Dynamics and interactions of spikes on smoothly curved boundaries for reaction–diffusion systems in 2D

It is known that for special types of reaction–diffusion Systems, such as the Gierer–Meinhardt model and the Gray-Scott model, stable stationary spike solutions exist on boundary points with maximal curvature. In this paper, we rigorously give the equation describing the motion of spike solutions along boundaries for general types of reaction–diffusion systems in R 2 . We also apply the general results to the Gierer–Meinhardt model and show that a single spike solution moves toward a boundary point with locally maximal curvature. Moreover, by showing the repulsive interaction of spikes along boundaries for solutions of the Gierer–Meinhardt model, we have stable multispike stationary solutions in the neighborhood of a boundary point with locally maximal curvature.


Introduction
In 1972, Gierer and Meinhardt [9] proposed model equations to describe pattern formations in biology, according to the mechanism of diffusion-induced instability described by Turing [14]. The model equation is as follows: Fig. 1 Profile of a spike solution with one peak in R 2 . Only the A-component is drawn with the Neumann boundary conditions, where d 1 and d 2 are positive constants, and a j ≥ 0 are nonnegative constants satisfying 0 < a 1 − 1 a 2 < a 3 a 4 + 1 .
In the model, d 1 is assumed to be sufficiently small, and we write it as d 1 = ε 2 for a sufficiently small ε > 0. One of the typical solutions of (1.1) is a spike solution. A spike solution has a profile such that the A-component is close to A(x) ∼ ε −η w(|x|/ε) for η > 0 and a radially symmetric function w(r ) ≥ 0. That is, the A-component has a sharp peak, as in Fig. 1. For the Gray-Scott model, similar spike solutions are observed [15,16].
There have been many works which studied the existence and stability of stationary spike solutions for (1.1) under appropriate conditions in one-or higher-dimensional spaces. We do not touch here on works related to one-dimensional problems of (1.1), because in this paper we consider (1.1) in two-dimensional spaces.
For higher-dimensional problems for (1.1), one of the most typical solutions is a boundary spike solution, which has peaks on the boundaries of its domain.
Related to boundary spike solutions in higher-dimensional spaces, a stationary spike solution with one peak on the boundary ∂ with globally maximum mean curvature was constructed in [13], and, later, a stationary spike solution with multiple peaks both inside and on the boundary ∂ was constructed in [10]. Recently, [11] and [12] proved the existence and stability of a stationary spike solution with more than one peak at points with locally maximal mean curvatures of ∂ under the condition a 3 = a 1 + 1 in (1.1). Thus, there has been much research and many results on stationary spike solutions with peaks on boundaries for (1.1), but we do not know a result on the dynamics of spikes along boundaries.
There have been several studies of the dynamics of solutions along boundaries for other models. Studies [1][2][3][4][5], dealt with the mass-conserving Allen-Cahn equations and/or the Cahn-Hilliard equations, and showed that small bubble solutions move along boundaries toward a boundary point with maximal mean curvature. This was proven using variational structures of systems together with other techniques.
On the other hand, many important examples of reaction-diffusion systems, such as the Gierer-Meinhard model and the Gray-Scott model, do not have such variational structures. But it is strongly expected that boundary spike solutions move toward points on boundaries with maximal mean curvature because many types of stable boundary spike solutions with peaks located at points with maximal mean curvature have been constructed.
In this paper, we give a general criteria necessary to study the dynamics and interactions of boundary spikes in a domain ⊂ R 2 , for general types of reaction-diffusion systems, and without assuming any variational structures. As one application of our results, we show the movement of a boundary spike solution for (1.1) toward a point on the boundary with locally maximal curvature. We also show the repulsive interaction between two boundary spikes and hence the existence of stable stationary solutions with two peaks in the neighborhood of a point with maximal curvature.
Here we note that there is a simplified version, called a shadow system, for the Gierer-Meinhardt model. It has a boundary spike solution, and a similar movement along the boundary is observed. But the treatment for the model is rather different from the one in this paper because the shadow system has nonlocal terms. This is reported in [7]. Now, we shall briefly discuss the results we present in this paper. Let be a bounded domain in R 2 , and we assume that the boundary ∂ is a sufficiently smooth closed curve given by { (s) ∈ R 2 ; 0 ≤ s ≤ s 0 , (0) = (s 0 )}, where s is the arc-length parameter of ∂ . Then we can take a tubular neighborhood of ∂ as x = (x, y) = (s) + zν(s), where ν = ν(s) is the inward normal unit vector of ∂ at (s). Define (x) and Z (x) by the functions satisfying x = ( (x))+Z (x)ν( (x)), and let κ = κ(s) be the curvature of ∂ at (s) measured in the direction of ν. We consider general types of reaction-diffusion systems: with the Neumann boundary condition. Here, First we consider with the Neumann boundary condition, where R 2 + := {ζ := (l, μ) ∈ R 2 ; −∞ < l < ∞, μ > 0}. Let 0 := (0, · · · , 0) ∈ R N . We assume as follows: (H1) F(0) = 0 is satisfied and (1.3) has a stationary radially symmetric solution, say S = S(r ), on the boundary ∂ R 2 + with asymptotic form S(r ) → 1 √ r e −αr a, where r = l 2 + μ 2 , α > 0 and a ∈ R N .
We assume the stability of 0 and S for (1.3). (H2) S(r ) is stable in the linearized sense, that is, the spectral set I (A) of A is I (A) ⊂ I 0 ∪ I 1 , where I 0 := {0} and I 1 ⊂ {Reλ < −γ 0 } for γ 0 > 0, and 0 is simple. (H2)' 0 is a stable equilibrium for (1.3) in the linearized sense. That is, the spectral set Note that A∂ l S = 0. Hereinafter we denote ∂ l S(r (l, μ)) simply by ∂ l S for r = l 2 + μ 2 . Other cases, such as when r is a different function, are treated similarly, while we denote d S dr by S r . Let A * be the adjoint operator of A, and φ * (x) be the eigenfunction satisfying We note that ∂ l S = cos θ S r = l r S r and that φ * is also given by φ * (ζ ) = ∂ l * = cos θ * r = l r * r for a radially symmetric function * (r ) under suitable conditions, where l = r cos θ and μ = r sin θ, which will be mentioned in the next section. Then the normalization ∂ l S, φ * 2 = π 2 implies that S r , * r R = 1, where U, V R := ∞ 0 r U, V dr for radially symmetric functions U = U (r ) and V = V (r ) ∈ R N . Now, coming back to the original problem (1.2) in , we show the following: Let Then the solution U (t, x) of (1.2) keeps close to S(r (x, h(t))/δ), and h(t) is governed by the dynamicsḣ whereḣ denotes dh dt . These results are proved by using invariant manifold theory (e.g. [5]), while we need to obtain explicit estimates of several objects, such as spectrum and resolvent, for (1.2). The above results are applied to the Gierer-Meinhardt model of the form where τ and a are nonnegative constants, and the a j are the same ones as (1.1). If both τ and a are positive or τ = a = 0 in (1.5), the case has been treated on the whole R 2 space in [8], and, by using the results, the constant M 0 is shown to be positive. This shows that h(t) approaches the point of the maximal κ(s), that is, the point with the maximal curvature of ∂ (Fig. 2). Thus, our results give the well-known results for a stable boundary spike solution of the Gierer-Meinhardt model [9,[11][12][13]16] from the viewpoint of dynamics. We can also show the existence of a stable stationary solution with two peaks in the neighborhood of a point with maximal curvature of ∂ by using the repulsive interaction (Fig. 3). The idea was presented in [8].
In this paper, only the two-dimensional case is considered. All the arguments and techniques can be applied to the higher dimensional cases, too but we do not do it here.

Main results
and ν(s) as periodic functions of s with period s 0 . We fix N 1 > 0 and represent = Hereinafter, c, c j , c j , and γ j denote general positive constants independent of δ. Let χ 0 (x) and In the tubular neighborhood 1 , define the coordinates s = (x) and Here, we extend r (x; h) to the whole domain so as to satisfy We add the following assumption. Let X R := {U = U (r ); U 2 R := U, U R < ∞} be a set of the radially symmetric functions, and define A R U := D(U rr + 1 and (1.4) uniformly for any t > 0 and sufficiently small δ > 0.
Let us consider the reduced ODE of (1.4) Next we give the results for the movement of multispike solutions on the boundary.
as long as

Remark 2.1
The restriction of the range of h 1 and h 2 is not necessary because we may consider h 1 and h 2 with mod s 0 . However, the statement of Theorem 2.2 then becomes complicated, and so, for simplicity, we restrict the range in the theorem. Consider the reduced system of (2.2) Suppose that M 0 , M 1 are positive and that h * is a stable equilibrium of (2.1) in the linearized sense. Then we can check that (2.3) has a linearly stable equilibrium, say The proof is given in Sect. 6.

Corollary 2.2 Under the above assumptions, there exists a stable stationary solution
has a profile with two peaks in the neighborhood of a boundary point with maximal curvature.

Corollary 2.3
For (1.5), suppose that both τ and a are positive, or both τ and a are equal to zero. If a 3 = 2 and 1 < a 1 < 3, or a 3 = a 1 + 1 for 1 < a 1 < ∞, then the constant M 0 is positive. That is, a spike on the boundary moves toward a point with maximal curvature. Moreover, M 1 is also positive, which implies the existence of a stable stationary solution of (1.5) with two peaks in the neighborhood of a point with maximal curvature, as stated in Corollary 2.2.

Formal derivation of (1.4)
In this section, we formally derive the ordinary differential equation (ODE) (1.4).
, and δμ := z. Since δ is sufficiently small, we can regard 1 as approximately R 2 Although K δ is defined only in 1 , we may assume it is appropriately extended in R 2 + , e.g. by multiplying K δ by cut-off functions disappearing outside of 1 and for sufficiently large μ. Then (3.1) is written as where l = r cos θ, μ = r sin θ, and E ⊥ : H 0 = 0 is easily shown. Considering terms of order δ, we have Taking the inner product of (3.4) with φ * (l, μ) = cos θ * r (r ), we have Proof Since ∂ μ S = sin θ S r and ∂ 2 l S = sin 2 θ r S r + cos 2 θ S rr , the direct calculation of the right-hand side of (3.5) gives H 1 = 0.
In order to obtain H 2 , we now consider terms of order δ 2 of (3.3). Then we have (3.7) Since V 1 ∈ E ⊥ is a unique solution of 0 = AV 1 + K 1 S and K 1 S is even with respect to l, V 1 = V 1 (l, z) is also even with respect to l.
. The right-hand side is directly calculated as  (x; h)) in , and X := C uni f ( ) with the sup-norm · .
First, we consider L(h) in 1 := \ 0 . In 1 , L(h) is expressed by using the coordinate (l, μ) of the tubular neighborhood as with the Neumann boundary condition at μ = 0 and the periodic boundary condition with respect to l. Here we note that the estimate δ K δ U = O( √ δ) U C 2 (I δ ) holds. Hence we appropriately extend the operator AU + δ K δ U to the one in R 2 + by extending the operator δ K δ U to R 2 + satisfying this estimate, which can be done by multiplying K δ by cut-off functions disappearing outside of I δ .
where ρ(A) denotes the resolvent set of A. Now we need the following proposition.
The proof of this proposition will be shown in the appendix.

We can write
. We may assume B 2 (h) is defined in with the same estimate. Then from (4.2), it follows in is invertible. Thus we find that . Let X ω be the fractional powered space of X for 1/2 < ω < 1. Then we note that ∇U ≤ c U ω for c > 0, where · ω is the norm of X ω . Let Q(h) := for 0 ≤ h ≤ s 0 (periodic with respect to h). This is proved in Appendices.
is a function of the form E(r (x; h)/δ, θ ) for some function E(r, θ) decaying exponentially with respect to r. Hence we can write h t = G 1 (W, h) by the calculation in Sect. 3, we have On the other hand, operating R(h) on (4.5), we have Thus, (4.5) is written as where L(h) : . Moreover it follows that for W, V ∈ W (D 1 ), Combining the above estimates and (4.4), we can show the existence of an attractive invariant manifold M : for an appropriately taken D 1 , D 2 in a way quite similar to [6]. That is, the solution U (t, x) of (1.2) is given by which completes the proof of the theorem.

Proof of Theorem 2.2
If h := h 2 − h 1 > βδ| log δ| for a large β > 0, then we have δ h e −αh/δ ≤ O(δ 4 ), which trivially implies the equation (2.2). Hence it suffices to consider the case when The rest of the proof is completed by combining the proofs of Theorem 2.1 [6,8].
In fact, we can show the existence of an exponentially attractive local invariant man-

Proof of Corollary 2.2
We omit the details here, but the proof is quite similar to the proof of Corollary 2.1. We substitute S(x; h 1 , h 2 ) + σ * (h 1 , h 2 )(x) into the equations ∂ t h 1 and ∂ t h 2 and use the implicit function theorem.

Proof of Corollary 2.3
The positivity of the constant M 1 was proved in [8]. Now we calculate the value of the constant M 0 . Under the assumptions of the corollary, there exists a spike solution S of (1.5) of the form S(r ) = (U (r/ε), V (r )) in R 2 + , where r := l 2 + μ 2 , and functions U (ζ ) and V (r ) are exponentially decaying positive functions. The function * (r ) in the definition of the constant M 0 is given by as ε ↓ 0 by the result of [8]. Hence the approximate value of M 0 is calculated as Assumptions (H1)-(H3) are also discussed in [8], but on the whole R 2 space. Then (H1)-(H3) in this paper are easily checked by the restriction to R 2 + . In fact, the semisimpleness of the 0 eigenvalue in R 2 [8] directly leads to (H2) and (H3).

Analysis of ODE (2.3)
In this section, we give the proof of the following theorem. Proof With the loss of generality, we may assume (0) ∈ ∂ is a point of maximal curvature. Equilibria of (2.3) satisfy where p = p 2 − p 1 . Equation (6.1) leads to Since p 1 , p 2 are small, we can expand κ s ( p 1 ) and κ s ( p 2 ) as Since p is also small, p 1 and p 2 can be expanded respectively as , for some constants a 1 and a 2 . Therefore (6.3) is rewritten as Thus (6.2) is rewritten as Then a 1 and a 2 are determined by and hence Here, holds by (6.1) and (6.3). Substituting (6.4) into (6.5), it follows that and hence Specially, the lowest-order part of (6.6) is Proposition 6.1 Equation (6.7) has a unique solution.