.14).

Preliminaries.
In this section we prepare some fundamental facts to give proofs of Proposition 1.1 and Theorem 1.1.
3.1. Definition of the elliptic functions. Let sn(x, k) and cn(x, k) be Jacobi's elliptic functions. The following properties hold: The complete elliptic integrals of the first, second and third kind are defined by and Π(ν, k) := respectively. We see that K(k) is monotone increasing in k, and E(k) is monotone decreasing in k, The following formulas for the complete elliptic integrals are fundamental: , It is easy to see that the following inequalities hold.
3.2. All exact solutions for (AP ;Ṽ ). In this section we show exact solution for (AP ;Ṽ ) and an exact expression of m(Ṽ , ε 2 ) defined by (1.12).
is the unique solution of the following system of transcendental equations Here, sn(·, ·), cn(·, ·) are Jacobi's elliptic function. K(·) is complete elliptic integral of the first kind.

Moreover, it holds that
is the complete elliptic integral of the first kind, and Π(·, ·) is the complete elliptic integral of the third kind.

Properties of fundamental functions.
The following lemmas are used in the proofs of Theorems A and B. We also use them in this paper. We have the following lemma by Lemma 3.2 and the proof of Lemma 3.4 in [2]. and In addition, It is easy to see the following lemmas (see, Lemmas 6.2 -6.5 in [7]).
5. Proof of Theorem 1.1. We prepare several propositions to prove Theorem 1.1. We will give proofs of them in subsequent sections.

Proposition 5.4.
Let J (h, s) be defined by (5.10). Then J (h, s) satisfies the following equation where Let us consider properties of F (h, s).
We have (5.7) by Thus, we complete the proof.

Proof of Proposition 5.2. We have (5.2) -(5.5). Multiplying M s by
Multiplying M h by A s . we get Hence we obtain by direct calculation. Thus (5.8) is obvious from (6.2). We show (5.13). We see from the proof of Lemma E of [7] that Thus we complete the proof.

Proofs of Propositions 5.3 and 5.4.
We prepare two lemmas to prove Proposition 5.3. Proof. Let h ∈ (0, 1) be fixed. We have We note that Hence we get Thus we obtain which implies (5.14).

Proof of Proposition 5.3. It is obvious from Lemmas 7.1 and 7.2.
Proof of Proposition 5.4. We have ∂ ∂s ) . (7.4) Hence, we obtain (5.16) by direct calculation. Thus, we complete proof.
8. Proof of Proposition 5.5. We begin with the following lemmas.
Hence we may show that by Lemma 3.1.
It is easy to see that where are defined by (5.18) -(5.23).
Proof of Proposition 5. 6 We obtain conclusions by Lemma 9.5 and Lemma 9.6.
Appendix A. Inequalities including complete elliptic integrals I. We prepare several lemmas to prove Proposition 5.6.
Proof. We may show that by Lemma 3.1.