SPIKE SOLUTIONS FOR A MASS CONSERVATION REACTION-DIFFUSION SYSTEM

Ni-Abstract. This article deals with a mass conservation reaction-diﬀusion system. As a model for studying cell polarity, we are interested in the existence of spike solutions and some properties related to its dynamics. Variational arguments will be employed to investigate the existence questions. The proﬁle of a spike solution looks like a standing pulse. In addition, the motion of such spikes in heterogeneous media will be derived.

Observe that adding (1.1) to (1.2) yields ∂u 1 ∂t thus for any solution (u 1 (x, t), u 2 (x, t)) of (1.1)-(1.4), the total mass Ω u 1 (x, t) + u 2 (x, t)dx (1.6) is conserved in all the time. As pointed out in literature [16,18,23], a mass conserved reaction-diffusion system is one of the fundamental principles of cell polarity. A stationary solution (û 1 (x),û 2 (x)) of (1.1)-(1.3) satisfieŝ From a simple observation, (dû 1 +û 2 ) is a bounded harmonic function on R N . Then the Liouville's Theorem shows that dû 1 +û 2 = c for some c ∈ R. Thus we are led to studying Motivated by [21,25,22] and cell biology, we are interested in localized patterns in mass conserved reaction-diffusion system. Variational methods will be employed to obtain non-constant stationary solutions of (1.1)-(1.3), as stated in the following theorem. As to capture an interesting phenomena in the cellular process, the generation of spike solutions has attracted a great deal of attention. For the FitzHugh-Nagumo equations, a spike solution has been obtained [2,5,6,11,25] by showing the existence of standing pulse solutions. The related stability questions were investigated in [10], using the Maslov index [8]. Such localized patterns [9,13,27] and waves [3,4,29] often result from a balance between dispersion and nonlinearity. For the existence of spike solutions of (1.8)-(1.9), the following result will be established. In Section 2 we start with the variational framework to employ the Mountain Pass Theorem. It will be shown Section 3 that if d is small, the critical points obtained by the Mountain Pass Theorem are non-constant stationary solutions. A further analysis in Section 4 shows that the profile of such a stationary solution possesses one spike. Moreover, it is known [17] that for certain types of f , the existence of a Lyapunov functional for the flow (u 1 (x, t), u 2 (x, t)) generated by (1.1)-(1.3) and we brief discuss the Lyapunov stability in Section 5. Heterogeneity is the most important and ubiquitous type of external perturbation observed in natural environments. When a reaction-diffusion system is perturbed by some small heterogeneity, some new interesting phenomena [7,28] have been observed. With zero being an eigenvalue due to the translation free mode by periodic boundary conditions [12], we study the associated eigenfunctions for the linearization and the adjoint operator in Section 6. Such eigenfunctions play important roles in studying the motion of spike in heterogeneous media, as to be demonstrated in Section 7.
2. Variational framework for stationary solutions. In this section variational methods will be used to show the existence of stationary solutions (u 1 (x), u 2 (x)) of (1.1)-(1.3). As to study a model related to cell polarity, we focus on some examples for N = 2, 3, while most of arguments can be applied to the case in other dimensions.
Setting f c (u) = f (u, c − du), we seek the solutions of the elliptic boundary value problem This functional is defined on the Hilbert space We will employ the critical point theory [24] to investigate the critical points of J to obtain the non-constant solutions of (2.1)-(2.2). There are many functions satisfy the conditions (f 1)-(f 4). For instance, f (u 1 , u 2 ) = −u 1 + u 3 1 + u 3 2 and f (u 1 , u 2 ) = (u 1 + u 2 ) 3 − u 1 . It is not difficult to check the following properties hold.
(i) If u ∈ E and u + is not zero almost everywhere, then lim t→∞ J 0 (tu) = −∞.
(ii) For any given small |c| ,there is a θ ∈ (0, 1 2 ) such that θf c (u)u − F c (u) is bounded below on R. (iii) There is a δ > 0 and a r > 0 such that inf u∈E, u =r J 0 (u) ≥ δ. (iv) J c (u) converges uniformly to J 0 (u) on any given bounded subset of E as c → 0.
A well-known critical point theory is the Mountain Pass Theorem [1]. In the next section we apply such a theorem to find non-constant stationary solutions of (1.1)-(1.3) when (i)-(iv) hold.
3. Existence of stationary solutions. Let d * be a small positive number and d < d * . We now investigate the existence of solutions of Such solutions are the critical points of J c defined by (2.3).
To simplify the presentation, we may consider f (u 1 , u 2 ) = −u 1 + u 3 1 + u 3 2 to illustrate the essence of detailed arguments. With it is clear that f 0 has three roots, namely − 1

SHIN-ICHIRO EI AND SHYUH-YAUR TZENG
Proof. Note that when c = 0, and J 0 (0) = 0. The other constant solution is 1 Hence u 0 is a non-constant solution.
Hence there is a continuous path γ, starting with 0 and passing through u and ending at se d , on which max 0≤θ≤1 J 0 (γ(θ)) = J 0 (u) holds. Now the proof is complete.
Next we turn to the case of c < 0 has a non-constant solution u c .
Proof. Observe that and J c (0) = c 4 4d . Pick a c * ∈ (−d, 0) with |c * | being sufficiently small. It is easy to verify, for c ∈ (c * , 0], that J c (0) < d 2 With J c being a C 2 functional on E, applying the standard Deformation Lemma [24] yields a Palais-Smale sequence {w n } in E such that as n → ∞ where E −1 is the dual of E. Moreover by (3.12) We claim: {w n } is uniformly bounded in E. Indeed for large n, by (3.13) and (3.14) and Then (3.16) and (3.17) give Note that 1 12 For any u(x) ∈ R, Thus combining (3.18) and (3.19) yields 20) which implies that {w n } is uniformly bounded in E. Therefore along a subsequence w n u c weakly for some u c ∈ E and w n → u c almost everywhere on Ω as n → ∞. Then (3.14) together with the Sobolev imbedding theorem gives for every ϕ ∈ E. Also, (3.16) and (3.17) imply Next we claim that with M 0 being a constant depending on d only. Note that there exists a γ ∈ it follows from (3.15) and (3.22) that This together with (3.22) yields Take a sequence c n → 0 as n → ∞. By (3.26), the sequence {u cn } is bounded in E. Hence along a subsequence u cn w 0 weakly for some w 0 ∈ E and u cn → w 0 almost everywhere on Ω. Since for ϕ ∈ E, invoking the Sobolev imbedding theorem gives Applyng the Sobolev imbedding theorem gives Since w 0 ∈ S, it follows from Lemma 3.2 that In fact, the above proof shows that (3.30) holds for any sequence c n → 0. Thus the proof is complete.

SHIN-ICHIRO EI AND SHYUH-YAUR TZENG
The same argument as in (3.20) shows that the set ∪ 0<|c|<c * A c is uniformly bounded in E. For each c > 0, pick a w c ∈ A c . Now consider a sequence {w cn } with c n → 0 as n → ∞. Then along a subsequence, w cn w 0 weakly for some w 0 ∈ E and w cn → w 0 almost everywhere on Ω. This together with Sobolev imbedding theorem implies J 0 (w 0 )(ϕ) = 0 for ϕ ∈ E. Moreover, It follows from Lemma 3.2 and the Sobolev imbedding theorem that as n → ∞. This together with Lemma 3.4 yields Observe that as n → ∞. Together with w cn w 0 weakly in E, we conclude that w cn − w 0 → 0 as n → ∞.
In (4.2),v, v and u 0 are functions depending on d. In what follows, we take d n = 1 n , n ∈ N, and denote Ω 1 n by Ω(n). We replacev by v n to clarify its dependence on d. Moreover by taking translation if necessary, we always assume that where z ∈ Z N , a lattice point of R N . Set It has been shown that the positive solution of is a radially symmetric and it is unique up to translation. We may assume that g is radially symmetric at the origin. Then It is known that g(|x|) is a decreasing function with exponentially decaying as |x| → ∞. Let E n = {u ∈ H 1 loc (R N )|u(x + ne i ) = u(x) for i = 1, 2, ..., N }. Lemma 4.1. Given ε > 0, there is a δ > 0 such that if 1 n < δ and u 0 ∈ A 0 then v n − g En < ε. (4.8) By (4.1) and Combining (4.9) and (4.10) gives Note that a slightly modified version of the proof of Lemma 3.2 shows that lim sup We claim there is a C 0 > 0 such that if n is large enough. Indeed applying the Holder Inequality and Solobev imbedding theorem, there is a C 1 > 0 such that It follows from (4.11),(4.12) and (4.15) that This together with (4.13) gives which completes the proof of (4.14). Next we show that lim inf By (4.11) and (4.13), the sequence { v n En |n ∈ N} is bounded. Hence along a subsequence v n converges weakly to g and v n → g pointwise on R N . Moreover for ϕ ∈ C 1 0 (R N ), as n → ∞. Together with (4.12) and Fatou's Lemma yields and consequently This toghther with (4.21) implies Then for large j, as n → ∞. Now the proof is complete.
In view of Lemma 3.5 and Lemma 4.1, the proof of Theorem 1.2 is complete.

Lyapunov stability. Consider the flow generated by (1.1)-(1.4). A class of functions
In particular, the existence results for the stationary solutions and spike solutions obtained in Section 3 and Section 4 are also applicable to f (u 1 , u 2 ) = (u 1 +u 2 ) 3 −u 1 , since Recall that by the mass conservation The Lyapunov stability for the stationary solutions of (1.1)-(1.3) follows from a result given by [17]: 6. Adjoint eigenfunctions. Let S(x) = (û 1 (x),û 2 (x)) (x ∈ Ω ⊂ R N ) be a stationary solution of (1 .1)-(1.3). We look at the linearization of (1.1)-(1.3) at (û 1 (x),û 2 (x)). With zero being an eigenvalue due to the translation free mode by periodic boundary conditions, we study the associated eigenfunctions and adjoint eigenfunctions. First, we rewrite (1.1)-(1.2) in the vector form as where U = t (u 1 , u 2 ), D = diag{d, 1} and a = t (1, −1). Then the linearized operator L with respect to S(x) is given by LU = D∆U + (∇f (S) · U )a, where ∇f (S) = t (f u1 (S), f u2 (S)), and the adjoint operator L * of L is L * U = D∆U + (a · U )∇f (S). Now, we derive the eigenfunctions and the adjoint eigenfunctions associated to 0 eigenvalue, which will be important in forthcoming papers to consider the various motions of spikes as in [15] and [19]. Here we note that s(x) satisfies ∆s = −f (S), which means that s(x) is given by for constants c 1 , c 2 . Differentiating (6.2) with respect to x j , we have LS xj = 0 (j = 1, · · · , N ). Thus the eigenspace associated to 0 is a N -dimensional space.

7.
Motion of a spike on heterogeneous media. In this section, we consider the equation (1.1)-(1.3) with inhomogeneity as follows: for 0 < ε << 1 and a function g, which is also written in a vector form as in (6.1). (7.1) has a conservative quantity Ω {u 1 + u 2 }dx. It is expressed as where ·, · 2 denotes the inner product in L 2 (Ω) and a * = 1 √ 2 t (1, 1) is taken such that a * L 2 = 1 and (a · a * ) = 0. We assume (7.1) with ε = 0, that is, (6.1) has a stable stationary solution S(x) = (û 1 (x),û 2 (x)). Then the function S(x −x) is also a stationary solution of (6.1) for anyx = t (x 1 , · · · ,x N ) ∈ Ω. For ε > 0, representing the solution as x −x(t)) and substituting it into (7.2), we have by That is, First, we intuitively derive the motion ofx(t) by assuming that |V |, | dx dt | are sufficiently small and omitting higher products of them formally. Then the lowest order equation of V is V t = LV + εg(z +x(t), S(z))a + ( dx dt · ∇ z S). (7.5) Taking L 2 inner product of (7.5) with the adjoint eigenfunction Φ * i , we get is a general form of the equation describing the motion ofx(t). In order to know more explicit dynamics, we assume a sufficiently large Ω, say Ω K = [−K, K] N for K >> 1. It implies that the stationary solution S(x) is close to a radially symmetric solution, sayŜ(r) for r = |x|. Precisely speaking, |S(x) −Ŝ(r)| is sufficiently small on a bounded domain including origin and |S(x)| ≤ O(e −γr ) holds. Hence by assuming that Ω K is sufficiently large, we may formally expect that the L 2 inner product on Ω K is replaced by the one on the whole space R N and S = S(r) = (û 1 (r),û 2 (r)). Under the above reduction, we can derive an explicit form of (7.6) as follows: Denoting the polar coordinate by x = re(Θ) for r ≥ 0 and e(Θ) = (e 1 (Θ), · · · , e N (Θ)) with |e(Θ)| = 1 and Θ = (θ 1 , · · · , θ N −1 ), we see the x j -derivative of the radially symmetric function s(r) is given by s xj = e j (Θ)s r . Then (6.7) is ∆Φ * j + e j (Θ)s r (r)D −1 ∇f (S(r)) = 0 and hence Φ * j (x) = Φ * j (re(Θ)) = e j (Θ)φ * (r) holds with a radially symmetric function φ * (r) satisfying ∆ R φ * + s r (r)D −1 ∇f (S(r)) = 0. Here ∆ R denotes the Laplacian restricted to radially symmetric functions, that is, we find Φ * j (x) = Ψ * xj . First, we calculate the matrix M . The component m ij is given by where m 0 = ∞ 0 r N −1 (φ * (r) · S r (r))dr and δ ij denotes the Kronecker's δ. Thus we know M = m 0 I N with identity matrix I N with order N . In order to compute b, we assume g(x, U ) = g 1 (U )g 2 (x), which is naturally derived as the first order term of Tayler expansion when a small heterogeneity is given to f as f (U, 1 + εg 2 (x)). In this case, b is given by As the simplest case, we put g 2 (x) = 1 πr δ(r) (Dirac function). Then b i = −g 1 (S(| −x(t)|)(a · Φ * i (−x(t))) = −g 1 (S(| −x(t)|)(a · Ψ * (−x(t))) xi holds. Thus we formally get the gradient flow like equation In the following, we give the rigorous result on the motion (7.6) of a spike. Let S(x) and L be a stationary solution of (7.1) with ε = 0 and the linearized operator with respect to S(x). L is given by LU = D∆U + (∇f (S) · U )a as mentioned previously. Since the quantity U, a * 2 is conserved in time t, we put X = {U ∈ {L 2 (Ω)} 2 ; U, a * 2 = 0}. Note that S xj ∈ X. We denote the restriction of L in X by L X = L| X and the adjoint operator of L X in X by L * X = (L X ) * . We assume the linear stability of S(x) by the following hypothesis: Hypothesis H1) L X satisfies Σ(L X ) = Σ 0 ∪ Σ 1 , where Σ(L X ) is a spectral set of L X and Σ 0 = {0}, Σ 1 ⊂ {z ∈ C| Re(z) < −γ 0 } for a positive constant γ 0 . 0 is a semi-simple eigenvalue.
Remark 3. This hypothesis has not been proved, but it is strongly suggested by Theorem 5.1.
By H1), the adjoint operator L * X also satisfies H1). Lemma 7.1. L * X is given by L * X U = L * U − L * U, a * 2 a * for U ∈ X, where L * U = D∆U + (a · U )∇f (S) as given in the previous section. The adjoint eigenfunctions Φ * Xj (j = 1, · · · , N ) associated with 0 eigenvalue of L * X are given by Φ * Xj = Φ * j − Φ * j , a * 2 a * , where Φ * j are the ones constructed in Section 5.
Lemma 7.1 is easily proved while we refer [19].