Multi-Spike Solutions to the Fractional Gierer-Meinhardt System in a One-Dimensional Domain
MMULTI-SPIKE SOLUTIONS TO THE FRACTIONAL GIERER-MEINHARDT SYSTEM IN AONE-DIMENSIONAL DOMAIN
DANIEL GOMEZ, JUN-CHENG WEI, AND WEN YANGA
BSTRACT . In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions. In particular we rigorously prove the existence of sym-metric and asymmetric two-spike solutions using a Lyapunov-Schmidt reduction. The linear stability of thesetwo-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain 2 × N -spike solutions using the method ofmatched asymptotic expansions. In addition, we explicitly consider examples of one- and two-spike solutionsfor which we numerically calculate their relevant existence and stability thresholds. By considering a one-spikesolution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectivelydestabilize or stabilize a single spike solution with respect to oscillatory instabilities. Furthermore, when consid-ering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutionsexist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expandedwith the introduction of fractional inhibitor diffusivity. However our calculations indicate that asymmetric two-spike solutions are always linearly unstable. Keywords : Gierer-Meinhardt system; eigenvalue; stability; fractional laplacian, localized solutions.
1. I
NTRODUCTION
The Gierer-Meinhardt (GM) model is a prototypical activator-inhibitor reaction-diffusion system thathas, since its introduction by Gierer and Meinhardt in 1972 gierer_1972 [6], been the focus of numerous mathematicalstudies. In the singularly perturbed limit for which the activator has an asymptotically small diffusivity theGM model is known to exhibit localized solutions in which the activator concentrates at a discrete collectionof points and is otherwise exponentially small. The analysis, both rigorous and formal, of the existence,structure, and linear stability of such localized solutions has been the focus of numerous studies over thelast two decades (see the book wei_2014_book [29]). The GM model in a one-dimensional domain has been particularlywell studied using both rigorous PDE methods wei_1998,wei_2007_existence [25, 28] as well as formal asymptotic methods iron_2001,ward_2002_asymmetric [13, 23]. Morerecent extensions to the classical one-dimensional GM model have considered the effects of precursors winter_2009,kolokolnikov_2020 [31, 14], bulk-membrane-coupling gomez_2019 [8], and anomalous diffusion nec_2012_sub,nec_2012_levi,wei_2019_multi_bump [19, 18, 30]. It is the latter of these extensionswhich motivates the following paper which focuses on extending the results obtained in nec_2012_levi,wei_2019_multi_bump [18, 30] for the fractional one-dimensional GM model.The analysis of localized solutions to the GM model fits more broadly into the study of pattern for-mation in reaction-diffusion systems. Such reaction-diffusion systems have widespread applicability inthe modelling of biological phenomena for which distinct agents diffuse while simultaneously undergo-ing prescribed reaction kinetics (see classic textbook by Murray murray_2003 [17]). While these models have typicallyassumed a normal (or Brownian) diffusion process for which the mean-squared-displacement (MSD) isproportional to the elapsed time, a growing body of literature has considered the alternative of anomalousdiffusion which may be better suited for biological processes in complex environments metzler_2004,oliveira_2019,reverey_2015 [16, 20, 21] (see also§7.1 in bressloff_2014 [1]). In contrast to normal diffusion, for anomalous diffusion the MSD and time are related by thepower law MSD ∝ ( time ) α where an exponent satisfying α > α < superdiffusion or sub-diffusion respectively. Studies of reaction-diffusion systems with subdiffusion and superdiffusion suggestthat anomalous diffusion can have a pronounced impact on pattern formation (see golovin_2008 [7] and the referencestherein). In particular studies have shown that both superdiffusion and subdiffusion can reduce the thresh-old for Turing instabilities when compared to the same systems with normal diffusion henry_2005,golovin_2008 [11, 7]. Likewiseit has been shown that the Hopf bifurcation threshold for spike solutions to the GM model with normaldiffusion for the inhibitor and superdiffusion, mainly with L´evy flights, for the activator is decreased nec_2012_levi [18] a r X i v : . [ n li n . PS ] F e b hereas it is increased in the case of subdiffusion for the inhibitor and normal diffusion for the activator nec_2012_sub [19].In this paper we consider the existence and stability of localized multi-spike solutions to the periodicone-dimensional GM model with L´evy flights for both the activator and the inhibitor. In particular weconsider the fractional Gierer-Meinhardt system u t + ε s ( − ∆ ) s u + u − u v =
0, for x ∈ ( −
1, 1 ) , τ v t + D ( − ∆ ) s v + v − u =
0, for x ∈ ( −
1, 1 ) , u ( x ) = u ( x + ) , v ( x ) = v ( x + ) , for x ∈ R , (1.1) where 0 < ε (cid:28) < D < ∞ and τ ≥ ε . We assume the exponentssatisfy 1/4 < s < < s <
1. The (nonlocal) fractional Laplacian ( − ∆ ) s replaces the classicalLaplacian as the infinitesimal generator of the underlying L´evy process for s < ( − ∆ ) s φ ( x ) ≡ C s (cid:90) ∞ − ∞ φ ( x ) − φ ( ¯ x ) | x − ¯ x | + s d ¯ x = C s (cid:90) − [ φ ( x ) − φ ( ¯ x )] K s ( x − ¯ x ) d ¯ x , (1.2a) eq:frac_lap_def_1 where C s ≡ s s Γ ( s + ) √ π Γ ( − s ) , K s ( z ) ≡ | z | + s + ∞ ∑ j = (cid:18) | z + j | + s + | z − j | + s (cid:19) , (1.2b) eq:frac_lap_def_2 and for which the second equality in ( eq:frac_lap_def_1eq:frac_lap_def_1 φ ( x ) . We remark that the system( nec_2012_levi [18] with the primary difference being that we consider theeffects of L´evy flights for both the activator and the inhibitor.Before outlining the structure of this paper we outline our contributions as follows. Using a Lyapunov-Schmidt type reduction we rigorously prove the existence of symmetric and asymmetric two-spike steadysolutions of ( ε s ( − ∆ ) s u + u − u v =
0, for x ∈ ( −
1, 1 ) , D ( − ∆ ) s v + v − u =
0, for x ∈ ( −
1, 1 ) , u ( x ) = u ( x + ) , v ( x ) = v ( x + ) , for x ∈ R . (1.3) and determine their linear stability by considering the spectrum of certain 2 × N -spike quasi-equilibrium solutionsand derive a system of ordinary differential equations governing their slow dynamics. We furthermoreillustrate the effects of anomalous diffusion on the stability of one- and two-spike solutions by calculatingthresholds for oscillatory and competition instabilities. In particular our results indicate that L´evy flights forthe activator and inhibitor have, respectively, a destabilizing and stabilizing effect on the stability of singlespike solutions. On the other hand we demonstrate that the stability of symmetric two-spike solutions withrespect to competition instabilities is independent of s and is stabilized when the inhibitor undergoes L´evyflights. Finally, we show that asymmetric two-spike solutions are always linearly unstable with respect tocompetition instabilities.The remainder of this paper is organized as follows. In § sec:main-resultssec:main-results sec:prelimsec:prelim sec:formal-resultssec:formal-results sec:proof-existencesec:proof-existence sec:proof-stabilitysec:proof-stability sec:conclusionsec:conclusion AIN RESULTS : E
XISTENCE AND S TABILITY sec:main-results
In this section we state the main results of this paper, which include the existence of two spike solutions(symmetric and asymmetric) to the steady problem of the fractional Gierer-Meinhardt system and their tability. Instead of studying the system ( u ( x ) by c ε u ( x ) and v ( x ) by c ε v ( x ) , and introducethe scaling x = ε y for the first equation of ( ( − ∆ ) sy u + u − u v =
0, for y ∈ (cid:16) − ε , ε (cid:17) , D ( − ∆ ) s v + v − c ε u =
0, for x ∈ ( −
1, 1 ) , u ( ε y ) = u ( ε y + ) , v ( x ) = v ( x + ) , for x , y ∈ R , (2.1) with c ε = (cid:18) ε (cid:90) R w ( y ) dy (cid:19) − and w being the unique solution of ( − ∆ ) s w + w − w = w ( x ) = w ( − x ) . (2.2) From now on, we shall focus on equation ( z ∈ ( −
1, 1 ) , let G D ( x , z ) be the function satisfying (cid:40) D ( − ∆ ) s G D ( x , z ) + G D ( x , z ) = δ ( x − z ) , for x ∈ ( −
1, 1 ) , G D ( x , z ) = G D ( x + z ) , for x ∈ R , (2.3) having the Fourier series expansion G D ( x , z ) = ∞ ∑ (cid:96) = − ∞ e i (cid:96) π ( x − z ) + D ( (cid:96) π ) s = + ∞ ∑ (cid:96) = cos ( (cid:96) π ( x − z )) + D ( (cid:96) π ) s .Let − < p < p < ( −
1, 1 ) where the spikes concentrate. We introduce several matricesfor later use. For p = ( p , p ) ∈ ( −
1, 1 ) we let G D be the 2 × ( G D ) ij = G D ( p i , p j ) . (2.4) Let us denote ∂∂ p i as ∇ p i . When i (cid:54) = j , we can define ∇ p i G D ( p i , p j ) in the classical way, while if i = j , since G D ( x , x ) is a constant due to the periodic boundary condition, we have ∇ p i G D ( p i , p i ) =
0. Next, we definethe matrix associated with the first and second derivatives of G as follows: ∇G D ( p ) = ( ∇ p i G D ( p i , p j )) , ∇ G D ( p ) = ( ∇ p i ∇ p j G D ( p i , p j )) . (2.5) We make the following two assumptions.(H1) There exists a solution ( ˆ ξ , ˆ ξ ) of the following equation ∑ j = G D ( p i , p j )( ˆ ξ j ) = ˆ ξ i , i =
1, 2. (2.6) (H2) / ∈ λ ( B ) , where λ ( B ) is the set of eigenvalues of the 2 × B with entries ( B ) ij = G D ( p i , p j ) ˆ ξ j (2.7) By the assumption ( H ) and the implicit function theorem, for p = ( p , p ) near p = ( p , p ) , thereexists a unique solution ˆ ξ ( p ) = ( ˆ ξ ( p ) , ˆ ξ ( p )) for the following equation ∑ j = G D ( p i , p j ) ˆ ξ j = ˆ ξ i , i =
1, 2. (2.8)
We define the following vector field: F ( p ) : = ( F ( p ) , F ( p )) , here F i ( p ) = ∑ j = ∇ p i G D ( p i , p j ) ˆ ξ j = ∑ j (cid:54) = i ∇ p i G D ( p i , p j ) ˆ ξ j , i =
1, 2. (2.9)
Set M ( p ) = ˆ ξ − i ∇ p j F i ( p ) . (2.10) The final assumption concerns the vector field F ( p ) .(H3) We assume that at p = ( p , p ) : F ( p ) = ( M ( p )) =
1. (2.11)
Next, let us calculate M ( p ) . Particularly, we shall show that it admits a zero eigenvalue. To computethe matrix M ( p ) , we have to derive the derivatives of ˆ ξ . It is easy to see that ˆ ξ ( p ) is C in p and from ( ∇ p j ˆ ξ i = ∑ l = G D ( p i , p l ) ˆ ξ l ∇ p j ˆ ξ l + ∑ l = ∂∂ p j G D ( p i , p l ) ˆ ξ l = ∑ l = G D ( p i , p l ) ˆ ξ l ∇ p j ˆ ξ l + ∇ p j G D ( p i , p j ) ˆ ξ j , if i (cid:54) = j ,2 ∑ l = G D ( p i , p l ) ˆ ξ l ∇ p j ˆ ξ l + ∑ l = ∂∂ p j G D ( p i , p l ) ˆ ξ l , if i = j , (2.12) where we used ∂ p i G D ( p i , p i ) =
0. Therefore, if we denote the matrix ∇ ξ = ( ∇ p j ˆ ξ i ) , (2.13) we have ∇ ξ ( p ) = ( I − G D H ) − ( ∇G D ) T H + O ( ∑ j = | F j ( p ) | ) , (2.14) where a superscript T denotes the transpose and where H is given by H ( p ) = (cid:0) ˆ ξ i ( p ) δ ij (cid:1) . (2.15) Let Q = ( q ij ) = ( ∇ p i ∇ p j G D ( p , p ) ∑ l (cid:54) = i ˆ ξ l ˆ ξ i δ ij ) . (2.16) We can compute M ( p ) by using ( M ( p ) = H − ( ∇ G D + Q ) H + H − ∇G D H ( I − G D H ) − ( ∇G D ) T H , (2.17) where A T means the transpose of A . To simplify our notation, we introduce the following matrices: P = ( I − G D H ) − . (2.18) Using ( M ( p ) as the following M ( p ) = ( ˆ ξ ) − ∇ p ∇ p G D ( p , p )( ˆ ξ ) ( ˆ ξ ) − ∇ p ∇ p G D ( p , p )( ˆ ξ ) ( ˆ ξ ) − ∇ p ∇ p G D ( p , p )( ˆ ξ ) ( ˆ ξ ) − ∇ p ∇ p G D ( p , p )( ˆ ξ ) . (2.19) It is easy to see that the summation of both rows is zero, thus M ( p ) is singular and admits a zero eigen-value. While the left non-zero eigenvalue can be represented as follows λ M ( p ) = ( ˆ ξ ) − ∇ p ∇ p G D ( p , p )( ˆ ξ ) + ( ˆ ξ ) − ∇ p ∇ p G D ( p , p )( ˆ ξ ) . (2.20) Our first result is the following: h1.exist Theorem 2.1.
Assume that (H1) and (H3) are satisfied. Then for ε (cid:28) problem ( has a 2-spike solution whichconcentrates at p ε , p ε . In addition,u ε ∼ c ε ∑ i = ˆ ξ i w (cid:18) x − p ε i ε (cid:19) and v ε ( p ε i ) ∼ c ε ∑ ˆ ξ i , i =
1, 2, and ( p ε , p ε ) → ( − , ) as ε → Remark:
In Theorem th1.existth1.exist symmetric and asymmetric two-spike solutions. In both cases the spike locations must satisfy ∇ p ε G D ( p ε , p ε ) = | p ε − p ε | =
1. As described in more detail in§ subsec:example-2subsec:example-2 ξ = ξ = G D (
0, 0 ) + G D (
1, 0 ) , and ξ = z G D (
0, 0 ) , ξ = z G D (
0, 0 ) ,for the symmetric and asymmetric cases respectively and where z and z are defined in terms of θ = G D (
1, 0 ) / G D (
0, 0 ) in ( eq:z1-z2eq:z1-z2 th1.existth1.exist th1.stability Theorem 2.2.
Assume that ε (cid:28) and let ( u ε , v ε ) be the solutions constructed in Theorem th1.existth1.exist B be defined in ( . (1) If min σ ∈ λ ( B ) σ > , then there exists τ such that ( u ε , v ε ) is linearly stable for ≤ τ < τ . (2) If min σ ∈ λ ( B ) σ < , then there exists τ such that ( u ε , v ε ) is linearly unstable stable for ≤ τ < τ . Remark : We shall prove Theorem th1.stabilityth1.stability B , which naturally appears in the study of large eigenvalue problem. Remark : To simplify the presentation, in the proof of Theorems th1.existth1.exist th1.stabilityth1.stability s = s = s . The arguments can be also applied for more general cases where s ∈ ( , 1 ) and s ∈ ( , 1 ) . 3. P RELIMINARIES sec:prelim
In this section we collect several key preliminary results needed for the existence and stability proofs in§ sec:proof-existencesec:proof-existence sec:proof-stabilitysec:proof-stability sec:formal-resultssec:formal-results w be the ground state solution satisfying (cid:40) ( − ∆ ) s w + w − w =
0, in R , w ( x ) → | x | → ∞ , (3.1) eq:core-problem we have the following result frank_2013_uniqueness [5] (also see Proposition 4.1 in wei_2019_multi_bump [30] and the references therein) pr3.1 Proposition 3.1.
Equation ( eq:core-problemeq:core-problem admits a positive, radially symmetric solution satisfying the following properties: (a) There exists a positive constant b s depending only on s such thatw ( x ) = b s | x | + s ( + o ( )) as | x | → ∞ . Moreover w (cid:48) ( x ) < for x > andw (cid:48) ( x ) = − ( + s ) b s x + s ( + o ( )) as x → ∞ .(b) Let L = ( − ∆ ) s + − w be the linearized operator. Then we have Ker ( L ) = span (cid:26) ∂ w ∂ x (cid:27) . c) Considering the following eigenvalue problem ( − ∆ ) s φ + φ − w φ + αφ = There is an unique positive eigenvalue α > app:nonlocalapp:nonlocal A. th3.stability Theorem 3.2.
Consider the following nonlocal eigenvalue problem ( − ∆ ) s φ + φ − w φ + γ (cid:82) R w φ dx (cid:82) R w dx w + αφ =
0. (3.2) (1) If γ < then there is a eigenvalue α to ( such that (cid:60) ( α ) > If γ > and s > , then for any nonzero eigenvalue α of ( , we have (cid:60) ( α ) ≤ − c < If γ (cid:54) = and α = , then φ = c ∂ x w for some constant c . In our application to the case when τ >
0, we have to deal with the situation when the coefficient γ is afunction of τα . Letting γ = γ ( τα ) be a complex function of τα let us suppose that γ ( ) ∈ R , | γ ( τα ) | ≤ C for α R ≥ τ ≥
0, (3.3) where C is a generic constant independent of τ , α . Then we have the following result. th3.2 Theorem 3.3.
Consider the following nonlocal eigenvalue problem ( − ∆ ) s φ + φ − w φ + γ ( τα ) (cid:82) R w φ dx (cid:82) R w dx w + αφ =
0, (3.4) where γ ( τα ) satisfies ( . Then there is a small number τ > such that for τ < τ , (1) if γ ( ) < then there is a positive eigenvalue to ( ; (2) if γ ( ) > and s > , then for any nonzero eigenvalue α of ( , we have (cid:60) ( α ) ≤ − c < Proof.
The above Theorem follows from Theorem th3.stabilityth3.stability α R ≥ < τ <
1, then | α | ≤ C , where C is a genericconstant (independent of τ ). In fact, multiplying ( φ - the conjugate of φ - and integrating by parts,we obtain that (cid:90) R ( | ( − ∆ ) s φ | + | φ | − w | φ | ) dx = − α (cid:90) R | φ | − γ ( τα ) (cid:82) R w φ dx (cid:82) R w dx (cid:90) R w φ dx . (3.5) From the imaginary part of ( | α I | ≤ C | γ ( τα ) | ,where α = α R + √− α I and C is a positive constant (independent of τ ). By assumption ( | γ ( τα ) | ≤ C and so | α I | ≤ C . Taking the real part of ( ≥ C (cid:90) R | φ | dx for some C ∈ R , (3.6) then we obtain that α R ≤ C where C is a positive constant (independent of τ > | α | isuniformly bounded and hence a perturbation argument gives the desired conclusion. (cid:3) We now consider the following system of linear operators L Φ : = ( − ∆ ) s Φ + Φ − w Φ + B (cid:18) (cid:90) R w Φ dx (cid:19) (cid:18) (cid:90) R w dx (cid:19) − w , (3.7) here B is given by ( Φ : = ( φ , φ ) T ∈ ( H s ( R )) . The conjugate operator of L under the scalarproduct in L ( R ) is L ∗ Ψ : = ( − ∆ ) s Ψ + Ψ − w Ψ + B T (cid:18) (cid:90) R w Ψ dx (cid:19) (cid:18) (cid:90) R w dx (cid:19) − w , (3.8) where Ψ : = ( ψ , ψ ) T ∈ ( H s ( R )) . We then have the following result. le3.1 Lemma 3.4.
Assume that ( H ) holds. Then Ker ( L ) = Ker ( L ∗ ) = X ⊕ X , (3.9) eq:L_L*_kernel where X = Span { w (cid:48) ( x ) } .Proof. We first prove Ker ( L ) ⊂ X ⊕ X . Suppose L Φ =
0. By the fact that G D is symmetry and H ( p ) is adiagonal matrix, we could diagonalize B . Let P − B P = J ,where P is an orthogonal matrix and J is diagonal form, i.e., J = (cid:18) σ σ (cid:19) with suitable real numbers σ i , i =
1, 2. Defining Φ = P ˜ Φ we have ( − ∆ ) s ˜ Φ + ˜ Φ − w ˜ Φ + (cid:18) (cid:90) R w dx (cid:19) − (cid:18) (cid:90) R w J ˜ Φ dx (cid:19) w =
0. (3.10)
For i =
1, 2 we look at the i -th equation of system ( ( − ∆ ) s ˜ Φ i + ˜ Φ i − w ˜ Φ i + σ i (cid:18) (cid:90) R w dx (cid:19) − (cid:18) (cid:90) R w ˜ Φ i dx (cid:19) w =
0. (3.11)
By Theorem th3.stabilityth3.stability ( H ) we know that 2 σ i (cid:54) =
1) ˜ Φ i ∈ X .We proceed similarly to prove Ker ( L ∗ ) ⊂ X ⊕ X . Using σ ( B ) = σ ( B T ) the i -th equation of the diago-nalized system is as follows ( − ∆ ) s ˜ Ψ i + ˜ Ψ i − w ˜ Ψ i + σ i (cid:18) (cid:90) R w dx (cid:19) − (cid:18) (cid:90) R w ˜ Ψ i dx (cid:19) w =
0. (3.12)
Multiplying the above equation by w and integrating over the real line, we obtain ( − σ i ) (cid:90) R w ˜ Ψ i =
0, (3.13) which together with the fact that 2 σ i (cid:54) = (cid:90) R w ˜ Ψ i = i =
1, 2.Thus all the nonlocal terms vanish and we have L ˜ Ψ i = i =
1, 2, which in turn implies that Ψ i ∈ X for i =
1, 2. On the other hand, it is obvious that X ⊕ X ⊂ Ker ( L ) and X ⊕ X ⊂ Ker ( L ∗ ) . Therefore, weconclude that ( eq:L_L*_kerneleq:L_L*_kernel (cid:3) le3.2 Lemma 3.5.
The operator L : ( H s ( R )) → ( L ( R )) is invertible if it is restricted as followsL : ( X ⊕ X ) ⊥ ∩ ( H s ( R )) → ( X ⊕ X ) ⊥ ∩ ( L ( R )) . Moreover, L − is bounded.Proof. This follows from the Fredholm Alternatives Theorem and Lemma le3.1le3.1 (cid:3)
Finally we study the eigenvalue problem (see ( L ) L Φ + α Φ =
0, (3.14) for which we have the following lemma. e3.3 Lemma 3.6.
Assume that all the eigenvalues of B are real. Then we have (1) If σ ∈ σ ( B ) σ > then for any nonzero eigenvalue of ( we must have (cid:60) ( α ) ≤ − c < If there exists σ ∈ σ ( B ) such that σ < , then there exists a positive eigenvalue of ( .Proof. We first prove (1). Let ( Φ , α ) satisfy ( σ ∈ σ ( B ) σ >
1. Suppose α R ≥ α (cid:54) = le3.2le3.2 ( − ∆ ) s Φ + Φ − w Φ + ( (cid:90) R w dx ) − ( (cid:90) R w J Φ ) w + α Φ =
0, (3.15) and the i -th equation of system ( ( − ∆ ) s Φ i + Φ i − w Φ i + σ i (cid:18) (cid:90) R w dx (cid:19) − (cid:18) (cid:90) R w Φ i (cid:19) w + α Φ i =
0. (3.16)
The first conclusion follows by Theorem th3.stabilityth3.stability σ i >
1. We conclude that either Φ = Φ = α ≤ − c <
0. Since Φ does not vanish and α <
0, thus (1) is proved.Next we prove (2) and assume that 2 σ i < σ i ∈ σ ( B ) . Then the equation corresponding to σ i becomes ( − ∆ ) s Φ i + Φ i − w Φ i + σ i (cid:18) (cid:90) R w (cid:19) − (cid:18) (cid:90) R w Φ i dx (cid:19) w + α Φ i = th3.stabilityth3.stability α > Φ such that L Φ + σ i (cid:18) (cid:90) R w dx (cid:19) − (cid:18) (cid:90) R w Φ dx (cid:19) w + α Φ =
0. (3.17)
Let us take Φ i = Φ and Φ j = j (cid:54) = i . Then ( Φ , α ) satisfies ( (cid:3)
4. F
ORMAL A NALYSIS OF N -S PIKE E QUILIBRIUM S OLUTIONS AND THEIR L INEAR S TABILITY sec:formal-results
Although the fractional Laplacian ( − ∆ ) s is nonlocal , the method of matched asymptotic expansions cannevertheless be used to construct leading order asymptotic approximations to equilibrium solutions of ( − < p < ... < p N < N ≥
1) are well separated in the sense that p + = O ( ) ,1 − p N = O ( ) , and | p i + − p i | = O ( ) for all i =
1, ..., N − eq:frac_lap_def_2eq:frac_lap_def_2 K s ( p i + ε y − p j − ε ¯ y ) = (cid:40) O ( ) , j (cid:54) = i , ε + s | y − ¯ y | + s + O ( ) , j = i , y , ¯ y = O ( ) .Moreover for any bounded and periodic function φ ( x ) such that φ ( x ) ∼ Φ ( y ) for x = p i + ε y and y = O ( )( − ∆ ) s φ ( x ) ∼ ε − s ( − ∆ ) s Φ + O ( ) , ( − ∆ ) s Φ ≡ C s (cid:90) ∞ − ∞ Φ ( y ) − Φ ( ¯ y ) | y − ¯ y | + s d ¯ y ,which effectively separates the inner region problems in the method of matched asymptotic expansions. Inthe remainder of this section we use the method of matched asymptotic expansions to formally constructmulti-spike equilibrium solutions to ( subsec:formal-equilibrium Multi-Spike Solutions and their Slow Dynamics.
With the separation of inner region problems asoutlined above, the construction of quasi-equilibrium solutions follows closely that for the classical casewhen s = s = iron_2001 [13]. In particular letting − < p < ... < p N < u ∼ ε − (cid:0) ξ i w s ( y ) + o ( ) (cid:1) , v ∼ ε − (cid:0) ξ i + o ( ) (cid:1) , for x = p i + ε y , y = O ( ) for each i =
1, ..., N where w s satisfies the core problem ( eq:core-problemeq:core-problem s = s and ξ i > − < x < eq:quasi-eq-sol u ( x ) ∼ ε − N ∑ i = ξ i w s ( ε − | x − p i | ) + o ( ε − ) , (4.1a) eq:quasi-eq-sol-u here the corrections due to the algebraic decay of the core solution don’t contribute until O ( ε s ) . More-over, in the sense of distributions we calculate the limit u → ε − ω s ∑ Nj = ξ j δ ( x − p j ) as ε → + from whichit follows that for all x such that | x − p i | (cid:29) ε for all i =
1, ..., N the inhibitor is given by v ∼ ε − ω s N ∑ j = ξ j G D ( x , p j ) + o ( ε − ) , ω s ≡ (cid:90) ∞ w s ( y ) dy , (4.1b) eq:quasi-eq-sol-v where G D ( · , · ) is the Green’s function satisfying ( s = s . Since v → ε − ( ξ i + o ( )) as x → p i weobtain the nonlinear algebraic system eq:quasi-eq ξξξ − ω s G D ξξξ =
0, (4.2a)where ξξξ = ( ξ , · · · , ξ N ) T , G D = ( G D ( p i , p j )) Ni , j = . (4.2b)When N = N = ξ = [ ω s G D ( p , p )] − .Given a fixed configuration − < p < ... < p N < eq:quasi-eqeq:quasi-eq ξ , ..., ξ N yielding quasi -equilibrium solution to ( eq:quasi-eq-soleq:quasi-eq-sol eq:quasi-eq-soleq:quasi-eq-sol O ( ) timescale the spike locations drift slowly over an O ( ε − ) timescaleaccording to the system of differential equations (see Appendix app:slow-dynamicsapp:slow-dynamics D for details) dp i dt = − ε κ s ξ − i ∑ j (cid:54) = i ξ j ∇ G D ( p i , p j ) , κ s ≡ (cid:82) ∞ − ∞ w s dy (cid:82) ∞ − ∞ w s dy (cid:82) ∞ − ∞ | dw s / dy | dy , (4.3) eq:slow-dynamics where ∇ denotes the derivative with respect to the first argument and we remark that this is to be solvedconcurrently with the algebraic system ( eq:quasi-eqeq:quasi-eq − < p < ... < p N < ∑ j (cid:54) = i ξ j ∇ G D ( p i , p j ) =
0, (4.4)for all i =
1, ..., N , then ( eq:quasi-eq-soleq:quasi-eq-sol equilibrium solution of ( th1.existth1.exist sec:proof-existencesec:proof-existence subsec:formal-stability Linear Stability of Multi-Spike Solutions.
We now consider the linear stability of the N -spike equi-librium solutions constructed above which we denote by u e and v e . Substituting u = u e + e λ t φ and v = v e + e λ t ψ where | φ | , | ψ | (cid:28) ε s ( − ∆ ) s φ + φ − v − e u e φ + v − e u e ψ + λφ = − < x <
1, (4.5a) eq:lin-stab-phi D ( − ∆ ) s ψ + ψ − u e φ + τλψ = − < x <
1, (4.5b) eq:lin-stab-psi where we assume in addition that both φ and ψ are 2-periodic. We focus first on the case where λ = O ( ) , the so-called large eigenvalues, and make a brief comment on the case of small eigenvalues for which λ = O ( ε ) at the end of this section. Proceeding with the method of matched asymptotic expansionsas in the previous section we deduce that φ ∼ φ i ( y ) + o ( ) when x = p i + ε y and y = O ( ) for each i =
1, ..., N . It follows that φ ∼ ∑ Nj = φ j ( ε − ( x − p j )) + o ( ) for all − < x < u e φ → ∑ Nj = ξ j (cid:82) ∞ − ∞ w s ( y ) φ j ( y ) dy δ ( x − p j ) as ε → + in the sense of distributions. Substituting this into ( eq:lin-stab-psieq:lin-stab-psi ψ ( x ) = N ∑ j = ξ j (cid:90) ∞ − ∞ w s ( y ) φ j ( y ) dyG λ D ( x , p j ) ,where G λ D ( x , z ) is the eigenvalue dependent Green’s function satisfying D ( − ∆ ) s G λ D + ( + τλ ) G λ D = δ ( x − z ) , − < x , z <
1, (4.6)with periodic boundary conditions. It follows that for x = p i + ε y equation ( eq:lin-stab-phieq:lin-stab-phi L φ i + w s N ∑ j = ξ j (cid:90) ∞ − ∞ w s ( y ) φ j ( y ) dyG λ D ( p i , p j ) + λφ i = or each i =
1, ..., N where L is the linear operator of Proposition pr3.1pr3.1 s = s . This system of equationsis conveniently rewritten as the system of NLEPs L Φ + w s (cid:82) ∞ − ∞ w s E λ Φ dy (cid:82) ∞ − ∞ w s dy + λ Φ =
0, (4.7a) eq:nlep-sys where Φ ≡ φ ( y ) ... φ N ( y ) , E λ = ˆ ξ G λ D ( p , p ) · · · ˆ ξ N G λ D ( p , p N ) ... . . . ...ˆ ξ G λ D ( p N , p ) · · · ˆ ξ N G λ D ( p N , p N ) , ˆ ξ i = ω s ξ i . (4.7b) eq:nlep-sys-def Letting ppp λ k and χ λ k be the eigenpairs of E λ satisfying E λ ppp λ k = χ λ k ppp λ k for each k =
0, 1, ..., N − eq:nlep-syseq:nlep-sys Φ = Φ k ppp λ k to get the decoupled system of NLEPs L Φ k + χ λ k w s (cid:82) ∞ − ∞ w s Φ k dy (cid:82) ∞ − ∞ w s dy + λ Φ k =
0. (4.8) eq:nlep-scalar An N -spike equilibrium solution is linearly stable with respect to the large eigenvalues provided that alleigenvalues of ( eq:nlep-scalareq:nlep-scalar (cid:60) ( λ ) < k =
0, ..., N −
1. Finally, we remark that the NLEP ( eq:nlep-scalareq:nlep-scalar A k ( λ ) ≡ χ λ k + F s ( λ ) = F s ( λ ) ≡ (cid:82) ∞ − ∞ w s ( L + λ ) − w s dy (cid:82) ∞ − ∞ w s dy , (4.9) eq:nlep-algebraic which will in general require the numerical evaluation of F s ( λ ) .The stability of a multi-spike equilibrium solution with respect to the small eigenvalues is closely relatedto the slow dynamics given by ( eq:slow-dynamicseq:slow-dynamics O ( ) timescale, the small eigenvalues are linked to the linear stability of thespike pattern with respect to the slow dynamics ( eq:slow-dynamicseq:slow-dynamics O ( ε − ) timescale. In thecase of two-spike equilibrium solutions Theorem th1.stabilityth1.stability sec:proof-stabilitysec:proof-stability subsec:example-1 Example: Symmetric N -Spike Solutions. By appropriately choosing the spike locations we can ex-plicitly calculate an N -spike solution that is symmetric in the sense that the local profile of each spike isidentical. Specifically, letting p i = − + N − ( i − ) , ξ i = ξ c ≡ (cid:18) ω s N − ∑ k = G D ( N − k , 0 ) (cid:19) − , for all i =
1, ..., N , (4.10)it is then straightforward to show that ( eq:quasi-eqeq:quasi-eq eq:slow-dynamicseq:slow-dynamics E λ defined by ( eq:nlep-sys-defeq:nlep-sys-def ppp λ k ≡ (cid:18) e i π kN , · · · , e i π ( N − ) kN (cid:19) T , χ λ k = ∑ N − j = H λ j e i π jkN ∑ N − j = H j , H λ k ≡ G λ D ( N − k , 0 ) (4.11)for each k =
0, ..., N − ward_2003 [24], we seek con-ditions under which ( eq:nlep-algebraiceq:nlep-algebraic k = (cid:60) ( λ ) > C R = { i λ I | − R ≤ λ I ≤ R } ∪ { Re i θ | − π /2 ≤ θ ≤ π /2 } traversed counterclockwise and noting that | χ λ | > .4 0.6 0.8 1.0 s h and h for D h ( s ) h ( s ) ( A ) fig:hopf_infty D h ( D , s , s ) s | s ( B ) fig:hopf_thresh u (0, t ) t ( C ) fig:hopf_examples F IGURE
1. Hopf bifurcation threshold for a one-spike solution in (A) the shadow limit D → ∞ , and (B) for finite D > s = s = ε = D =
2, and τ = ( s , s ) = ( ) , ( ) , and ( ) for the top, middle,and bottom plots respectively. fig:hopf_thresholds for all λ with (cid:60) ( λ ) ≥ F s ( λ ) has a simple pole on the positive half-plane corresponding to theprincipal eigenvalue of L we find that the number Z of unstable solutions to ( eq:nlep-algebraiceq:nlep-algebraic π i lim R → ∞ (cid:73) C R d A / d λ A d λ = Z − χ λ ∼ O ( λ s − ) and therefore A ( λ ) ∼ O ( λ − s ) for | λ | (cid:29) A over the semi-circle part of the contour is (cid:0) − s (cid:1) π from which it follows that Z = − s − π arg A ( i λ I ) (cid:12)(cid:12) ∞ λ I = .We note that arg A ( i λ I ) → ( − s ) as λ I → ∞ whereas A ( ) = − L − w s = − w s . Furthermorenumerical evidence suggests that (cid:60)A ( i λ I ) is monotone increasing in λ I and so there exists a unique value0 < λ (cid:63) I < ∞ such that (cid:60)A ( i λ (cid:63) I ) =
0. It then follows that either Z = Z = (cid:61)A ( i λ (cid:63) I ) > (cid:61)A ( i λ (cid:63) I ) < A ( i λ I ) = τ = τ h ( D , s , s ) and λ I = λ h ( D , s , s ) . By first considering the limit D → ∞ for which χ λ → ( + τλ ) − we calculate the Hopf bifurcation threshold τ ∞ h ( s ) and accompany-ing eigenvalue λ ∞ h ( s ) , both of which are independent of s and are plotted in Figure fig:hopf_inftyfig:hopf_infty τ ∞ h is monotone increasing with s and therefore the introduction of L´evy flights for the acti-vator destabilizes the single spike solution as previously observed in nec_2012_levi [18]. This behaviour persists for finitevalues of D > s andtherefore introducing L´evy flights for the inhibitor stabilizes the single spike solution. This behaviour isillustrated in Figure fig:hopf_threshfig:hopf_thresh
1b for which we plot the Hopf bifurcation threshold as a function of D for select valuesof s and s . We remark in addition that the Hopf bifurcation’s dependence on the inhibitor diffusivity D remains qualitative unchanged with the introduction of L´evy flights: τ h ( D , s , s ) decreases monotonicallywith D .To illustrate the above observations, mainly the destabilization (resp. stabilization) of the single-spikesolution with decreasing s (resp. s ), we numerically solve ( p = ε = D =
2, and τ = ( s , s ) = ( ) , ( ) ,and ( ) . See Appendix app:numericalapp:numerical B for details on the numerical calculation. From the numerically calculatedthreshold we find τ h (
2, 0.8, 0.7 ) ≈ τ h (
2, 0.8, 0.9 ) ≈ τ h (
2, 0.4, 0.7 ) ≈ τ h = u ( t ) in Figure fig:hopf_examplesfig:hopf_examples
1c support these predictions. ubsec:example-2 Example: Symmetric and Asymmetric Two-Spike Solutions.
When s = s = asymmetric solutions consisting of spikeswith different heights ward_2002_asymmetric,wei_2007_existence [23, 28]. The gluing method for constructing such asymmetric N -spike solutions reliescrucially on the locality of the classical Laplace operator. However, since the fractional Laplace operator ( − ∆ ) s is nonlocal for s < eq:quasi-eqeq:quasi-eq N = eq:quasi-eqeq:quasi-eq − < p < p < eq:slow-dynamicseq:slow-dynamics d ( p − p ) dt = − ε κ s ξ + ξ ξ ξ G (cid:48) D ( | p − p | , 0 ) ,where G (cid:48) D ( z , 0 ) = dG D ( z , 0 ) / dz . By numerically evaluating G D ( z , 0 ) (see Appendix app:greens-funcapp:greens-func C) we observe that itis monotone decreasing for 0 < z <
1, attains its global minimum at z =
1, and is monotone increasingfor 1 < z <
2. Any stationary solution of ( eq:slow-dynamicseq:slow-dynamics p − p = z ≡ ω s G D (
0, 0 ) ξ , z ≡ ω s G D (
0, 0 ) ξ , θ ≡ G D (
1, 0 ) / G D (
0, 0 ) ,the algebraic system ( eq:quasi-eqeq:quasi-eq z − z − θ z = z − θ z − z =
0. (4.12) eq:z-system
This system always admits the symmetric solution for which z = z = z c where z c = ( + θ ) − recoveringthe result from the previous example for N =
2. One the other hand, assuming z (cid:54) = z we may subtractthe first equation from the second to obtain z = ( − θ ) − − z . Substituting this expression for z backinto the first equation in ( eq:z-systemeq:z-system z which is readily solved to obtain z = − θ (cid:18) + (cid:114) − θ + θ (cid:19) , z = − θ (cid:18) − (cid:114) − θ + θ (cid:19) . (4.13) eq:z1-z2 We immediately deduce that an asymmetric two-spike solution exists if an only if θ < fig:two-spike-structfig:two-spike-struct θ depending only on D and the inhibitor exponent s .We conclude this section by considering the linear stability of two-spike solutions with respect to com-petition instabilities, neglecting the possibility of Hopf bifurcations by assuming that τ is sufficiently small.In view of ( eq:nlep-scalareq:nlep-scalar th3.stabilityth3.stability E = (cid:18) z θ z θ z z (cid:19) .When z = z = z c it is easy to see that E has eigenvectors ppp = (
1, 1 ) and ppp = ( − ) with correspondingeigenvalues χ = χ = ( − θ ) / ( + θ ) . Since χ > th3.stabilityth3.stability k = k = χ < θ < z and z aregiven by ( eq:z1-z2eq:z1-z2 E are given by χ = − θ (cid:18) + (cid:115) θ − θ + + θ (cid:19) , χ = − θ (cid:18) − (cid:115) θ − θ + + θ (cid:19) ,from which we deduce that χ > χ < −√ < for all 0 < θ < th3.stabilityth3.stability k = fig:two-spike-structfig:two-spike-struct
2a we indicate the values of θ where the two-spike solution islinearly stable (resp. unstable) with respect to competition instabilities by solid (resp. dashed) curves. Bynumerically solving θ = D as a function of s we can calculate the competition instability threshold D = D ( s ) for the symmetric two-spike solution and this is shown in Figure fig:two-spike-existencefig:two-spike-existence fig:two-spike-examplesfig:two-spike-examples
2c we illustratethe onset of competition instabilities when s = ε = τ = s = .0 0.2 0.4 0.60.00.20.40.60.81.0 Two-Spike Solution Heights z c z z ( A ) fig:two-spike-struct s Competition Threshold D ( s ) ( B ) fig:two-spike-existence u ( 0.5, t ) and u (0.5, t ) t ( C ) fig:two-spike-examples F IGURE
2. (A) Bifurcation diagram showing the rescaled spike heights z i = ω s G D (
0, 0 ) ξ i versus θ . Solid (resp. dashed) lines indicate the resulting two-spike solution is linearlystable (resp. unstable) with respect to competition instabilities. (B) The competition insta-bility threshold for a symmetric two-spike solution. (C) Spike heights at x = − x = s = ε = τ = D = D ( s ) where s = D = D ( s ) . fig:two-spike D = × D ( s ) by performing full numerical simulations of ( app:numericalapp:numerical B for details).We remark that the accuracy of the leading order approximation to the competition instability calculatedabove grows increasingly inaccurate as s → ε >
0. Indeed, as described in moredetail in the derivation of the slow dynamics found in Appendix app:slow-dynamicsapp:slow-dynamics
D, the first order correction to the quasi-equilibrium solution is O ( ε s − ) and this tends to O ( ) as s → s = wei_2019_multi_bump [30]) and we anticipate that the method ofmatched asymptotic expansions will lead to an asymptotic expansion in powers of ν = −
1/ log ε as is oftenthe case for singularly perturbed reaction-diffusion systems in two-dimensions kolokolnikov_2009,chen_2011 [15, 2].5. R IGOROUS PROOF OF THE EXISTENCE RESULTS sec:proof-existence
In this section we shall prove the existence theorem, i.e., Theorem th1.existth1.exist th1.existth1.exist
Study of the Approximate Solutions.
Let − < p < p < ( H ) − ( H ) . Let ˆ ξ = ( ˆ ξ , ˆ ξ ) be the solution of ( p = ( p , p ) . We shall construct an approxi-mate solution to ( − < p < p < p = ( p , p ) ∈ B ε s − ( p ) . Set r =
110 min (cid:26) p +
1, 1 − p , 12 | p − p | (cid:27) and define a cut-off function χ ( x ) such that χ ( x ) = | x | < χ ( x ) = | x | >
2. Letting w i ( y ) = w (cid:16) y − p i ε (cid:17) χ (cid:18) ε y − p i r (cid:19) , (5.1) here w is the ground state solution of ( eq:core-problemeq:core-problem ( − ∆ ) sy w i ( y ) + w i ( y ) − w i ( y ) = h . o . t ., (5.2) where h . o . t . refers to terms of order ε + s in L (cid:16) − ε , ε (cid:17) . Let ˆ ξ ( p ) = ( ˆ ξ , ˆ ξ ) be defined as in ( H ) . Fix anyfunction u ∈ H s (cid:16) − ε , ε (cid:17) and define T [ u ] to be the solution of (cid:40) D ( − ∆ ) s T [ u ] + T [ u ] − c ε u = x ∈ ( −
1, 1 ) , T [ u ]( x ) = T [ u ]( x + ) , x ∈ R , (5.3) where c ε = (cid:18) ε (cid:90) R w ( y ) dy (cid:19) − . (5.4) Letting p ∈ B ε s − ( p ) we define w ε , p = ∑ i = ˆ ξ i w i ( y ) and using ( τ i : = T [ w ε , p ]( p i ) = ε c ε (cid:90) ε − ε G D ( p i , ε y ) w ε , p ( y ) dy = ε c ε ∑ j = ˆ ξ j (cid:90) ε − ε G D ( p i , ε y ) w j ( y ) dy = ε c ε ∑ j = ˆ ξ j (cid:18) G D ( p i , p j ) (cid:90) R w ( y ) dy (cid:19) + P i = ∑ j = G D ( p i , p j ) ˆ ξ j + P i , (5.5) where G D ( x , y ) is defined in ( P i is a number with order ε s − . Thus, we have obtained the followingsystem of equations: τ i = ∑ j = G D ( p i , p j ) ˆ ξ j + P i . (5.6) According to the assumption ( H ) - ( H ) and the implicit function theorem, we have the above equationhas a unique solution τ i = ˆ ξ i + ϑ i , i =
1, 2, ϑ i = O ( ε s − ) .Hence T [ w ε , p ]( p i ) = ˆ ξ i + O ( ε s − ) .Now for x = p i + ε z we calculate T [ w ε , p ]( x ) − T [ w ε , p ]( p i ) = c ε (cid:90) − [ G D ( x , ζ ) − G D ( p i , ζ )] w ε , p (cid:18) ζε (cid:19) d ζ = c ε ˆ ξ i (cid:90) − [ G D ( x , ζ ) − G D ( p i , ζ )] w i (cid:18) ζε (cid:19) d ζ + c ε ∑ j (cid:54) = i ˆ ξ j (cid:90) − [ G D ( x , ζ ) − G D ( p i , ζ )] w j (cid:18) ζε (cid:19) d ζ = c ε ˆ ξ i (cid:90) R [ G D ( ε y − ε z ) − G D ( ε y )] w ( y ) dy + c ε ∑ j (cid:54) = i ˆ ξ j (cid:90) − [ G D ( x , ζ ) − G D ( p i , ζ )] w j (cid:18) ζε (cid:19) d ζ + h . o . t . = P i ( z ) + ε ∑ j (cid:54) = i (cid:16) ˆ ξ j z ∇ p i G D ( p i , p j ) + O ( ε z ) (cid:17) + h . o . t ., (5.7) where P i ( z ) = c ε ˆ ξ i (cid:90) R [ G D ( ε y − ε z ) − G D ( ε y )] w ( y ) dy is an even function and of order ε s − .Next we define S [ u ] : = ( − ∆ ) sy u + u − u T [ u ] , (5.8) or which we calculate S [ w ε , p ]( y ) = ( − ∆ ) sy w ε , p + w ε , p − w ε , p T [ w ε , p ]= ∑ j = ˆ ξ j χ (cid:18) ε y − p j r (cid:19) ( − ∆ ) s w (cid:18) y − p j ε (cid:19) + ∑ j = ˆ ξ j w j − w ε , p T [ w ε , p ] + h . o . t . = (cid:34) ∑ j = ˆ ξ j w j − ( ∑ j = ˆ ξ j w j ) T [ w ε , p ] (cid:35) + h . o . t . = E + E + h . o . t . in L (cid:18) − ε , 1 ε (cid:19) , (5.9) where E = ∑ j = ˆ ξ j w j − ( ∑ j = ˆ ξ j w j ) T [ w ε , p ]( p i ) , and E = ( ∑ j = ˆ ξ j w j ) T [ w ε , p ]( p i ) − ( ∑ j = ˆ ξ j w j ) T [ w ε , p ]( x ) .Using ( E = ∑ j = ˆ ξ j w j − ( ∑ j = ˆ ξ j w j ) T [ w ε , p ]( p i ) = ∑ j = (cid:32) ˆ ξ j − ˆ ξ j ˆ ξ i + ϑ i (cid:33) w j = O ( ε s − ) ∑ j = ˆ ξ j w j ,and therefore (cid:107) E (cid:107) L ( − ε , ε ) = O ( ε s − ) . (5.10) In addition since x is close to p i we see that E can be decomposed into two parts: one part of order ε s − and symmetric in x − p i , and the other part of order ε . Next we calculate E = ∑ j = ( ˆ ξ j w j ) ( T [ w ε , p ]( p i )) ( T [ w ε , p ]( x ) − T [ w ε , p ]( p i )) (cid:32) + ∞ ∑ n = (cid:18) T [ w ε , p ]( p i ) − T [ w ε , p ]( x ) T [ w ε , p ]( p i ) (cid:19) n (cid:33) = ∑ j = ( ˆ ξ j w j ) ( T [ w ε , p ]( p i )) P i ( z ) (cid:32) + ∞ ∑ n = (cid:18) P i ( z ) T [ w ε , p ]( p i ) (cid:19) n (cid:33) + ε ∑ j = w j ∑ l (cid:54) = i ˆ ξ l z ∇ p i G ( p i , p l ) + h . o . t . = E + E + h . o . t ., (5.11) where E = O ( ε s − ) is symmetry in x − p i , i =
1, 2, and (cid:107) E (cid:107) L ( − ε , ε ) = O ( ε ) . (5.12) We have thus established the following lemma. le4.1
Lemma 5.1.
For x = p i + ε z, | ε z | < r , we have the decomposition for S [ w ε , p ]( x ) ,S [ w ε , p ] = S + S , where S ( z ) = ε ∑ j = w j ∑ l (cid:54) = i ˆ ξ l z ∇ p i G ( p i , p l ) + h . o . t ., and S ( z ) = ( ˆ ξ i w i ) ( T [ w ε , p ]( p i )) R i ( z ) + h . o . t ., where R i ( z ) is even in z and (cid:107) S (cid:107) L ( − ε , ε ) ≤ C ε s − . Furthermore,S [ w ε , p ] = h . o . t . for | x − p i | ≥ r , i =
1, 2. .2. The Liapunov-Schmidt Reduction Method.
In this subsection, we use the Liapunov-Schmidt reduc-tion method to solve the problem S [ w ε , p + φ ] = ∑ j = c j ∂ w j ∂ y (5.13) for real constants c j and a perturbation φ ∈ H s (cid:16) − ε , ε (cid:17) which is small in the corresponding norm. Toproceed we first need to study the linearized operator˜ L ε , p φ : = S (cid:48) ε [ w ε , p ] φ = ( − ∆ ) sy φ + φ − w ε , p T [ w ε , p ] φ + w ε , p ( T [ w ε , p ]) ( T (cid:48) [ w ε , p ] φ ) .For a given function φ ∈ L ( Ω ) we introduce T (cid:48) [ w ε , p ] φ as the unique solution of (cid:40) D ( − ∆ ) s ( T (cid:48) [ w ε , p ] φ ) + T (cid:48) [ w ε , p ] φ − c ε w ε , p φ = x ∈ ( −
1, 1 ) , ( T (cid:48) [ w ε , p ] φ )( x ) = ( T (cid:48) [ w ε , p ] φ )( x + ) , x ∈ R . (5.14) The approximate kernel and co-kernel are respectively defined by K ε , p : = Span (cid:26) ∂ w j ∂ y (cid:12)(cid:12)(cid:12) j =
1, 2 (cid:27) ⊂ H s (cid:18) − ε , 1 ε (cid:19) , C ε , p : = Span (cid:26) ∂ w j ∂ y (cid:12)(cid:12)(cid:12) j =
1, 2 (cid:27) ⊂ L (cid:18) − ε , 1 ε (cid:19) .From the definition of the linear operator L in ( le3.2le3.2 L : ( X ⊕ X ) ⊥ ∩ ( H s ( R )) → ( X ⊕ X ) ⊥ ∩ ( L ( R )) is invertible with a bounded inverse. We shall see that the linear operator L is a limit of the operator ˜ L ε , p as ε →
0. First we introduce the projection π ⊥ ε , p : L (cid:16) − ε , ε (cid:17) → C ⊥ ε , p and study the operator L ε , p : = π ⊥ ε , p ◦ ˜ L ε , p .Letting ε → L ε , p : K ⊥ ε , p → C ⊥ ε , p is invertible with a bounded inverse provided ε is smallenough. This result is contained in the following proposition. pr5.1 Proposition 5.2.
There exists positive constants ε , δ , C such that for all ε ∈ ( ε ) , ( p , p ) ∈ ( −
1, 1 ) with min ( | + p | , | − p | , | p − p | ) > δ , (cid:107) L ε , p φ (cid:107) L ( − ε , ε ) ≥ C (cid:107) φ (cid:107) H s ( − ε , ε ) . Furthermore, the map L ε , p : K ⊥ ε , p → C ⊥ ε , p (5.15) is surjective.Proof. The proof follows the standard method of Liaypunov-Schmidt reduction which was also used in gui_1999,gui_2000,wei_2001_gm_2d_weak,wei_2002_gm_2d_strong,wei_2007_existence [9, 10, 26, 27, 28]. Suppose the proposition is not true. Then there exist sequences { ε k } , { p k } , φ k satisfying ε k → k → p k ∈ ( −
1, 1 ) , min ( | + p k | , | − p k | , | p k − p k | ) > δ , and φ k = φ ε k ∈ K ⊥ ε k , p k for all k ≥ (cid:107) φ k (cid:107) H s ( − ε , ε ) = (cid:107) L ε k , p k φ k (cid:107) L ( − ε , ε ) →
0, as k → ∞ . (5.16) We define φ ki , i =
1, 2 and φ k as follows: φ ki ( y ) = φ k ( y ) χ (cid:18) ε y − p i r (cid:19) , i =
1, 2, φ k ( y ) = φ k ( y ) − ∑ i = φ ki ( y ) , y ∈ (cid:18) − ε , 1 ε (cid:19) . (5.17) Although each φ ki is defined only in (cid:16) − ε , ε (cid:17) . By a standard result they can be extended to R such that theirnorm in H s ( R ) is still bounded by a constant independent of ε and p for ε small enough. In the followingwe shall study the corresponding problem in R . To simplify our notation, we keep the same notation for he extension. Since { φ ki } is bounded in H s loc ( R ) it has a weak limit in H s loc ( R ) and therefore also a stronglimit in L ( R ) and L ∞ loc ( R ) . We denote the limit by φ i . Then Φ = ( φ , φ ) T solves the system L Φ = le3.1le3.1 Φ ∈ Ker ( L ) = X ⊕ X . Since φ k ⊥ K ⊥ ε k , p k , by taking k → ∞ we get φ ∈ ( X ⊕ X ) ⊥ andtherefore φ = (cid:107) φ ki (cid:107) H s ( R ) → k → ∞ for i =
1, 2. Furthermore, φ k → φ in H s ( R ) ,where φ solves ( − ∆ ) s φ + φ = R . (5.18) Therefore, we conclude φ = (cid:107) φ k (cid:107) H s ( R ) → k → + ∞ . This contradicts (cid:107) φ k (cid:107) H s (cid:16) − ε k , ε k (cid:17) = pr5.1pr5.1 L ε , p (denoted by L ∗ ε , p ) is injective from K ⊥ ε , p to C ⊥ ε , p . Note that L ∗ ε , p = π ε , p ◦ ˜ L ∗ ε , p with˜ L ∗ ε , p ψ = ( − ∆ ) sy ψ + ψ − w ε , p T [ w ε , p ] ψ + T (cid:48) [ w ε , p ] (cid:32) w ε , p ( T [ w ε , p ]) ψ (cid:33) .The proof for L ∗ ε , p follows exactly the same as the one of L ε , p and we omit the details. (cid:3) Now we are in position to solve the problem π ⊥ ε , p ◦ S ε ( w ε , p + φ ) =
0. (5.19)
Since L ε , p | K ⊥ ε , p is invertible (call the inverse L − ε , p ) we can rewrite the above problem as φ = − ( L − ε , p ◦ π ⊥ ε , p ◦ S ε ( w ε , p )) − ( L − ε , p ◦ π ⊥ ε , p ◦ N ε , p ( φ )) ≡ M ε , p ( φ ) , (5.20) where N ε , p ( φ ) = S ε ( w ε , p + φ ) − S ε ( w ε , p ) − S (cid:48) ε ( w ε , p ) φ and the operator M ε , p is defined by φ ∈ H s (cid:16) − ε , ε (cid:17) . We are going to show that the operator M ε , p is acontraction map on B ε , σ : = (cid:26) φ ∈ H s (cid:18) − ε , 1 ε (cid:19) (cid:12)(cid:12)(cid:12) (cid:107) φ ε (cid:107) H s ( − ε , ε ) < σ (cid:27) (5.21) if σ and ε are small enough. We have by the discussion in last section and Proposition pr5.1pr5.1 (cid:107) M ε , p ( φ ) (cid:107) H s ( − ε , ε ) ≤ C (cid:16) (cid:107) π ⊥ ε , p ◦ N ε , p ( φ ) (cid:107) L ( − ε , ε ) + (cid:107) π ⊥ ε , p ◦ S ε ( w ε , p ) (cid:107) L ( − ε , ε ) (cid:17) ≤ C ( c ( σ ) σ + ε s − ) , (5.22) where C > σ > ε > c ( σ ) → σ →
0. Similarly we show that (cid:107) M ε , p ( φ ) − M ε , p ( φ ) (cid:107) H s ( − ε , ε ) ≤ C ( c ( σ ) σ ) (cid:107) φ − φ (cid:107) H s ( − ε , ε ) ,where c ( σ ) → σ →
0. If we choose σ = ε α for α ≤ s − ε > M ε , p is acontraction map on B ε , σ . The existence then follows by the standard the fixed point theorem and φ ε , p is asolution to ( le5.2 Lemma 5.3.
There exists ε > δ > such that for every pair of ε , p with < ε < ε , p ∈ ( −
1, 1 ) , and min { + p , 1 − p , | p − p |} > δ , there is a unique φ ε , p ∈ K ⊥ ε , p satisfying S ε ( w ε , p + φ ε , p ) ∈ C ε , p . Furthermore, we have the estimate (cid:107) φ ε , p (cid:107) H s ( − ε , ε ) ≤ C ε α for any α ≤ s − . ore refined estimates for φ ε , p are needed. We recall from the discussion in last section that S [ w ε , p ] can bedecomposed into the two parts S and S if x is close to the center of spike, where S is in leading orderan odd function and S is in leading order a radially symmetric function. We can similarly decompose φ ε , p as in the following lemma. le5.3 Lemma 5.4.
Let φ ε , p be defined in Lemma le5.2le5.2 = p i + ε z, | ε z | < δ , i =
1, 2 , we have the decomposition φ ε , p = φ ε , p ,1 + φ ε , p ,2 , (5.23) where φ ε , p ,2 is an even function in z which satisfies φ ε , p ,2 = O ( ε s − ) in H s (cid:18) − ε , 1 ε (cid:19) , (5.24) and φ ε , p ,1 = O ( ε ) in H s (cid:18) − ε , 1 ε (cid:19) . (5.25) Proof.
We first solve S [ w ε , p + φ ε , p ,2 ] − S [ w ε , p ] − ∑ j = S (cid:18) y − p j ε (cid:19) ∈ C ε , p , (5.26) for φ ε , p ,2 ∈ K ⊥ ε , p . Then we solve S [ w ε , p + φ ε , p ,2 + φ ε , p ,1 ] − S [ w ε , p + φ ε , p ,2 ] − ∑ j = S (cid:18) y − p j ε (cid:19) ∈ C ε , p , (5.27) for φ ε , p ,1 ∈ K ⊥ ε , p . Using the same proof as in Proposition pr5.1pr5.1 ε (cid:28)
1. By uniqueness, φ ε , p = φ ε , p ,1 + φ ε , p ,2 , and it is easy to see that φ ε , p ,1 and φ ε , p ,2 havethe required properties. (cid:3) The Reduced Problem.
In this subsection, we solve the reduced problem which will will complete theproof of Theorem th1.existth1.exist pr5.1pr5.1 p ∈ B ε s − ( p ) there exists an unique solution φ ε , p ∈ K ⊥ ε , p such that S [ w ε , p + φ ε , p ] = v ε , p ∈ C ε , p . (5.28) To complete the proof of Theorem th1.existth1.exist p ε = ( p ε , p ε ) near p such that S [ w ε , p + φ ε , p ] ⊥C ε , p , which in turn implies that S [ w ε , p + φ ε , p ] =
0. To this end, let W ε : = ( W ε ,1 ( p ) , W ε ,2 ( p )) : B ε s − ( p ) → R where W ε , i ( p ) : = ε − (cid:90) ε − ε S [ w ε , p + φ ε , p ] ∂ w i ∂ y dy , i =
1, 2.Then W ε ( p ) is a map which is continuous in p and our problem is reduced to finding a zero of the vectorfield W ε ( p ) . Let us now calculate W ε ( p ) W ε , i ( p ) = ε − (cid:90) ε − ε S ε [ w ε , p + φ ε , p ] ∂ w i ∂ y dy = ε − (cid:90) ε − ε (cid:34) ( − ∆ ) s ( w ε , p + φ ε , p ) + ( w ε , p + φ ε , p ) − ( w ε , p + φ ε , p ) T [ w ε , p ] + ψ ε , p (cid:35) ∂ w i ∂ y dy = ε − (cid:90) ε − ε (cid:34) ( − ∆ ) s ( w ε , p + φ ε , p ) + ( w ε , p + φ ε , p ) − ( w ε , p + φ ε , p ) T [ w ε , p ] (cid:35) ∂ w i ∂ y dy − ε − (cid:90) ε − ε (cid:34) ( w ε , p + φ ε , p ) T [ w ε , p ] + ψ ε , p − ( w ε , p + φ ε , p ) T [ w ε , p ] (cid:35) ∂ w i ∂ y dy = I + I , (5.29) here I , I are defined by the last equality and ψ ε , p satisifies D ( − ∆ ) s ψ ε , p + ψ ε , p − c ε w ε , p φ ε , p − c ε φ ε , p =
0. (5.30)
For I , we have by Lemma le5.3le5.3 I = ε − (cid:32) (cid:90) ε − ε (cid:34) ( − ∆ ) s ( w ε , p + φ ε , p ) + ( w ε , p + φ ε , p ) − ( w ε , p + φ ε , p ) T [ w ε , p ]( p i ) (cid:35) ∂ w i ∂ y dy + (cid:90) ε − ε ( w ε , p + φ ε , p ) ( T [ w ε , p ]( p i )) ( T [ w ε , p ]( p i + ε y ) − T [ w ε , p ]( p i )) ∂ w i ∂ y dy (cid:33) + O ( ε s − )= ε − (cid:32) (cid:90) ε − ε (cid:34) ( − ∆ ) s ( ˆ ξ i w i + φ ε , p ) + ( ˆ ξ i w i + φ ε , p ) − ( ˆ ξ i w i + φ ε , p ) T [ w ε , p ]( p i ) (cid:35) ∂ w i ∂ y dy (cid:33) + ε − (cid:32) (cid:90) ε − ε ( ˆ ξ i w i + φ ε , p ,2 ) ( T [ w ε , p ]( p i )) ( T [ w ε , p ]( p i + ε y ) − T [ w ε , p ]( p i )) ∂ w i ∂ y dy (cid:33) + O ( ε s − ) . (5.31) Note that, by Lemma le5.3le5.3 (cid:90) ε − ε [( − ∆ ) s φ ε , p + φ ε , p − w i φ ε , p ] ∂ w i ∂ y dy = (cid:90) ε − ε φ ε , p ,1 ∂∂ y (cid:16) ( − ∆ ) s w i + w i − w i (cid:17) dy + O ( ε + s ) = O ( ε + s ) ,(5.32) and (cid:90) ε − ε φ ε , p ∂ w i ∂ y dy = (cid:90) ε − ε φ ε , p ,1 φ ε , p ,2 ∂ w i ∂ y dy + h . o . t . = O ( ε s ) . (5.33) Now by Lemma le5.3le5.3 I = ε − (cid:90) ε − ε w i ( T [ w ε , p ]( p i + ε z ) − T [ w ε , p ]( p i )) ∂ w i ∂ y dy + O ( ε s − )= ε − (cid:90) ε − ε w i (cid:32) P i ( z ) + ε ∑ j (cid:54) = i ˆ ξ j z ∇ p i G D ( p i , p j ) (cid:33) ∂ w i ∂ y dy + O ( ε s − )= − (cid:90) R w ( y ) dy ∑ j (cid:54) = i ˆ ξ j ∇ p i G D ( p i , p j ) + O ( ε s − ) . (5.34) Similarly, we calculate I = ε − (cid:90) ε − ε (cid:34) ( w ε , p + φ ε , p ) T [ w ε , p ] + ψ ε , p − ( w ε , p + φ ε , p ) T [ w ε , p ] (cid:35) ∂ w i ∂ y dy = − ε − (cid:90) ε − ε ( w ε , p + φ ε , p ) ( T [ w ε , p ]) ψ ε , p ∂ w i ∂ y dy + O ( ε s − )= − ε − ˆ ξ i (cid:90) ε − ε ∂ w ∂ y ( ψ ε , p − ψ ε , p ( p i )) dy + O ( ε s − ) . (5.35) Since ψ ε , p satisifies ( le5.3le5.3 ψ ε , p ( p i + ε z ) − ψ ε , p ( p i ) = c ε (cid:90) − ( G D ( p i + ε z , ζ ) − G D ( p i , ζ )) (cid:18) w ε , p (cid:18) ζε (cid:19) φ ε , p (cid:18) ζε (cid:19) + φ ε , p (cid:18) ζε (cid:19)(cid:19) d ζ = o (cid:32) ε ∑ j (cid:54) = i ˆ ξ j z ∇ p i G D ( p i , p j ) (cid:33) + ˆ P i ( z ) + h . o . t ., (5.36) where ˆ P i ( z ) is an even function in z = y − p i ε . Substituting ( I = o ( ∑ j (cid:54) = i ˆ ξ j ∇ p i G D ( p i , p j )) + o ( ε s − ) . (5.37) ombining the estimates for I and I , we obtain W ε , i ( p ) = − (cid:90) R w ( y ) dy ∑ j (cid:54) = i ˆ ξ j ∇ p i G D ( p i , p j )( + o ( )) + O ( ε s − ) = − F i ( p ) (cid:90) R w ( y ) dy + O ( ε s − ) ,(5.38) where F i ( p ) is defined in ( F ( p ) = p − p =
1. By symmetry we conclude that if there exists p = ( p , p ) such thateither one of W ε ,1 ( p ) = W ε ,2 ( p ) = W ε ( p ) =
0. For W ε , i we have W ε , i ( p ) = − (cid:90) R w ( y ) dy (cid:16) ( p − p ) ˆ ξ ∇ p ∇ p G D ( p , p ) + ( p − p ) ˆ ξ ∇ p ∇ p G D ( p , p ) (cid:17) + O ( | p − p | + ε s − ) .By assumption (H3) we have ∇ p ∇ p G D ( p , p ) (cid:54) =
0. As a consequence, we can apply Brouwer’s fixedpoint theorem to show that for ε (cid:28) p ε such that W ε ( p ε ) = p ε ∈ B ε s − ( p ) . Thuswe have proved the following proposition pr6.1 Proposition 5.5.
For ε sufficiently small there exist points p ε with p ε → p such that W ε ( p ε ) = Proof of Theorem th1.existth1.exist
By above Proposition, there exists p ε → p such that W ε ( p ε ) =
0. In other words, S [ w ε , p ε + φ ε , p ε ] =
0. Let u ε = c ε ( w ε , p ε + φ ε , p ε ) , v ε = c ε T [ w ε , p ε + φ ε , p ε ] . By the Maximum principle, u ε > v ε >
0. Moreover ( u ε , v ε ) satisfies all the properties of Theorem th1.existth1.exist (cid:3)
6. R
IGOROUS PROOF OF THE STABILITY ANALYSIS sec:proof-stability
The linear stability of the two-spike solution constructed above is determined by two classes of eigen-values: the large and small eigenvalues satisfying λ ε = O ( ) and λ ε → ε → Stability Analysis: Large Eigenvalues.
In this subsection, we consider the stability of the steady state ( u ε , v ε ) constructed in Theorem th1.existth1.exist u = u ε + φ ε ( x ) e λ ε t , v = v ε + ψ ε e λ ε t = T [ u ε ] + ψ ε e λ ε t , (6.1) and substituting the result into (GM) we deduce the following eigenvalue problem (cid:40) ( − ∆ ) sy φ ε + φ ε − u ε T [ u ε ] φ ε + u ε ( T [ u ε ]) ψ ε + λ ε φ ε = D ( − ∆ ) s ψ ε + ψ ε − c ε u ε φ ε + τλ ε ψ ε =
0, (6.2) where λ ε is some complex number. In this section, we study the large eigenvalues, i.e. those for which wemay assume that there exists c > | λ ε | ≥ c > ε small. If (cid:60) ( λ ε ) < − c then we are done (sincethese eigenvalues are always stable) and we therefore assume that (cid:60) ( λ ε ) ≥ − c . For a subsequence ε → λ ε → λ we shall derive a limiting NLEP satisfied by λ .We first present the case τ =
0. At the end, we shall explain how we proceed when τ > ψ ε = T (cid:48) [ u ε ]( φ ε ) . Let us assume that (cid:107) φ ε (cid:107) H s ( − ε , ε ) = φ ε as follows: φ ε , i ( y ) = φ ε ( y ) χ ( ε y − p ε i r ) , (6.3) where χ ( x ) is a given in ( sec:proof-existencesec:proof-existence
5. Using Lemma le5.3le5.3 (cid:60) ( λ ε ) ≥ − c , the asymptotic expansionof u ε given in Theorem th1.existth1.exist w given in Proposition pr3.1pr3.1 φ ε = ∑ i = φ ε , i + h . o . t .. in H s (cid:18) − ε , 1 ε (cid:19) . (6.4) Then by standard procedure we extend φ ε , i to a function defined on R such that (cid:107) φ ε , i (cid:107) H s ( R ) ≤ C (cid:107) φ ε , i (cid:107) H s ( − ε , ε ) , i =
1, 2. (6.5) ithout loss of generality we may assume that (cid:107) φ ε (cid:107) H s ( R ) = ε , we mayalso assume that φ ε , i → φ i strongly as ε → L ∩ L ∞ for i =
1, 2, on compact subsets of R . Therefore wealso have w φ ε , i → w φ i as ε →
0, strongly in L ∞ ( R ) . (6.6) It is known that ψ ε ( x ) = c ε (cid:90) − G D ( x , ζ ) u ε (cid:18) ζε (cid:19) φ ε (cid:18) ζε (cid:19) d ζ . (6.7) Now we use the expansion of u ε to calculate the value of ψ ε at x = p ε i for each i =
1, 2 ψ ε ( p ε i ) = c ε (cid:90) − G D ( p ε i , ζ ) ∑ j = ˆ ξ j w (cid:32) ζ − p ε j ε (cid:33) χ (cid:18) εζ − p j r (cid:19) φ ε (cid:18) ζε (cid:19) d ζ + h . o . t . = ε c ε ∑ j = ˆ ξ j G D ( p i , p j ) (cid:90) R w φ j dy + o ε ( ) . (6.8) Substituting ( ε →
0, we obtain the nonlocal eigenvalue prob-lem ( − ∆ ) s φ i + φ i − w φ i + (cid:18) (cid:90) R w ( y ) dy (cid:19) − (cid:32) (cid:90) R ∑ j = ˆ ξ j G D ( p i , p j ) w φ j dy (cid:33) w + λ φ i = i =
1, 2. (6.9)
We can rewrite ( ( − ∆ ) s Φ + Φ − w Φ + (cid:18) (cid:90) R w ( y ) dy (cid:19) − (cid:18) (cid:90) R w B Φ dy (cid:19) w + λ Φ = where B is the matrix introduced in ( Φ = ( φ , φ ) T ∈ ( H s ( R )) . We then have the followingconclusion th7.1 Theorem 6.1.
Let λ ε be an eigenvalue of ( such that (cid:60) ( λ ε ) > − c for some c > Suppose that for suitable sequences ε n → we have λ ε n → λ (cid:54) = . Then λ is an eigenvalue of the problemgiven in ( . (2) Let λ (cid:54) = with (cid:60) ( λ ) > be an eigenvalue of the problem given in ( . Then for ε sufficiently small,there is an eigenvalue λ ε of ( with λ ε → λ as ε → Proof.
The proof of (1) follows from a similar asymptotic analysis to that used in § sec:proof-existencesec:proof-existence th7.1th7.1 dancer_2001_hopf [3]. We assume that λ (cid:54) = (cid:60) ( λ ) > ψ ε , we can express ψ ε in terms of φ ε as in ( φ ε = − R ε ( λ ε ) (cid:20) u ε φ ε v ε − u ε v ε ψ ε (cid:21) ,where R ε ( λ ε ) is the inverse of ( − ∆ ) s + ( + λ ε ) in H s ( R ) and ψ ε = T (cid:48) ε [ u ε ]( φ ε ) is given in the secondequation of ( R ε ( λ ε ) is a Fredholm type operator if ε is sufficiently small.The rest of the argument follows as in dancer_2001_hopf [3]. (cid:3) By diagonalizing B we see that the eigenvalue problem ( ( − ∆ ) s ˆ φ i + ˆ φ i − w ˆ φ i + σ i (cid:82) R w ˆ φ i dy (cid:82) R w dy w + λ ˆ φ i =
0, ˆ φ i ∈ H s ( R ) , i =
1, 2, (6.11) where σ and σ are the two eigenvalues of B .We now study the stability of ( σ ∈ λ ( B ) σ <
1. (6.12)
Then by Theorem th3.stabilityth3.stability th7.1th7.1 λ ε of ( (cid:60) ( λ ε ) > c for some positive number c . This implies that ( u ε , v ε ) s unstable. On the other hand if 2 min σ ∈ λ ( B ) σ > th3.stabilityth3.stability λ isstable. Therefore by Theorem th7.1th7.1 ε small enough all nonzero eigenvalues λ ε of ( | λ ε | ≥ c > (cid:60) ( λ ε ) ≤ − c < ε small enough.Finally we comment that when τ (cid:54) = τ is small. We shall apply the results of Theorem th3.2th3.2 B will have to be replaced by the matrix B τλ ε which depends on τε . In particular theGreen’s function G D is replaced by the Green’s function G λ D satisfying D ( − ∆ ) s G λ D + ( + τλ ε ) G λ D = δ z , G λ D ( x + z ) = G λ D ( x , z ) . (6.13) It is then easy to check that the eigenvalues of B τλ ε satisfy the same properties as those of B provided that τ is sufficiently small.6.2. Stability Analysis: Small Eigenvalues.
We now study the eigenvalue problem ( λ ε → ε →
0. Let¯ u ε = w ε , p ε + φ ε , p ε , ¯ v ε = T [ w ε , p ε + φ ε , p ε ] , (6.14) where p ε = ( p ε , p ε ) . After rescaling, the eigenvalue problem ( (cid:40) ( − ∆ ) sy φ ε + φ ε − u ε ¯ v ε φ ε + ¯ u ε ¯ v ε ψ ε + λ ε φ ε = D ( − ∆ ) s ψ ε + ψ ε − c ε ¯ u ε φ ε + τλ ε ψ ε =
0, (6.15) where c ε is given by ( τ = τλ ε (cid:28) τ finite, this is due to the fact that the small eigenvalue are of the order O ( ε ) , we shall prove it in thissubsection.We cut off ¯ u ε as follows ˜ u ε , i ( y ) = χ (cid:18) ε y − p ε i r (cid:19) ¯ u ε ( y ) , i =
1, 2, (6.16) where χ ( x ) and r are given in § sec:proof-existencesec:proof-existence
5. Similarly to the § sec:proof-existencesec:proof-existence K ε , p , new : = Span (cid:8) ˜ u (cid:48) ε , i | i =
1, 2 (cid:9) ⊂ H s (cid:18) − ε , 1 ε (cid:19) , C ε , p , new : = Span (cid:8) ˜ u (cid:48) ε , i | i =
1, 2 (cid:9) ⊂ L (cid:18) − ε , 1 ε (cid:19) .Then it is easy to see that ¯ u ε ( y ) = ∑ i = ˜ u ε , i ( y ) + h . o . t .. (6.17) Note that ˜ u ε , i ( y ) ∼ ˆ ξ i w (cid:18) y − p ε i ε (cid:19) in H s ( −
1, 1 ) and ˜ u ε , i satisfies ( − ∆ ) s ˜ u ε , i + ˜ u ε , i − ˜ u ε , i ¯ v ε + h . o . t . =
0. (6.18)
Thus ˜ u (cid:48) ε , i : = d ˜ u ε , i dy satisfies ( − ∆ ) sy ˜ u (cid:48) ε , i + ˜ u (cid:48) ε , i − u ε , i ¯ v ε ˜ u (cid:48) ε , i + ε ˜ u ε , i ¯ v ε ¯ v (cid:48) ε + h . o . t . =
0, (6.19) and we have ˜ u (cid:48) ε , i = ˆ ξ i ∂ w ∂ y (cid:18) y − p ε i ε (cid:19) ( + o ( )) .Let us now decompose φ ε = ∑ i = a ε i ˜ u (cid:48) ε , i + φ ⊥ ε , (6.20) here a ε i are complex numbers and φ ⊥ ε ⊥ K ε . Similarly, we can decompose ψ ε = ∑ i = a ε i ψ ε , i + ψ ⊥ ε , (6.21) where ψ ε , i satisfies D ( − ∆ ) s ψ ε , i + ψ ε , i − c ε ¯ u ε ˜ u (cid:48) ε , i = i =
1, 2, (6.22) and ψ ⊥ ε satisfies D ( − ∆ ) s ψ ⊥ ε + ψ ⊥ ε − c ε ¯ u ε φ ⊥ ε =
0. (6.23)
We impose periodic boundary conditions in both of these equations.Suppose that (cid:107) φ ε (cid:107) H s ( − ε , 1 ε ) =
1. Then | a ε i | ≤ C since a ε i = (cid:82) ε − ε φ ε ∂ ˜ u ε , i ∂ y dy ˆ ξ i (cid:82) R w dy + o ( ) .Substituting the decompositions of φ ε and ψ ε into ( ( − ∆ ) sy φ ⊥ ε + φ ⊥ ε − u ε ¯ v ε φ ⊥ ε + ¯ u ε ¯ v ε ψ ⊥ ε + λ ε φ ⊥ ε − ε ∑ i = a ε i (cid:32) ˜ u ε , i ¯ v ε ¯ v (cid:48) ε − ε ¯ u ε ¯ v ε ψ ε , i (cid:33) + h . o . t . = − λ ε ∑ i = a ε i ˜ u (cid:48) ε , i . (6.24) Let us first compute J : = ε ∑ i = a ε i (cid:32) ˜ u ε , i ¯ v ε ¯ v (cid:48) ε − ε ¯ u ε ¯ v ε ψ ε , i (cid:33) = ε ∑ i = a ε i (cid:32) ˜ u ε , i ¯ v ε ( ¯ v (cid:48) ε − ε ψ ε , i ) (cid:33) − ∑ i = a ε i ∑ j (cid:54) = i ˜ u ε , j ¯ v ε ψ ε , i + h . o . t . = ε ∑ i = a ε i ˜ u ε , i ¯ v ε (cid:18) − ε ψ ε , i + ¯ v (cid:48) ε (cid:19) − ∑ i = ∑ j (cid:54) = i a ε j ψ ε , j ˜ u ε , i ¯ v ε + h . o . t ..We rewrite J as follows J = − ε ∑ i = ∑ j = a ε j ˜ u ε , i ¯ v ε (cid:18) ε ψ ε , j − ¯ v (cid:48) ε δ ij (cid:19) + h . o . t .. (6.25) Let us also set ˜ L ε φ ⊥ ε : = − ( − ∆ ) sy φ ⊥ ε − φ ⊥ ε + u ε ¯ v ε φ ⊥ ε − ¯ u ε ¯ v ε ψ ⊥ ε (6.26) and a ε : = ( a ε , a ε ) T . (6.27) Multiplying both sides of ( u (cid:48) ε , l and integrating over (cid:16) − ε , ε (cid:17) , we obtain r . h . s . = − λ ε ∑ i = a ε i (cid:90) ε − ε ˜ u (cid:48) ε , i ˜ u (cid:48) ε , l dy = − λ ε a ε l ˆ ξ l (cid:90) R ( w (cid:48) ( y )) dy ( + O ( ε s + )) , (6.28) and l . h . s . = (cid:32) ∑ i = ∑ j = a ε j (cid:90) ε − ε ˜ u ε , i ¯ v ε ( ψ ε , j − ε ¯ v (cid:48) ε δ ij ) ˜ u (cid:48) ε , l dy + (cid:90) ε − ε ˜ u ε , l ¯ v ε ψ ⊥ ε ˜ u (cid:48) ε , l dy − ε (cid:90) ε − ε ˜ u ε , l ¯ v ε ¯ v (cid:48) ε φ ⊥ ε dx (cid:33) ( + o ( ))= ( J l + J l + J l )( + o ( )) , (6.29) where J i , l , i =
1, 2, 3, are defined by the last inequality.We define the vectors J i = ( J i ,1 , J i ,2 ) T , i =
1, 2, 3. (6.30) eq:J-def
To give estimates on each J i ( i =
1, 2, 3) we need the following three lemmas. e8.1 Lemma 6.2.
We have ( ψ ε , j − ε ¯ v (cid:48) ε δ ji )( p ε i ) = − ε ∇G TD H + O ( ε ) . (6.31) Proof.
Note that for i (cid:54) = j , we have ( ψ ε , j − ε ¯ v (cid:48) ε δ ji )( p ε i ) = ψ ε , j ( p ε i ) = c ε (cid:90) − G D ( p ε i , ζ ) ¯ u ε ˜ u (cid:48) ε , j d ζ = − ε ˆ ξ j ∇ p ε j G D ( p ε i , p ε j ) + O ( ε + s ) . (6.32) Next we compute ψ ε , i − ¯ v (cid:48) ε near p ε i :¯ v ε ( x ) = c ε (cid:90) − G D ( x , ζ ) ¯ u (cid:18) ζε (cid:19) d ζ = ε c ε (cid:90) ∞ − ∞ G D ( x , ε z ) ˜ u ε , i ( z ) dz + c ε ∑ j (cid:54) = i (cid:90) − G D ( x , ζ ) ˜ u ε , j (cid:18) ζε (cid:19) d ζ + O ( ε + s ) .So ¯ v (cid:48) ε ( x ) = ε c ε (cid:90) ∞ − ∞ ∇ x G D ( x , ε z ) ˜ u ε , i ( z ) dz + c ε ∑ j (cid:54) = i (cid:90) − ∇ x G D ( x , ζ ) ˜ u ε , j (cid:18) ζε (cid:19) d ζ + O ( ε + s ) .Thus ψ ε , i ( x ) − ε ¯ v (cid:48) ε ( x ) = ε c ε (cid:90) ∞ − ∞ G D ( x , ε z ) ˜ u ε , i ˜ u (cid:48) ε , i dz − ε c ε (cid:90) ∞ − ∞ ∇ x G D ( x , ε z ) ˜ u ε , i ( z ) dz − ε c ε ∑ j (cid:54) = i (cid:90) − ∇ x G D ( x , ζ ) ˜ u ε , j (cid:18) ζε (cid:19) d ζ + O ( ε + s ) .Therefore we have, ψ ε , i ( p ε i ) − ε ¯ v (cid:48) ε ( p ε i ) = ε c ε (cid:90) ∞ − ∞ G D ( p ε i , ε z ) ˜ u ε , i ˜ u (cid:48) ε , i dz − ε c ε (cid:90) ∞ − ∞ ∇ p ε i G D ( p ε i , ε z ) ˜ u ε , i ( z ) dz − ε c ε ∑ j (cid:54) = i (cid:90) ∞ − ∞ ∇ p ε i G D ( p ε i , ε z ) ˜ u ε , j ( z ) dz + O ( ε + s )= − ε ˆ ξ i ∇ p ε i G D ( p ε i , p ε i ) − ε ∑ j (cid:54) = i ˆ ξ j ∇ p ε i G D ( p ε i , p ε j ) + o ( ε )= − ε ˆ ξ i ∇ p ε i G D ( p ε i , p ε i ) + o ( ε ) . (6.33) Equation ( (cid:3) le8.2
Lemma 6.3.
Let q ji be defined as in ( . Then we have ( ψ ε , i − ε ¯ v (cid:48) ε δ ji )( p ε j + ε z ) − ( ψ ε , i − ε ¯ v (cid:48) ε δ ji )( p ε j ) = − ε z (cid:16) ∇ p ε j ∇ p ε i G D ( p ε j , p ε i ) + q ji δ ji (cid:17) ˆ ξ i + o ( ε ) . (6.34) We next study the asymptotic expansion of φ ⊥ ε . Let us first define φ ε , i = ∑ j = ∇ p ε i ˆ ξ j w (cid:32) y − p ε j ε (cid:33) , φ ε : = − ε ∑ i = a ε i φ ε , i . (6.35) Then we have the following lemma. le8.3
Lemma 6.4.
Let be ε sufficiently small. Then (cid:107) φ ⊥ ε − φ ε (cid:107) H s ( − ε , ε ) = o ( ε ) . (6.36) Proof.
We first derive a relation between ψ ⊥ ε and φ ⊥ ε . Note that similar to the proof of Proposition pr5.1pr5.1 L ε isinvertible from K ε , p , new to C ε , p , new . By Lemma le8.1le8.1 L ε is invertible, we deduce that (cid:107) φ ⊥ ε (cid:107) H s ( − ε , ε ) = O ( ε ) . (6.37) Let us decompose ˜ φ ε , i = φ ⊥ ε ε χ (cid:18) ε y − p ε i r (cid:19) . (6.38) hen φ ⊥ ε = ε ∑ i = ˜ φ ε , i + h . o . t . (6.39) Suppose that ˜ φ ε , i → φ i in H (cid:18) − ε , 1 ε (cid:19) . (6.40) By the equation for ψ ⊥ ε (similar to the proof of Lemma le8.1le8.1 ψ ⊥ ε ( p ε i ) = ε c ε ∑ j = (cid:90) − G D ( p ε i , z ) ¯ u ε ˜ φ ε , j ( z ) dz + o ( ε ) = ε ∑ j = G D ( p ε i , p ε j ) ˆ ξ j (cid:82) R w φ j dx (cid:82) R w dx + o ( ε ) , (6.41) and therefore ( ψ ⊥ ε ( p ε ) , ψ ⊥ ε ( p ε )) T = ε G D H (cid:82) R w Φ dx (cid:82) R w dx + o ( ε ) , (6.42) where Φ = ( φ , φ ) T . Substituting ( le8.1le8.1 Φ satisfies ( − ∆ ) s Φ + Φ − w Φ + G D H (cid:82) R w Φ dx (cid:82) R w dx w − ( ∇G D ) T H a w =
0, (6.43) where a = lim ε → a ε = lim ε → ( a ε , a ε ) T .Thus Φ = − ( I − G D H ) − ( ∇G D ) T H a w = −P ( ∇G D ) T H a w , (6.44) where P = ( I − G D H ) − .Now we compare Φ with φ ε . By definition φ ε = − ε ∑ j = ∑ i = a ε i ∇ p ε i ˆ ξ j w (cid:32) y − p ε j ε (cid:33) . (6.45) On the other hand, φ ⊥ ε = ε ∑ i = ˜ φ ε , i + h . o . t . = ε ∑ i = φ i (cid:18) y − p ε i ε (cid:19) + o ( ε ) . (6.46) The lemma is proved by using ( (cid:3)
From Lemma le8.3le8.3 ( ψ ⊥ ε ( p ε ) , ψ ⊥ ε ( p ε )) T = − ε G D HP ( ∇G D ) T H a + o ( ε ) (6.47) and ψ ⊥ ε ( p ε i + ε z ) − ψ ⊥ ε ( p ε i ) = ε z ∑ j = ∇ p ε i G D ( p ε i , p ε j ) ˆ ξ j (cid:82) R w φ j dx (cid:82) R w dx + ε Π i ( y ) + o ( ε ) , (6.48) where Π i ( y ) is an even function in y .With the above three lemmas we can now derive the following results concerning the three terms J , J , J defined in ( eq:J-defeq:J-def le8.4 Lemma 6.5.
Let G D , H , Q , and a ε be given by ( , ( , ( , ( respectively. Then J = c ε H ( ∇ G D + Q ) H a ε + o ( ε ) , J = c ε H∇G D HP ( ∇G D ) T H a ε + o ( ε ) , (6.49) and J = o ( ε ) , where c = (cid:82) R w dy. roof. The computation of J follows from Lemma le8.2le8.2 v (cid:48) ε = o ( ) J l = ∑ j = a ε j (cid:90) ε − ε ˜ u ε , l ¯ v ε ( ψ ε , j − ε ¯ v (cid:48) ε δ jl ) ˜ u (cid:48) ε , l + h . o . t . = ∑ j = a ε j (cid:90) ε − ε ˜ u ε , l ¯ v ε (cid:16) [ ψ ε , j ( y ) − ε ¯ v (cid:48) ε ( y ) δ jl ] − [ ψ ε , j ( p ε l ) − ε ¯ v (cid:48) ε ( p ε l ) δ jl ] (cid:17) ˜ u (cid:48) ε , l dy + o ( ε )= − ε ˆ ξ l (cid:90) R ( yw w (cid:48) ( y )) dy ∑ j = a ε j (cid:16) ∇ p ε l ∇ p ε j G D ( p ε l , p ε j ) + q lj δ lj (cid:17) ˆ ξ j + o ( ε )= c ε ˆ ξ l ∑ j = a ε j (cid:16) ∇ p ε l ∇ p ε j G D ( p ε l , p ε j ) + q lj δ lj (cid:17) ˆ ξ j + o ( ε ) , (6.50) which, by Lemma le8.1le8.1 J follows from J l = (cid:90) ε − ε ˜ u ε , l ¯ v ε ψ ⊥ ε ˜ u (cid:48) ε , l dy = (cid:90) ε − ε ˜ u ε , l ¯ v ε ψ ⊥ ε ( p ε l ) ˜ u (cid:48) ε , l dy + (cid:90) ε − ε ˜ u ε , l ¯ v ε ( ψ ⊥ ε ( x ) − ψ ⊥ ε ( p ε l )) ˜ u (cid:48) ε , l dy = (cid:90) ε − ε ˜ u ε , l ¯ v ε ( ψ ⊥ ε ( x ) − ψ ⊥ ε ( p ε l )) ˜ u (cid:48) ε , l dx + o ( ε ) , (6.51) together with ( J l follows from Lemma le8.3le8.3 v ε ( p ε l ) = ˆ ξ l + O ( ε s − ) at p ε l and the leading order of ¯ v (cid:48) ε ( p ε l + ε y ) − ¯ v (cid:48) ε ( p ε l ) is an odd function of order ε . (cid:3) We can now provide an estimate on the small eigenvalue. From Lemma le8.4le8.4 J + J + J = c ε H (cid:16) ( ∇ G D + Q ) H + ∇G D HP ( ∇G D ) T H (cid:17) a ε + o ( ε )= c ε H M ( p ε ) a ε + o ( ε ) ,and therefore by combining ( c ε H M ( p ε ) a ε + o ( ε ) = − λ ε H a ε (cid:90) R ( w (cid:48) ( y )) dy ( + o ( )) . (6.52) From this equation we see that λ ε = − ε c λ M ( p ) ( + o ( )) ,where c is a positive constant and λ M ( p ) is the non-zero left eigenvalue of M ( p ) given in ( app:greens-funcapp:greens-func C we derive a quickly converging series expression for the Green’s function for which we caninterchanging summation and differentiation to calculate its second derivatives. Numerical calculationsthen indicate that ∂ x G D ( x , 0 ) > x = M ( p ) is positive. The small λ ε is therefore negative (stable) so that the two spike pattern islinearly stable with respect to the small eigenvalues. In particular, linear stability is determined solely bythe eigenvalues of B and the proof of Theorem th1.stabilityth1.stability ONCLUSION AND O PEN P ROBLEMS sec:conclusion
In this paper we have proven the existence and rigorously analyzed the stability of both symmetric andasymmetric two spike equilibrium solutions of the fractional one-dimensional Gierer-Meinhardt system( N -spike quasi-equilibrium solutionsand derived a system of ODEs governing their slow dynamics on an O ( ε − ) timescale as well as a systemof NLEPs governing their linear stability on an O ( ) timescale. Our findings indicate that a single spike so-lution may be destabilized or stabilized with respect to oscillatory instabilities by decreasing the fractionalexponents for the activator, s , or inhibitor, s , respectively. On the other hand we found that decreasing he fractional exponent for the inhibitor, s , has a stabilizing effect on the stability of symmetric two-spikesolutions with respect to competition instabilities. Finally we determined that asymmetric two-spike solu-tions are always linearly unstable with respect to competition instabilities. In all one- and two-spike caseswe found that the equilibrium spike patterns are linearly stable with respect to the slow dynamics and thatthis is a consequence of the choice of periodic boundary conditions.We conclude this section with an outline of open problems and directions for future research. The firstopen problem is to prove the existence and to provide a complete classification of all N -spike equilib-rium solutions to the fractional one-dimensional Gierer-Meinhardt model. In particular a key question iswhether, as in the classical Gierer-Meinhardt model wei_2007_existence [28], asymmetric N -spike solutions are generated bysequences of spikes of two types. In addition it would be interesting to extend our results to the fractionalGierer-Meinhardt model with different boundary conditions (e.g. Neumann or Dirichlet) for which ouranalysis needs to be extended in order to provide regularity estimates at the boundaries x = ±
1. Anotherinteresting direction for future research is to investigate the behaviour of solutions to the fractional GMmodel in the D (cid:28) EFERENCES bressloff_2014 [1] P. Bressloff.
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A. T
HE NONLOCAL EIGENAVLUE PROBLEM app:nonlocal
In this section, we prove Theorem th3.stabilityth3.stability ( − ∆ ) s φ + φ − w φ + γ (cid:82) R w φ dx (cid:82) R w dx + αφ = φ ∈ H s ( R ) . (A.1) a.1 Our aim is to show that the above eigenvalue problem has an eigenvalue with real part when γ ∈ (
0, 1 ) and the real part of the eigenvalue is always negative if γ > s > .Before we give the proof of Theorem th3.stabilityth3.stability pra.1 Proposition A.1 ( frank_2013_uniqueness [5]) . The eigenvalue problem ( − ∆ ) s φ + φ − w φ + µφ = R , φ ∈ H s ( R ) (A.2) a.eigen admits the following set of eigenvalues: µ > µ = µ < · · · . (A.3) a.eigen1 Moreover, the eigenfunction corresponding to µ is radial and of costant sign.Proof of Theorem th3.stabilityth3.stability The original problem is equivalent to finding a positive zero root of the function F ( α ) defined by F ( α ) = (cid:90) R w dx + γ (cid:90) R w ( L + α ) − w dx ,where L φ = ( − ∆ ) s φ + φ − w φ .By the above proposition, L has a unique eigenvalue µ > F ( α ) in the interval ( µ ) . Since L − w = − w , we deduce that F ( ) = ( − γ ) (cid:90) R w dx >
0, (A.4) a.small provided γ <
1. Next, as α → µ − , we have that (cid:90) R w ( L + α ) − w dx → − ∞ . (A.5) a.small1 Hence, we get from ( a.small1a.small1
A.5) that { ( α ) → − ∞ as α → µ − , (A.6) a.small2 when γ ∈ (
0, 1 ) . By ( a.smalla.small A.4), ( a.small2a.small2
A.6) and the continuity of { ( α ) , we can find a α ∈ ( µ ) such that f ( α ) = γ ∈ (
0, 1 ) . (cid:3) ext, we shall study ( a.1a.1 A.1) when γ >
1. We shall prove that the real part of the eigenvalue is negative inany case. To this end, we introduce some notation and make some preparations. Set L φ : = L φ + γ (cid:82) R w φ dx (cid:82) R w dx w , φ ∈ H s ( R ) . (A.7) a.2 According to the definition of L , we can easily see that L is not self-adjoint. Let X : = kernel ( L ) = Span (cid:26) ∂ w ∂ x (cid:27) .Then L w = − w , L (cid:18) w + s x · ∇ w (cid:19) = − w . (A.8) a.3 Hence (cid:90) R ( L − w ) wdx = (cid:90) R (cid:18) − s x · ∇ w − w (cid:19) wdx = − s s (cid:90) R w dx , (A.9) a.4 and (cid:90) R ( L − w ) w p dx = − (cid:90) R L − wL wdx = − (cid:90) R w dx . (A.10) a.5 Before we give the proof of Theorem th3.stabilityth3.stability lea.2
Lemma A.2.
Let L be an operator defined byL φ = L φ + (cid:82) R w φ dx (cid:82) R w dx w + (cid:82) R w φ dx (cid:82) R w dx w − (cid:82) R w dx (cid:82) R w φ dx (cid:0) (cid:82) R w dx (cid:1) w . (A.11) a.6 Then we have (1) L is self-adjoint and the kernel of L (denoted by X ) is Span (cid:110) w , ∂ w ∂ x (cid:111) . (2) There exists a positive constant a > such thatL ( φ , φ ) : = (cid:90) R (cid:16) | ( − ∆ ) s φ | + φ − w φ (cid:17) dx + (cid:82) R w φ dx (cid:82) R w φ dx (cid:82) R w dx − (cid:82) R w dx (cid:0) (cid:82) R w φ dx (cid:1) (cid:0) (cid:82) R w dx (cid:1) ≥ a d L ( R ) ( φ , X ) , (A.12) a.7 for all φ ∈ H s ( R ) , where d L ( R ) means the distance in L -norm.Proof. By ( a.7a.7
A.12), L is self-adjoint. It is easy to see that w , ∂ w ∂ y ∈ Kernel ( L ) . On the other hand, if φ ∈ Kernel ( L ) , then by Proposition pra.1pra.1 A.1 L φ = − c ( φ ) w − c ( φ ) w = c ( φ ) L ( w + s x · ∇ w ) + c ( φ ) L ( w ) ,where c ( φ ) = (cid:82) R w φ dx (cid:82) R w dx − (cid:82) R w dx (cid:82) R w φ dx (cid:0) (cid:82) R w dx (cid:1) , c ( φ ) = (cid:82) R w φ dx (cid:82) R w dx .Hence φ − c ( φ )( w + s x · ∇ w ) − c ( φ ) w ∈ kernel ( L ) . (A.13) a.8 Note that c ( φ ) = c ( φ ) (cid:82) R w (cid:16) w + s x · ∇ w (cid:17) dx (cid:82) R w dx − c ( φ ) (cid:82) R w dx (cid:82) R w ( w + s x · ∇ w ) dx (cid:0) (cid:82) R w dx (cid:1) = c ( φ ) − c ( φ )( − s ) (cid:82) R w dx (cid:82) R w dx by ( a.4a.4 A.9) and ( a.5a.5
A.10). This implies that c ( φ ) = s > . By ( a.8a.8 A.13) and Proposition pra.1pra.1
A.1, we prove the firstconclusion. t remains to prove (2). Suppose it is not true. Then by the first conclusion there exists ( α , φ ) such that ( i ) α is real and positive, ( ii ) φ ⊥ w , φ ⊥ ∂ w ∂ x and ( iii ) L ( φ ) + αφ = ( ii ) and ( iii ) we have ( L + α ) φ + (cid:82) R w φ dx (cid:82) R w dx w =
0. (A.14) a.9
First we claim that (cid:82) R w φ (cid:54) =
0. In fact if (cid:82) R w φ =
0, then − α < L . By Proposition pra.1pra.1 A.1, − α = µ and φ has costant sign. This contradicts with the fact that φ ⊥ w . Therefore − α (cid:54) = µ , 0 andhence L + α is invertible in X ⊥ . So ( a.9a.9 A.14) implies φ = − (cid:82) R w φ dx (cid:82) R w ( L + α ) − w .Thus (cid:90) R w φ dx = − (cid:82) R w φ dx (cid:82) R w dx (cid:90) R (( L + α ) − w ) w dx ,which implies (cid:90) R w dx = − (cid:90) R (( L + α ) − w ) w dx = (cid:90) R (( L + α ) − w )(( L + α ) w − α w ) dx ,hence (cid:90) R (( L + α ) − w ) wdx =
0. (A.15) a.10
Let h ( α ) = (cid:82) R (( L + α ) − w ) wdx , then h ( ) = (cid:90) R ( L − w ) wdx = − (cid:90) R ( w + s x · ∇ w ) w = ( s − ) (cid:90) R w dx < s < . Moreover, h (cid:48) ( α ) = − (cid:90) R ( L + α ) − ww = − (cid:90) R (( L + α ) − w ) dx < h ( α ) < α ∈ ( µ ) . Clearly, h ( α ) > α ∈ ( µ , ∞ ) since lim α → + ∞ h ( α ) = a.10a.10 A.15) and we finish the proof. (cid:3)
Proof of Theorem th3.stabilityth3.stability
We now finish the proof of Theorem th3.stabilityth3.stability α = α R + i α I and φ = φ R + i φ I . Since α (cid:54) =
0, we can choose φ ⊥ kernel ( L ) . Then we can obtain twoequations L φ R + γ (cid:82) R w φ R dx (cid:82) R w dx w = − α R φ R + α I φ I , L φ I + γ (cid:82) R w φ I dx (cid:82) R w dx w = − α R φ I − α I φ R , (A.16) a.11 Multiplying the first equation of ( a.11a.11
A.16) by φ R and the second one of ( a.11a.11 A.16) by φ I and adding them together,we obtain − α R (cid:90) R ( φ R + φ I ) dx = L ( φ R , φ R ) + L ( φ I , φ I ) + (cid:82) R w dx (cid:0) (cid:82) R w dx (cid:1) (cid:34)(cid:18) (cid:90) R w φ R dx (cid:19) + (cid:18) (cid:90) R w φ I dx (cid:19) (cid:35) + ( γ − ) (cid:82) R w φ R dx (cid:82) R w φ R dx + (cid:82) R w φ I dx (cid:82) R w φ I dx (cid:82) R w dx (A.17) a.12 Multiplying both equations of ( a.11a.11
A.16) by w and adding together, we get (cid:90) R w φ R dx − γ (cid:82) R w φ R dx (cid:82) R w dx (cid:90) R w dx = α R (cid:90) R w φ R dx − α I (cid:90) R w φ I dx , (cid:90) R w φ I dx − γ (cid:82) R w φ I dx (cid:82) R w dx (cid:90) R w dx = α R (cid:90) R w φ I dx + α I (cid:90) R w φ R dx . (A.18) a.13 e multiply the first equation of ( a.13a.13 A.18) by (cid:82) R w φ R dx and the second one of ( a.13a.13 A.18) by (cid:82) R w φ I dx and addthem together, we obtain (cid:90) R w φ R dx (cid:90) R w φ R dx + (cid:90) R w φ I dx (cid:90) R w φ I dx = (cid:32) α R + γ (cid:82) R w dx (cid:82) R w dx (cid:33) (cid:18) ( (cid:90) R w φ R dx ) + ( (cid:90) R w φ I dx ) (cid:19) .(A.19) a.14 Therefore, we have − α R (cid:90) R ( φ R + φ I ) dx = L ( φ R , φ R ) + L ( φ I , φ I ) + (cid:82) R w dx ( (cid:82) R w dx ) (cid:20) ( (cid:90) R w φ R dx ) + ( (cid:90) R w φ I dx ) (cid:21) + ( γ − ) (cid:32) α R + γ (cid:82) R w dx (cid:82) R w dx (cid:33) ( (cid:82) R w φ R dx ) + ( (cid:82) R w φ I dx ) dx (cid:82) R w dx . (A.20) a.15 Set φ R = c R w + φ ⊥ R , φ ⊥ R ⊥ X , φ I = c I w + φ ⊥ I , φ ⊥ I ⊥ X .Then (cid:90) R w φ R dx = c R (cid:90) R w dx , (cid:90) R w φ I dx = c I (cid:90) R w dx ,and d L ( R ) ( φ R , X ) = (cid:107) φ ⊥ R (cid:107) L , d L ( R ) ( φ I , X ) = (cid:107) φ ⊥ I (cid:107) L .By some simple computations we have L ( φ R , φ R ) + L ( φ I , φ I ) + α R ( γ − )( c R + c I ) (cid:90) R w dx + ( c R + c I )( γ − ) (cid:90) R w dx + α R ( (cid:107) φ ⊥ R (cid:107) L + (cid:107) φ ⊥ I (cid:107) L ) = lea.2lea.2 A.2, α R ( γ − )( c R + c I ) (cid:90) R w dx + ( γ − ) ( c R + c I ) (cid:90) R w dx + α R ( (cid:107) φ ⊥ R (cid:107) L + (cid:107) φ ⊥ I (cid:107) L ) ≤ γ <
1, we have α R <
0, which proves Theorem th3.stabilityth3.stability φ satisfies ( − ∆ ) s φ + φ − w φ + γ (cid:82) R w φ dx (cid:82) R w dx w =
0. (A.21) a.16
Then L φ = − c ( φ ) w , where c ( φ ) = γ (cid:82) R w φ dx (cid:82) R w dx . Hence φ − c ( φ ) w ∈ Kernel ( L ) . Thus c ( φ ) γ = γ (cid:82) R w φ dx (cid:82) R w dx = c ( φ ) . (A.22) a.17 So if γ (cid:54) =
1, we get c ( φ ) =
0. Then φ ∈ Kernel ( L ) and we complete the proof. (cid:3) A PPENDIX
B. O
VERVIEW OF N UMERICAL C ALCULATIONS app:numerical
In this section we briefly outline the numerical calculation of solutions to the core problem ( eq:core-problemeq:core-problem huang_2013 [12]. When discretizing ( eq:core-problemeq:core-problem pr3.1pr3.1 ruuth_1995 [22]. In the remainder of this section we provide additional details for both of these cases.First we consider the numerical calculation of solutions to the core problem ( eq:core-problemeq:core-problem eq:core-problemeq:core-problem − ∞ < y < ∞ we need to both truncate and then discretize the truncated domain to obtain anumerical calculation. Outside of the truncated domain we use the far-field behaviour from Proposition pr3.1pr3.1 o impose a Dirichlet boundary condition. Specifically, letting L > eq:core-problemeq:core-problem ( − ∆ ) s U + U − U = | y | < L , U ( y ) = U ( L )( L / y ) + s , | y | ≥ L ,where we have replaced b s with U ( L ) L + s since we do not yet know the value of b s . To account for thenonlocal contributions outside of the truncated domain we discretize a computational domain that extendsbeyond the truncated domain. Specifically we discretize the computational domain − L ≤ y ≤ L byletting y i = ih for i = − N , ..., 2 N where h = N . Seeking symmetric solutions we impose U i = U | i | for all i = − N , ..., 2 N which reduces the unknown values to U , ..., U N . Note in addition that U i = ( L / y | i | ) + s U N for all N < | i | ≤ N . The fractional Laplacian can then be approximated by (see §5 in huang_2013 [12]) ( − ∆ ) s U ( y i ) ≈ ( − ∆ h ) s U i = N ∑ j = − N ( U i − U | i − j | ) w j + C II U i − C IIIi U N , i =
0, ..., N (B.1) eq:core-problem-discretized where the first term accounts for integration inside of the truncated domain and w j = C s s ( s − ) h s (cid:40) − s − + ( − s ) − s , j = ± | j + | − s − | j | − s + | j − | − s , otherwise, ( j = ± ±
2, ... ) , (B.2) eq:discretized_weights where we note that the value of w is never needed in the discretization. The remaining two terms C II and C IIIi account for contributions outside of the computational domain and are respectively given by C II = C s s ( L ) s , C IIIi = C s L s + ( s + )( L ) s + (cid:18) F (cid:0) s +
1, 4 s +
1; 4 s + y i L (cid:1) + F (cid:0) s +
1, 4 s +
1; 4 s + − y i L (cid:1)(cid:19) ,where F is the Gaussian hypergeometric function.With the above discretization it is then possible to approximate solutions to ( eq:core-problemeq:core-problem eq:core-problem-discretizedeq:core-problem-discretized B.1) for the N + U , ..., U N . To numerically solve this nonlinear system weuse the fsolve function in the Python 3.6.8 SciPy library. Our initial guess for the nonlinear solver isobtained by numerical continuation in s starting with s = w s = ( + y ) is known. In this way we may numerically calculate the core solution for an arbitrary value of s and inFigure fig:core_problem-solfig:core_problem-sol
3a we plot the resulting core solutions for select values of s where we have used N = b s and this is plotted in Figure fig:core_problem-constantfig:core_problem-constant eq:core-problemeq:core-problem s ≤ frank_2013_uniqueness [5]) and our numerical computations failed to yield solutions for s ≈ − < x < N uniformlydistributed points given by x i = − + ih for i =
0, ..., N − h = N . Assuming that φ ( x ) isa 2 − periodic function on − < x < φ i ≡ φ ( x i ) for each i =
0, ..., N − huang_2013 [12]) ( − ∆ ) s φ ( x i ) ≈ ( − ∆ h ) s φ i = ∞ ∑ j = − ∞ ( φ i − φ i − j ) w j = N − ∑ j = W i − j ( φ i − φ j ) , (B.3) eq:discretized_gm_1 where the final equality follows from the periodicity of φ and where W σ ≡ w σ + ∞ ∑ k = ( w σ + Nk + w σ − Nk ) ,with each weight w i ( i ∈ Z ) being given by ( eq:discretized_weightseq:discretized_weights B.2). In our numerical calculations we truncate the sumafter 500 terms. From ( eq:discretized_gm_1eq:discretized_gm_1
B.3) it is then straightforward to deduce the entries of the matrix ( − ∆ h ) s which weremark is dense in contrast to the tridiagonal matrix obtained by applying a finite-difference approximation y Core Solution w s ( y ) s ( A ) fig:core_problem-sol s Decay Coefficient s ( B ) fig:core_problem-constant F IGURE
3. (A) Sample plots of numerically computed solutions to the core problem ( eq:core-problemeq:core-problem b s in the core problem ( eq:core-problemeq:core-problem fig:core_problem to the one-dimensional Laplacian. With this spatial discretization we can then approximate ( N -dimensional system of ODEs d Φ dt + A Φ + N ( Φ ) =
0, (B.4) eq:discretized_gm_2 where Φ ( t ) = ( u ( t ) , ..., u N − ( t ) , v ( t ) , ..., v N − ( t )) T , A = diag ( ε s ( − ∆ h ) s , τ − D ( − ∆ h ) s ) , and N ( Φ ) isthe 2 N -dimensional array that accounts for the nonlinearities in ( eq:discretized_gm_2eq:discretized_gm_2 B.4) we employ asecond-order semi-implicit backwards difference scheme (2-SBDF) ruuth_1995 [22] that uses second-order backwarddifference time-stepping for the fractional Laplace term and explicit (forward) time-stepping for the non-linear term. Specifically, given a time-step size ∆ t > Φ n = Φ ( t n ) where t n = n ∆ t the2-SBDF scheme becomes ( I − ∆ tA ) Φ n + = Φ n − Φ n − + ∆ t N ( Φ n ) − ∆ t N ( Φ n − ) , (B.5)where I is the 2 N × N identity matrix. Given an initial condition Φ (based on the asymptotic approxi-mations of § sec:formal-resultssec:formal-results
4) we also need Φ to initiate time-stepping with 2-SBDF. We calculate Φ by using a first-ordersemi-implicit backwards difference scheme (1-SBDF) ruuth_1995 [22] given by ( I − ∆ t A ) Φ n + = Φ n + ∆ t N ( Φ n ) , (B.6)with which we perform five time steps with a step size that is one-fifth that used in our main 2-SBDFscheme. Throughout the numerical simulations of ( subsec:example-1subsec:example-1 subsec:example-2subsec:example-2 N = ∆ t = PPENDIX
C. A R
APIDLY C ONVERGING S ERIES FOR THE F RACTIONAL G REEN ’ S F UNCTION app:greens-func
In this section we provide a quickly converging series expansion of the Green’s function G D ( x , z ) satis-fying ( | x − z | s − and | x − z | s − asoutlined below we obtain the series expansion G D ( x , z ) = a s (cid:0) | x − z | s − − s (cid:1) − b s (cid:0) | x − z | s − − s (cid:1) − (cid:0) ( s − ) a s − ( s − ) b s (cid:1)(cid:0) | x − z | − (cid:1) + + D ∞ ∑ n = (cid:18) + D ( n π ) s (cid:19) − cos n π | x − z | ( n π ) s + ∞ ∑ n = (cid:18) b s b n ( n π ) s − a s a n ( n π ) s (cid:19) cos n π | x − z | , (C.1) eq:greens-fast-series where a s = − π D s Γ ( − s ) sin ( π s ) , b s ≡ − π D s Γ ( − s ) sin ( π s ) , (C.2) nd a n = ( s − )( s − ) (cid:90) ∞ n π x s − cos xdx , (C.3) eq:an-bn-def b n = − ( s − )( s − )( s − ) (cid:18) ( − ) n ( n π ) s − + ( s − ) (cid:90) ∞ n π x s − cos xdx (cid:19) . (C.4)The key reason for considering this expansion is that the coefficients of cos n π | x − z | converge to zerosufficiently fast to allow the order of summation and second-differentiation to be interchanged. In particularusing ( eq:greens-fast-serieseq:greens-fast-series C.1) we can numerically calculate that ∂ x G D ( x , 0 ) is strictly positive at x = eq:greens-fast-serieseq:greens-fast-series C.1) we use integration by parts to calculate the coefficients in the Fourier series | x | β − = β + ∞ ∑ n = c n , β ( π n ) β cos n π x , c n , β = (cid:90) n π x β − cos xdx , (C.5)where β = s ∈ (
1, 2 ) or β = s ∈ (
2, 4 ) . Specifically we calculate c n ,2 s = − ( s − ) (cid:90) n π x s − sin xdx = − ( s − ) (cid:90) ∞ x s − sin xdx + ( s − ) (cid:90) ∞ n π x s − sin xdx = − ( s − ) (cid:90) ∞ x s − sin xdx + ( − ) n ( s − )( n π ) s − + a n ,for β = s and c n ,4 s =( s − )( s − )( s − ) (cid:90) ∞ x s − sin xdx + ( − ) n ( s − )( n π ) s − + b n ,for β = s and where a n and b n are defined by ( eq:an-bn-defeq:an-bn-def C.3). The definite integrals appearing in c n ,2 s and c n ,4 s canthen be written in terms of a s and b s respectively by using the integral representation of the Gamma function (cid:82) ∞ x z − sin xdx = Γ ( z ) sin (cid:0) π z (cid:1) for − < (cid:60) ( z ) < z Γ ( z ) Γ ( − z ) = − π / sin π z for all z / ∈ Z (see equations 5.9.7 and 5.5.3 in NIST:DLMF [4] respectively).A
PPENDIX
D. D
ERIVATION OF THE S LOW D YNAMICS app:slow-dynamics
In this appendix we outline the derivation of the system of ODEs ( eq:slow-dynamicseq:slow-dynamics subsec:formal-equilibriumsubsec:formal-equilibrium x = x i + ε y with y = O ( ) weobtain ( eq:quasi-eq-sol-veq:quasi-eq-sol-v eq:greens-fast-serieseq:greens-fast-series C.1) (with s = s ) v ∼ ε − ω s (cid:18) N ∑ j = ξ j G D ( x i , x j ) + a s ξ i ε s − | y | s − + ε b i y + O ( ε min { s − } ) (cid:19) , (D.1)where b i ≡ ∑ j (cid:54) = i ξ j ∇ G D ( x i , x j ) . It follows that the first order correction term in the inner expansion mustbe O ( ε s − ) and in particular for x = x i + ε y and y = O ( ) u ∼ ε − (cid:0) ξ i w s ( y ) + ε s − U i + o ( ε s − ) (cid:1) , v ∼ ε − (cid:0) ξ i + ε s − V i + o ( ε s − ) (cid:1) ,By repeatedly using the method of matched asymptotic expansions we determine that the fractional power ε s − initiates a chain of corrections at powers of ε that are multiples of 2 s −
1. In particular for each i =
1, ..., N the inner expansion when x = x i + ε y with y = O ( ) takes the form u ∼ ε − (cid:0) ξ i w s ( y ) + k max − ∑ k = ε k ( s − ) U ik + ε U ik max + o ( ε ) (cid:1) , (D.2) eq:higher-order-inner v ∼ ε − (cid:0) ξ i + k max − ∑ k = ε k ( s − ) V ik + ε V ik max + o ( ε ) (cid:1) (D.3)where k max is the smallest integer such that k max ( s − ) ≥
1. Importantly, since V ik ∼ C k | y | s − as | y | → ∞ for 1 ≤ k < k max each of these corrections are even in y . On the other hand when k = k max wehave the far-field behaviour V ik max ∼ ω s b i y + δ k max ( s − ) C k max | y | s − | y | → ∞ , (D.4) here δ i , j is the discrete Kronecker delta function. Therefore we can write V ik max = ω s b i y + V eik max where V eik max is an even function in y . Assuming that each x i = x i ( t ) and substituting ( eq:higher-order-innereq:higher-order-inner D.2) into ( x = x i + ε y we obtain − ε ξ i dw s dy dx i dt + k max − ∑ k = ε k ( s − ) L U ik + ε L U ik max + N ε + ε w s (cid:0) ω s b i y + V eik max (cid:1) + o ( ε ) =
0, (D.5) eq:slow-temp-eq-1 where N ε is an even function of y that consists of the residual nonlinear combinations of U ik and V ik for1 ≤ k < k max . Recalling that dw s / dy spans the kernel of L we impose a solvability condition on ( eq:slow-temp-eq-1eq:slow-temp-eq-1 D.5) bymultiplying it with dw s / dy and integrating to obtain dx i dt = ε ω s b i (cid:82) ∞ − ∞ w s dw s dy ydy ξ i (cid:82) ∞ − ∞ | dw s / dy | dy = − ε ω s (cid:82) ∞ − ∞ w s ξ i (cid:82) ∞ − ∞ | dw s / dy | dy b i where we have used integration by parts to obtain the second equality. This establishes ( eq:slow-dynamicseq:slow-dynamics D ANIEL G OMEZ , D
EPARTMENT OF M ATHEMATICS , T HE U NIVERSITY OF B RITISH C OLUMBIA , V
ANCOUVER , BC C
ANADA
V6T1Z2
Email address : [email protected] J UN - CHENG W EI , D EPARTMENT OF M ATHEMATICS , T HE U NIVERSITY OF B RITISH C OLUMBIA , V
ANCOUVER , BC C
ANADA
V6T1Z2
Email address : [email protected] W EN Y ANG , W
UHAN I NSTITUTE OF P HYSICS AND M ATHEMATICS , C
HINESE A CADEMY OF S CIENCES , P.O. B OX UHAN
HINA
Email address : [email protected]@wipm.ac.cn