STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL
GIERER-MEINHARDT SYSTEM
JUNCHENG WEI AND MATTHIAS WINTER
A BSTRACT. We consider the Gierer-Meinhardt system with precursor inhomogeneity and two
small diffusivities in an interval

2

x ∈ (−1, 1), t > 0,
At = ε2 A′′ − µ(x)A + AH ,



τ Ht = DH ′′ − H + A2 ,
x ∈ (−1, 1), t > 0,



 ′
A (−1) = A′ (1) = H ′ (−1) = H ′ (1) = 0,
√
where 0 < ε ≪ D ≪ 1,
τ ≥ 0 and τ is independent of ε.
A spike cluster is the combination of several spikes which all approach the same point in the
singular limit. We rigorously prove the existence of a steady-state spike cluster consisting of N
spikes near a non-degenerate local minimum point t0 of the smooth inhomogeneity µ(x), i.e. we
assume that µ′ (t0 ) = 0, µ′′ (t0 ) > 0. Here N is an arbitrary positive integer. Further, we show
that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of order
O(1)) and small (of order o(1)). The main features of studying the Gierer-Meinhardt system in
this setting are as follows: (i) it is biologically relevant since it models a hierarchical process
(pattern formation of small-scale structures induced by a pre-existing large-scale inhomogeneity); (ii) it contains three different spatial
√ scales two of which are small: the O(1) scale of the
precursor inhomogeneity µ(x), the O( D) scale of the inhibitor diffusivity and the O(ε) scale
of the activator diffusivity; (iii) the expressions can be made explicit and often have a particularly
simple form.
1. I NTRODUCTION
In his pioneering work [31] in 1952, Turing studied how pattern formation could start from
an unpatterned state. He explained the onset of pattern formation by the presence of spatially
varying instabilities combined with the absence of spatially homogeneous instabilities. This approach is now commonly called Turing diffusion-driven instability. Since then many reactiondiffusion systems in biological modeling have been proposed and the occurrence of pattern
formation has been investigated based on the approach of Turing instability [31]. One of the
most widely used class of biological pattern-formation models consists of the activator-inhibitor
University of British Columbia, Department of Mathematics, Vancouver, B.C., Canada V6T 1Z2
(jcwei@math.ubc.ca).
Brunel University London, Department of Mathematics, Uxbridge UB8 3PH, United Kingdom
(matthias.winter@brunel.ac.uk).
1
2
JUNCHENG WEI AND MATTHIAS WINTER
type models which are based on real-world interactions such as those encountered in experiments and observations on seashells, animal skin patterns, embryological development, cell signalling pathways and many more. Among these, one of the most popular models is the GiererMeinhardt system [9], [16], [19], which in one dimension with a precursor-inhomogeneity and
two small diffusivities can be stated as follows:

2
x ∈ (−1, 1), t > 0,
At = ε2 ∆A − µ(x)A + AH ,



(1.1)
τ Ht = D∆H − H + A2 ,
x ∈ (−1, 1), t > 0,


 ′
A (−1) = A′ (1) = H ′ (−1) = H ′ (1) = 0,
where
0<ε≪
√
D ≪ 1,
τ ≥ 0 and τ is independent of ε.
In the standard Gierer-Meinhardt system without precursor it is assumed that µ(x) ≡ 1.
Precursor gradients in reaction-diffusion systems have been investigated in earlier work. The
original Gierer-Meinhardt system [9], [16], [19] has been introduced with precursor gradients.
These precursors were proposed to model the localization of the head structure in the coelenterate Hydra. Gradients have also been used in the Brusselator model to restrict pattern formation
to some fraction of the spatial domain [13]. In that example, the gradient carries the system in
and out of the pattern-forming part of the parameter range (across the Turing bifurcation), thus
effectively confining the domain where peak formation can occur. A similar localization effect
has been used to model segmentation patterns in the fruit fly Drosophila melanogaster in [15]
and [12].
In this paper, we study the Gierer-Meinhardt system with precursor and prove the existence
and stability of a cluster, which consists of N spikes approaching the same limiting point.
More precisely, we prove the existence of a steady-state spike cluster consisting of N spikes
near a non-degenerate local minimum point t0 of the inhomogeneity µ(x) ∈ C 3 (Ω), i.e. we
assume that µ′ (t0 ) = 0, µ′′ (t0 ) > 0. Further, we show that this solution is linearly stable.
We explicitly compute all eigenvalues, both large (of order O(1)) and small (of order o(1)).
The main features of studying the Gierer-Meinhardt system in this setting are as follows: (i) it
is biologically relevant since it models a hierarchical process (pattern formation of small-scale
structures induced by a pre-existing inhomogeneity)
(ii) it is important to note that this system contains three different spatial scales two of which
are small (i.e. o(1)):
(a) The O(1) scale of the precursor µ(x),
√
(b) The O( D) scale of the inhibitor diffusivity,
(c) The O(ε) scale of the activator diffusivity.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
Consequently there are the two small quantities
throughout the paper.
√
D and
√ε
D
3
which play an important role
(iii) the expressions can be made explicit and often have a particularly simple form.
Let us now summarize the analytical approach employed in our paper. The existence proof
is based on Liapunov-Schmidt reduction. The stability of the cluster is shown by first separating the eigenvalues into two cases: large eigenvalues which tend to a nonzero limit and small
eigenvalues which tend to zero in the limit D → 0 and √εD → 0. Large eigenvalues are then
explored by deriving suitable nonlocal eigenvalue problems and studying them using results of
[36] and a compactness argument of Dancer [4]. Small eigenvalues are calculated explicitly by
using asymptotic analysis with rigorous error estimates.
We shall establish the existence and stability of positive N−peaked steady-state spike clusters
to (1.1). The steady-state problem for positive solutions of (1.1) is the following:
 2 ′′
2
ε A − µ(x)A + AH = 0
x ∈ (−1, 1),





 DH ′′ − H + A2 = 0
x ∈ (−1, 1),
(1.2)


A(x) > 0, H(x) > 0,
x ∈ (−1, 1),



 ′
A (−1) = A′ (1) = H ′ (−1) = H ′ (1) = 0.
To have a nontrivial spike cluster, we assume throughout the paper that
N ≥ 2.
(1.3)
Before stating our main results, let us review some precious results on pattern formation for the
Gierer-Meinhardt system (1.1) , in particular concerning spiky patterns.
1. I. Takagi [30] proved the existence of N-spike steady-state solutions of (1.1) in an interval
for homogeneous coefficients (i.e. µ(x) = 1) in the regime ε ≪ 1 and D ≫ 1, where N is an
arbitrary positive integer. For these solutions, the spikes are identical copies of each other and
their maxima are located at
tj = −1 +
2j − 1
,
N
j = 1, . . . , N,
The proof in [30] is based on symmetry and the implicit function theorem.
2. In [14] (using matched asymptotic expansions) and [43] (based on rigorous proofs), the
following stability result has been shown: for the N-spike steady-state solution derived in item
1 and 0 ≤ τ < τ0 (N), where τ0 (N) > 0 is independent of ε, there are numbers D1 > D2 >
· · · > DN > · · · (which have been computed explicitly) such that the N-spike steady-state is
stable for for D < DN and unstable for D > DN .
4
JUNCHENG WEI AND MATTHIAS WINTER
3. In [14] (using matched asymptotic expansions) and [33] (based on rigorous analysis)
the following existence and stability results have been shown: for a certain parameter range
of D, the Gierer-Meinhardt system (1.1) with µ(x) = 1 has asymmetric N−spike steadystate solutions, which consist of exact copies of precisely two different spikes with distinct
amplitudes. They can be considered as bifurcating solutions from those in item 1 such that
the amplitudes start to differ at the bifurcation point (saddle-node bifuraction). The stability of
these asymmetric N−peaked solutions has been studied in [33].
4. In [45] the existence and stability of N−peaked steady states for the Gierer-Meinhardt
system with precursor inhomogeneity has been shown. These spikes have different amplitudes.
In particular, the results imply that precursor inhomogeneities can induce instabilities. Singlespike solutions for the Gierer-Meinhardt system with precursor including spike motion have
been studied in [32].
5. In [42] the existence of symmetric and asymmetric multiple spike clusters in an interval
has been shown.
Compared to each of the items listed above, the setting and results in our paper have marked
differences. We now consider two small parameters, D and √εD which results in new types of
behavior. The leading-order asymptotic expression of the large and small eigenvalues depend
on the index of the eigenvalue quadratically, whereas in items 1 and 2 this relation is oscillatory
(involving trigonometric functions).
In our study, the spikes in leading order have equal amplitudes and uniform spacing, although
there is precursor inhomogeneity in the system, in contrast to item 3. The amplitudes, positions
and eigenvalues in our study can be characterized explicitly and have a simpler form than in
item 4. We can also prove the stability of clusters not merely their existence as in item 5. In
particular, we show here that the clusters may be stable, whereas in item 5 they are expected to
be unstable.
In the shadow system case (D = ∞) the existence of single- or N-peaked solutions has
been established in [10, 11, 21, 22] and other papers. It is interesting to remark that symmetric
and asymmetric patterns can also be obtained for the Gierer-Meinhardt system on the real line,
see [5, 6]. We refer to [23] for the SLEP approch for the existence and stability of multilayered solutions for reaction-diffusion systems. For two-dimensional domains the existence
and stability of multi-peaked steady states has been proved in [38, 39, 40] and results similar to
items 1 and 2 have been derived. Hopf bifurcation has been established in [4, 34, 35, 40]. The
repulsive dynamics of multiple spikes has been studied in [7].
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
5
Another study with three different spatial scales, two of which are small, considers a consumer chain model allowing for a novel type of spiky clustered pattern which is stable for
certain parameters [46].
The model in our paper shows some similarity to variational models for material microstructure [1, 20, 48]. In both models the solutions have two small scales. However, in our case we
have two parameters to control each of them independently, whereas in the microstructure case
they are expressions of different orders depending on the same small parameter and so they are
related to one another.
Results on the existence and stability of multi-spike steady states have been reviewed and put
in a general context in [47].
This paper has the following structure: In Section 2 we state our main results on existence
and stability and present four highlights of their proofs. In Section 3 we introduce some preliminaries. In Sections 4–5 and Appendices A-B we prove the existence of steady-state spike
clusters: in Section 4 we introduce suitable approximate solutions, in Appendix A we compute
their error, in Appendix B we use the Liapunov-Schmidt method to reduce the existence of solutions of (1.2) to a finite-dimensional problem, in Section 5 we solve this finite-dimensional
reduced problem. In Sections 6–7 and Appendix C we prove the stability of these steady-state
spike clusters: in Section 6 we study the large eigenvalues of the linearized operator and show
that it has diagonal form. We give a complete description of their asymptotic behavior which
is stated in Lemma 12. In Section 7 we characterize the small eigenvalues of the linearized
operator and show that they all have negative real part. This includes deriving the eigenvalues
of a matrix which is needed to compute the small eigenvalues explicitly. We give a complete
description of their asymptotic behavior in leading order which can be found in Lemma 13. Our
approach here is to interpret the main matrix as the finite-difference approximation of a suitable
ordinary differential equation, compute the solution of this approximation explicitly and get the
eigenvectors by taking the values of this solution at uniformly spaced points. In Appendix C we
perform the technical analysis needed to derive the small eigenvalues. In Section 8 we conclude
with a discussion of our results with respect to the bridging of length scales and the hierarchy
of multi-stage biological processes.
Acknowledgements: This research is supported by NSERC of Canada. MW thanks the Department of Mathematics at UBC for their kind hospitality.
2. M AIN R ESULTS
ON
E XISTENCE
AND
S TABILITY
In this section, we state our main results on existence and stability of solutions and present
four highlights of our approach.
6
JUNCHENG WEI AND MATTHIAS WINTER
We first need to introduce some essential notation. Let L2 (−1, 1) and H 2 (−1, 1) denote
Lebesgue and Sobolev space, respectively. Let the function w be the unique solution (ground
state) of the problem
 ′′
w − w + w 2 = 0, y ∈ R,



w > 0,
w(0) = maxy∈R w(y),
(2.1)



w(y) → 0
as |y| → ∞.
Then w(y) can explicitly be written as
w(y) =
Elementary calculations give
ˆ
w 2 (z) dz = 6,
ˆ
R
y 3
cosh−2
.
2
2
3
w (z) dz = 7.2,
R
ˆ
(2.2)
2
(w ′ ) (z) dz = 1.2.
(2.3)
R
Let
Ω = (−1, 1).
For z ∈ (−1, 1), let GD (x, z) be the Green’s function defined by
(
DG′′D (x, z) − GD (x, z) + δz (x) = 0, x ∈ (−1, 1),
G′D (−1, z) = G′D (1, z) = 0,
where G′D (x, z) =
∂
G (x, z)
∂x D
(2.4)
(and the lefthand and righthand limits are considered for x = z).
We calculate
GD (x, z) =
(
θ
sinh(2θ)
cosh[θ(1 + x)] cosh[θ(1 − z)], −1 < x < z,
θ
sinh(2θ)
cosh[θ(1 − x)] cosh[θ(1 + z)], z < x < 1,
(2.5)
where
θ = D −1/2 .
Let t0 ∈ (−1, 1) and set
µ0 = µ(t0 ).
(2.6)
1
ξˆ0 = √
.
2 DGD (t0 , t0 )(µ0 )3/2
(2.7)
√
2 D
ξε := ´ 2
.
ε R w (z) dz
(2.8)
Let ξˆ0 be defined by
We set
Our first result is about the existence of an N-spike cluster solution near a non-degenerate
minimum point of the precursor.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
7
Theorem 1. (Existence of an N-spike cluster.)
Let N be a positive integer and t0 ∈ (−1, 1). We assume that µ ∈ C 3 (−1, 1) and
µ′ (t0 ) = 0, µ′′ (t0 ) > 0.
Then, for ε ≪
√
(2.9)
D ≪ 1, problem (1.2) has an N-spike cluster solution which concentrates
at t0 . In particular, it satisfies
Aε (x) ∼
N
X
ξε ξˆ0 µ0 w
k=1
Hε (tεk ) ∼ ξε ξˆ0 ,
tεk → t0 ,
p
µ0
x − tεk
ε
,
k = 1, . . . , N,
k = 1, . . . , N,
(2.10)
(2.11)
(2.12)
where µ0 has been defined in (2.6), ξˆ0 has been introduced in (2.7) and ξε has been defined in
(2.8).
Next we state our second result which concerns the stability of the N-spike cluster steady
states given in Theorem 1.
Theorem 2. (Stability of an N-spike cluster.)
√
For ε ≪ D ≪ 1, let (Aε , Hε ) be an N-spike cluster steady state given in Theorem 1. Then
√
there exists τ0 > 0 independent of ε and D such that the N-spike cluster steady state (Aε , Hε )
is linearly stable for all 0 ≤ τ < τ0 .
Remark 3. For the stability, we assume that 0 ≤ τ < τ0 for some τ0 > 0. Stability in the
case where τ is large has been investigated in [35] for single spikes and those results on Hopf
bifurcation are expected to carry over to the case of an N-spike cluster considered here. We
remark that stability in the case of large τ for the shadow system has been studied in [4, 34]. It
turns out that this Hopf bifurcation leads to oscillations of the amplitudes.
Remark 4. Previous studies of the precursor case can be found among others in [2], [27], [28].
We also refer to results for the dynamics of pulses in heterogeneous media [24], [49].
The proofs of both Theorems 1 and 2 will follow the approach in [47], where we reviewed
and discussed many results on the existence and stability of multi-spike steady states.
Next we present some highlights of the proofs of Theorems 1 and 2 in an informal manner.
We will give reference to the full proofs which will follow in later sections.
Highlight 1: For the proof in Theorem 1 we use Liapunov-Schmidt reduction to derive a
reduced problem which will determine the positions of the spikes. This reduced problem in
8
JUNCHENG WEI AND MATTHIAS WINTER
leading order is given by
W0 (t) ∼ c1
X
k,|k−s|=1
√
−|ts −tk |/ D
e
√
ts − tk
−
+ c2 Dµ′′ (t0 )(ts − t0 ),
|ts − tk |
s = 1, . . . , N,
(2.13)
where c1 , c2 > 0 are constants which are independent of the small parameters and t = (t1 , . . . , tN )
are the positions of the spikes (compare (5.4)). We need to solve W0 (t) = 0, which implies
√
1
ts − ts−1 ∼ D log , s = 2, . . . , N,
D
(compare (5.6)). The distance between neighboring spikes in the cluster is small (converging to
zero) and in leading order it is the same between any pair of neighbors.
Highlight 2: The large eigenvalues with λε → λ0 6= 0 and their corresponding eigenfunctions
x − ti
φε,i (y) → φi (y), y =
, i = 1, . . . , N,
ε
where φε,i (y) is the
restriction of the rescaled eigenfunction of the activator Aε near ti , in the
limit max
√ε , D
D
→ 0 solve the nonlocal eigenvalue problem (NLEP)
´
2 R wφi dy 2
∆y φi − φi + 2wφi − ´ 2
w = λ0 φi , i = 1, . . . , N,
w dy
R
(see (6.6)). This nonlocal eigenvalue problem has diagonal form. Thus each spike only interacts
with itself and not with the other spikes.
It follows that the spike cluster is stable with the respect to large eigenvalues.
Highlight 3: The small eigenvalues λε → 0 in leading order are given by the eigenvalues of
the matrix
−ε2 c3 M(t0 ),
where c3 > 0 is independent of the small parameters, t0 = (t0 , . . . , t0 ) and
′′ 0
M(t0 )N
i,j=1 ∼ µ (t )
1
× log [−(i − 1)(N + 1 − i)δi,j−1 − i(N − i)δi,j+1 + [(i − 1)(N + 1 − i) + i(N − i)]δi,j ]+4δi,j
D
with δN,0 = δ1,N +1 = 0 (compare 7.13)).
Highlight 4: We determine all the eigenvalues of the matrix M(t0) (see Highlight 3) ex-
plicitly by a method based on exactly finding a finite-difference approximation to a suitable
ordinary differential equation.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
9
These eigenvalues are given by
1
c3 µ′′ (t0 )n(n + 1),
D
Further, there is an eigenvalue of smaller size given by
λn,ε ∼ −ε2 log
n = 1, . . . , N − 1.
λ0,ε ∼ −ε2 4c3 µ′′ (t0 )
(compare Lemma 13).
This implies that the spike cluster is stable with respect to small eigenvalues.
3. P RELIMINARIES :
SCALING PROPERTY,
G REEN ’ S
FUNCTION AND EIGENVALUE
PROBLEMS
In this section we will provide some preliminaries which will be needed later for the existence
and stability proofs.
Let w be the ground state solution given in (2.1). By a simple scaling argument, the function
√
wa (y) = aw( a y)
(3.1)
is the unique solution of the problem
( ′′
wa − awa + wa2 = 0 y ∈ R,
wa > 0,
wa (y) → 0 as |y| → ∞.
wa (0) = maxy∈R wa (y),
(3.2)
We compute
ˆ
wa2 (y) dy
3/2
=a
R
ˆ
R
ˆ
2
w (z) dz,
ˆ
5/2
=a
R
(wa′ )2 (y) dy = a5/2
R
wa3 (y) dy
ˆ
ˆ
w 3 (z)dz,
R
(w ′ )2 (z) dz.
(3.3)
R
We set
1 − √1 |x−z|
KD (|x − z|) = √ e D
(3.4)
2 D
to be the singular part of GD (x, z). Let the regular part HD of GD be defined by HD = KD −
GD . Note that HD (x, z) belongs to the space C ∞ ((−1, 1) × (−1, 1)).
By (2.5) we have
√ 0 0
−2(d0 −η0 )/ D
,
(3.5)
GD (t , t ) = KD (0) 1 + O e
where d0 = min(1 − t0 , t0 + 1) and η0 > 0 is an arbitrary but fixed constant.
For ξˆ0 , we estimate
√ 1
1
ξˆ0 = √
= 0 3/2 + O e−2(d0 −η0 )/ D
(µ )
2 DGD (t0 , t0 )(µ0 )3/2
by (3.4), (3.5).
(3.6)
10
JUNCHENG WEI AND MATTHIAS WINTER
Let us denote ∂t∂i as ∇ti . When i 6= j, we can define ∇ti G(ti , tj ) in the classical way because
the function is smooth. When i = j, then KD (|ti − tj |) = KD (0) = 2√1D is a constant
independent of ti and we define
Similarly, we define
∂ ∇ti GD (ti , ti ) := −
H(x, ti ).
∂x x=ti
∇ti ∇tj GD (ti , tj ) =
(
∂
∂
− ∂x
|x=ti ∂y
|y=ti HD (x, y) if i = j,
∇ti ∇tj GD (ti , tj ) if i 6= j.
(3.7)
For convenience and clarity, we introduce a re-scaled version of the Green’s function which has
a finite limit as D → 0. Thus we set
√
ĜD (x, z) = 2 DGD (x, z),
√
K̂D (x, z) = 2 DKD (x, z),
√
ĤD (x, z) = 2 DHD (x, z).
(3.8)
(3.9)
(3.10)
Next we consider the stability of a system of nonlocal eigenvalue problems (NLEPs). We first
recall the following result:
Theorem 5. Consider the nonlocal eigenvalue problem
´
wφ dy
′′
φ − φ + 2wφ − γ ´R 2 w 2 = αφ.
w dy
R
(3.11)
(1) (Appendix E of [14].) If γ < 1, then there is a positive eigenvalue to (3.11).
(2) (Theorem 1.4 of [36].) If γ > 1, then for any nonzero eigenvalue α of (3.11) we have
Re(α) ≤ −c < 0.
(3) If γ 6= 1 and α = 0, then
φ = c0 w ′
for some constant c0 .
Next we consider the following system of nonlocal eigenvalue problems:
´
wΦ dy
′′
LΦ := Φ − Φ + 2wΦ − 2 ´R 2 w 2 ,
w dy
R
where




Φ=



φ1


φ2 

 ∈ (H 2 (R))N .
.. 
. 

φN
(3.12)
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
11
Set
L0 u := u′′ − u + 2wu,
where u ∈ H 2 (R).
Then the conjugate operator of L under the scalar product in L2 (R) is given by
´ 2
w Ψ dy
L∗ Ψ = Ψ′′ − Ψ + 2wΨ − 2 ´R 2
w,
w dy
R
where




Ψ=



Then we have the following result.
ψ1
(3.13)
(3.14)


ψ2 

 ∈ (H 2 (R))N .
.. 
. 

ψN
Lemma 6. [43] We have
Ker(L) = X0 ⊕ X0 ⊕ · · · ⊕ X0 ,
(3.15)
where
X0 = span {w ′(y)}
and
Ker(L∗ ) = X0 ⊕ X0 ⊕ · · · ⊕ X0 .
(3.16)
Proof. The system (3.12) is in diagonal form. Suppose
LΦ = 0.
For l = 1, 2, . . . , N the l-th equation of system (3.12) is given by
´
wΦl dy 2
w = 0.
Φ′′l − Φl + 2wΦl − 2 ´R 2
w dy
R
(3.17)
By Theorem 5 (3) with γ = 2, we have
Φl ∈ X0 ,
l = 1, . . . , N
(3.18)
and (3.15) follows.
To prove (3.16), we proceed in a similar way for L∗ . The l-th equation of (3.14) is given as
follows:
Ψ′′l
− Ψl + 2wΨl − 2
´
w 2 Ψl dy
w = 0.
w 2 dy
R
R
´
Multiplying (3.19) by w and integrating, we obtain
ˆ
w 2 Ψl dy = 0.
R
(3.19)
12
JUNCHENG WEI AND MATTHIAS WINTER
Thus all the nonlocal terms vanish and we have
L0 Ψl = 0,
l = 1, . . . , N.
(3.20)
By Theorem 5 (3) with γ = 0, this implies
Ψ l ∈ X0 ,
l = 1, . . . , N.
As a consequence of Lemma 6, we have
Lemma 7. [43] The operator
´
wΦ dy
L : (H (R)) → (L (R)) , LΦ = Φ − Φ + 2wΦ − 2 ´R 2 w 2 ,
w dy
R
2
N
2
N
′′
is invertible if it is restricted as follows
L : (X0 ⊕ · · · ⊕ X0 )⊥ ∩ (H 2 (R))N → (X0 ⊕ · · · ⊕ X0 )⊥ ∩ (L2 (R))N .
Moreover, L−1 is bounded.
Proof. This result follows from the Fredholm Alternative and Lemma 6.
Finally, we study the eigenvalue problem for L:
LΦ = αΦ.
(3.21)
We have
Lemma 8. For any nonzero eigenvalue α of (3.21) we have Re(α) ≤ −c < 0.
Proof. Let (Φ, α) satisfy the system (3.21). Suppose Re(α) ≥ 0 and α 6= 0. Then the l-th
equation of (3.21) becomes
´
wΦl
Φ′′l − Φl + 2wΦl − 2 ´R 2 w 2 = αΦl .
w
R
By Theorem 5 (2) we conclude that
Re(α) ≤ −c < 0.
Throughout the paper, let C, c denote generic constants which may change from line to line.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
4. E XISTENCE
PROOF
I:
13
APPROXIMATE SOLUTIONS
Let t0 ∈ (−1, 1) be a non-degenerate minimum point of the precursor inhomogeneity, i.e. we
assume that (2.9) is satisfied. In this section, we construct an approximation to a spike cluster
solution to (1.2) which concentrates at t0 .
√
i
The approximate cluster consists of the spikes µi w µi x−t
which are centered at the points
ε
ti and have the scaling factors µi = µ(ti ), where i = 1, . . . , N.
Let Ωη denote the set of all t = (t1 , t2 , . . . , tN ) ∈ ΩN with −1 < t1 < t2 · · · < tN < 1
satisfying (4.1) and (4.2), where
′′ 0 t − t
1
5µ
(t
)
1
s
s−1
− log + log log + log
+ log[(s − 1)(N + 1 − s)] ≤ η
√
0
D
D
16µ
D
(4.1)
for s = 2, . . . , N,
N
1 X
1
tk − t0 ≤ η log
N
D
k=1
(4.2)
and η > 0 is a constant which is small enough and will be chosen in Section 5. The reason for
assuming (4.1) and (4.2) will become clear in Section 5 when we solve the reduced problem.
We further denote
t0 = (t0 , t0 , . . . , t0 )
(4.3)
Ω0 = {t0 }.
(4.4)
and set
To simplify our notation, for t ∈ Ωη and k = 1, . . . , N, we set
√ x − ti
wk (x) = µk w
µk
ε
(4.5)
and
x − tk x − tk
,
· χ w̃k (x) = µk w
µk
ε
δε where χ is a smooth cut-off function which satisfies the conditions
1
χ(x) = 1 for |x| < ,
2
√
3
χ(x) = 0 for |x| > ,
4
χ ∈ C0∞ (R)
(4.6)
(4.7)
and
10
1
ε ≪ δε ≪ p ε log .
0
ε
µ
√
Using (4.1), we have |ti − t0 | = O
D log D1 for i = 1, . . . , N. This implies
2 !
1
|µ(ti ) − µ(t0 )| = O D log
,
D
(4.8)
(4.9)
14
JUNCHENG WEI AND MATTHIAS WINTER
2 !
√
1
1
′
′′ 0
0
µ (ti ) = µ (t )(ti − t ) + O D log
D log
=O
,
D
D
√
1
′′
′′ 0
= O(1), µ′′ (t0 ) = O(1).
µ (ti ) = µ (t ) + O
D log
D
µ′′′ (ti ) = O(1), µ′′′ (t0 ) = O(1).
(4.10)
(4.11)
(4.12)
To simplify notation, we set
µi = µ(ti ),
i = 1, . . . , N.
(4.13)
Further, we compute, using (2.5),
√
√
ĜD (ti , ti ) = K̂D (0) 1 + O(e−2(d0 −η0 )/ D ) = 1 + O(e−2(d0 −η0 )/ D ),
(4.14)
where d0 = min(1 − t0 , t0 + 1), η0 > 0 is an arbitrary but fixed constant (compare (3.5)). We
have
1
1
ĜD (ti , ts ) = O D log
, K̂D (ti , ts ) = O D log
for |i − s| = 1,
(4.15)
D
D
2 !
2 !
1
1
ĜD (ti , ts ) = O
D log
, K̂D (ti , ts ) = O
D log
for |i − s| = 2. (4.16)
D
D
Generally, we have
ĜD (ti , ts ) = O
1
D log
D
|i−s| !
|i−s| !
1
, K̂D (ti , ts ) = O
D log
for |i − s| ≥ 1.
D
(4.17)
For the derivatives, we estimate
!
|i−s|
∂k
1
D −k/2 for |i − s| ≥ 1,
Ĝ (t , t ) = O
D log
k D i s
D
∂ti
!
|i−s|
∂k
1
D −k/2 for |i − s| ≥ 1,
K̂D (ti , ts ) = O
D log
k
D
∂ti
k = 1, 2, . . .
(4.18)
k = 1, 2, . . .
(4.19)
and analogous results hold for the mixed derivatives.
By rescaling  = ξε A, Ĥ = ξε H with ξε defined in (2.8), it follows that (1.2) is equivalent
to the following system for the rescaled functions Â, Ĥ:

2

ε2 Â′′ − µ(x)Â + ÂĤ = 0, x ∈ (−1, 1),




 D Ĥ ′′ − Ĥ + ξ Â2 = 0, x ∈ (−1, 1),
ε


Â(x) > 0, Ĥ(x) > 0 in (−1, 1),



 ′
 (−1) = Â′ (1) = Ĥ ′ (−1) = Ĥ ′ (1) = 0.
(4.20)
From now on, we shall work with (4.20) and drop the hats. Next we rewrite (4.20) as a single
equation with a nonlocal term.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
For a function A ∈ H 2 (−1, 1), we define T [A] to be the solution of
(
D(T [A])′′ − T [A] + ξε A2 = 0, −1 < x < 1,
(T [A])′ (−1) = (T [A])′ (1) = 0.
15
(4.21)
It is easy to see that the solution T [A] is unique and positive. Then (4.20) becomes
A2
= 0, A > 0, A′ (−1) = A′ (1) = 0.
T [A]
For t ∈ Ωη , we define an approximate solution to (4.22) as follows:
Sε [A] := ε2 A′′ − A +
A(x) = wε,t(x) =
N
X
ξˆk w̃k (x),
k=1
x ∈ Ω,
(4.22)
(4.23)
where t ∈ Ωη and w̃k has been defined in (4.6).
Next we are now going to determine the amplitudes ξˆk to leading order. Let us first compute
τs := T [wε,t](ts ).
(4.24)
From (4.21), we have
ˆ
= ξε
ˆ
1
−1
1
2
τs = ξε
GD (ts , z)wε,t
(z) dz
−1
" N
#
X
X
GD (ts , z)
ξˆk2 w̃k2 (z) +
ξˆk ξˆl w̃k (z)w̃l (z) dz = I1 ,
k=1
(4.25)
k6=l
where I1 is defined by the last equality.
We have
2
ˆ
ˆ
√ x − tk
2
µk
dx (1 + O(ε10)).
ξε GD (ts , x)w̃k (x) dx = ξε GD (ts , x) µk w
ε
Ω
Ω
For k 6= s, we compute
ˆ
ˆ
ε
2
3/2
2
ξε GD (ts , x)w̃k (x) dx = ξε ε(µk ) GD (ts , tk )
w (y) dy + O √
D
Ω
R
ε
= (µk )3/2 ĜD (ts , tk ) 1 + O √
D
"
2 !#
ε
1
= (µ0 )3/2 ĜD (ts , tk ) 1 + O √ + D log
D
D
"
2 !#
1
ε
1
= (µ0 )3/2 O D log
1 + O √ + D log
D
D
D
1
= O D log
,
(4.26)
D
using (2.8), (3.8), (4.9) and (4.17). For k = s, we have
ˆ
ξε GD (ts , x)w̃s2 (x) dx
Ω
16
JUNCHENG WEI AND MATTHIAS WINTER
1 −|ts −x|/√D
√ e
− HD (ts , x) w̃s2 (x) dx
= ξε
Ω 2 D
ˆ
ε
3/2
2
= ξε ε(µs ) GD (ts , ts )
w (y) dy + O √
D
R
ε
3/2
= (µs ) ĜD (ts , ts ) + O √
D
2 !
1
ε
= (µ0 )3/2 ĜD (ts , ts ) + O √ + D log
,
D
D
ˆ using (2.8), (3.8) and (4.9). Next, for k 6= l, we have
ˆ
ξε ĜD (ts , z)w̃k (z)w̃l (z) dz = 0
(4.27)
(4.28)
Ω
by (4.6). Combining (4.26), (4.27) and (4.28), we have
I1 =
N
X
ξˆk2 (µk )3/2 ĜD (ts , tk )
k=1
=
ξˆs2 (µs )3/2 ĜD (ts , ts )
ε
1+O √
D
ε
1
1 + O √ + D log
.
D
D
(4.29)
Substituting (4.29) into (4.25), we conclude that
ε
1
2
3/2
ˆ
T [wε,t ](ts ) = τs = ξs (µs ) ĜD (ts , ts ) 1 + O √ + D log
D
D
"
= ξˆs2 (µ0 )3/2 1 + O
2 !#
ε
1
√ + D log
.
D
D
We now choose ξˆs such that τs = ξˆs . Then (4.30) has a unique solution which satisfies
"
2 !#
1
ε
1
.
ξˆs = 0 3/2 1 + O √ + D log
(µ )
D
D
(4.30)
(4.31)
This concludes the construction of the approximate solution.
The next two parts of the existence proof have been moved to the appendix since they are
quite technical and follow the approaches in previous papers:
Appendix A: Existence Proof II – error terms
Appendix B: Existence Proof III – Liapunov Schmidt reduction.
In the next section, we continue with the discussion of the reduced problem, which concludes
the Proof of Theorem 1.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
5. E XISTENCE
PROOF
IV:
17
REDUCED PROBLEM
In this section, we solve the reduced problem. This completes the proof of our main existence
result given by Theorem 1.
⊥
By Lemma 19, for every t ∈ Ωη , there exists a unique solution φǫ,t ∈ Kǫ,t
such that
Sε [wε,t + φε,t] = vε,t ∈ Cǫ,t .
(5.1)
We need to determine tε = (tε1 , tε2 , . . . , tεN ) ∈ Ωη such that
Sε [wε,tε + φε,tε ] ⊥ Cǫ,tε ,
(5.2)
which implies Sε [wε,tε + φε,tε ] = 0.
To this end, let
1
Wǫ,s (t) := √
ε D log D1
ˆ
Ω
Sε [wε,t + φε,t]
dw̃s
dx,
dx
Wǫ (t) := (Wǫ,1(t), . . . , Wǫ,N (t)) : Ωη → RN .
Then the map Wǫ (t) is continuous in t ∈ Ωη and it remains to find a zero of the vector field
Wε (t).
We compute
ˆ
1
dw̃s
√
S
[w
+
φ
]
dx
ε
ε,t
ε,t
dx
ε D log D1 Ω
#
ˆ "
dw̃s
Sε [wε,t ] + Sε′ [wε,t ](φε,t ) + O(kφε,tk2H 2 (Iε ) )
dx.
dx
Ω
1
= √
ε D log D1
We first compute the main term given by
ˆ
1
dw̃s
√
Sε [wε,t ]
dx = cs
1
dx
ε D log D Ω
Let x = ts + εy. By (9.10), we have
ˆ
1
dw̃s
√
Sε [wε,t ]
dx = cs,1 + cs,2,
1
dx
ε D log D Ω
where
cs,1
1
=−
D log D1
=
√
X
x − ts 2 dws
ts − tk
3/2 ˆ2 −|ts −tk |/ D
w̃s
dx
(µk ) ξk e
−
ε
dx
|ts − tk |
Ω
k6=s
√
ε
−2(d0 −η0 )/ D
+O √ + e
D
ˆ
√
X
1
ts − tk
3
3/2 ˆ2 −|ts −tk |/ D
w dy
(µk ) ξk e
−
3 R µs
|ts − tk |
k6=s
√
ε
−2(d0 −η0 )/ D
+O √ + e
D
ˆ
1
D log D1
(5.3)
18
JUNCHENG WEI AND MATTHIAS WINTER
1
1
=
(µs )5/2
1
D log D 3
ˆ
3
w dy
R
k6=s
√
(µk )3/2 ξˆk2e−|ts −tk |/ D
ts − tk
−
|ts − tk |
√
ε
−2(d0 −η0 )/ D
+O √ + e
D
√
X
ts − tk
2.4
5/2
3/2 ˆ2 −|ts −tk |/ D
(µs )
(µk ) ξk e
−
=
|ts − tk |
D log D1
k6=s
√
ε
−2(d0 −η0 )/ D
+O √ + e
D
√
X
√
ε
ts − tk
2.4
1
4 ˆ0 2
−|ts −tk |/ D
(µ0 ) (ξ )
e
−
+ O √ + D log
=
|t
D
D log D1
D
s − tk |
k6=s
and
X
cs,2
ˆ
√
1
x − ts dw̃s
ε
1
′′ 0
0 ˆ
=− √
µ (t )(ts − t )ξs ε
w̃s
dx + O √ + D log
ε
dx
D
ε D log D1
D
Ω
ˆ
√
1
dwµs
ε
1
′′ 0
0 ˆ
µ (t )(ts − t )ξs ywµs
dy + O √ + D log
= −√
dy
D
D log D1
D
R
ˆ
√
1
1
ε
3/2 ′′ 0
0 1ˆ
2
(µs ) µ (t )(ts − t ) ξs w dy + O √ + D log
=√
2
D
D log D1
D
R
√
2
ε
1
3/2 ˆ ′′ 0
0
√
(µ
)
ξ
µ
(t
)(t
−
t
)
+
O
+
=√
D log
s
s
s
1
D
D log D
D
√
3
1
ε
0 3/2 ˆ0 ′′ 0
0
=√
(µ ) ξ µ (t )(ts − t ) + O √ + D log
.
D
D log D1
D
In summary, we have
√
X
2.4
ts − tk
4
0 2 −|ts −tk |/ D
ˆ
cs =
(µ0 )
(ξ ) e
−
|ts − tk |
D log D1
k6=s
√
√
ε
1
0 3/2 ˆ0 ′′ 0
0
+3 D(µ ) ξ µ (t )(ts − t ) + O √ + D log
, s = 1, . . . , N.
(5.4)
D
D
Next we estimate
ˆ
1
dw̃s
√
Sε′ [wε,t ](φε,t )
dx
1
dx
ε D log D Ω
#
ˆ "
2
w
1
2w
dw̃s
ε,t
ε,t
= √
φε,t −
(T ′[wε,t ]φε,t )
dx
ε2 φ′′ε,t − µ(x)φε,t +
2
1
T [wε,t]
(T [wε,t])
dx
ε D log D Ω
#
ˆ "
2
w
1
2w
dw̃s
ε,t
ε,t
2 ′′
′
= √
ε
φ
−
µ
φ
+
φ
−
(T
[w
]φ
)
dx
s
ε,t
ε,t
ε,t
ε,t
ε,t
2
T [wε,t ]
(T [wε,t ])
dx
ε D log D1 Ω
ˆ
1
dw̃s
√
+
−(µ(x) − µ(ts ))φε,t
dx
1
dx
ε D log D Ω
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
19
ˆ "
#
ˆs w̃s )2
1
1
(
ξ
dw̃s
2ξˆs w̃s φε,t −
−
(T ′ [wε,t ]φε,t )
dx
2
T [wε,t ] ξˆs
(T [wε,t ])
dx
Ω
ˆ
dw̃s
1
−(µ(x) − µ(ts ))φε,t
+ √
dx
1
dx
ε D log D Ω
ε
=O √
D
which follows from (10.9) and (10.10). This implies
√
X
2.4
ts − tk
4
0 2 −|ts −tk |/ D
ˆ
(µ0 )
(ξ ) e
Wε,s (t) =
−
|ts − tk |
D log D1
k6=s
√
√
ε
1
0 3/2 ˆ0 ′′ 0
0
+3 D(µ ) ξ µ (t )(ts − t ) + O √ + D log
, s = 1, . . . , N.
(5.5)
D
D
Now, for given small ε > 0, we have to determine tε ∈ Ωη such that Wε,s (tε ) = 0 for s =
1, . . . , N.
1
= √
ε D log D1
We first consider the limiting case which only takes into account the leading terms and set
√
X
1
ts − tk
0 4 ˆ0 2
−|ts −tk |/ D
(µ ) (ξ )
e
W0 (t) = 2.4
−
|ts − tk |
D log D1
k,|k−s|=1
√
+3 D(µ0 )3/2 ξˆ0 µ′′ (t0 )(ts − t0 ).
We compute W0 (t∗ ) = 0, where t∗ satisfies
t∗s − t∗s−1
1
1
√
= log − log log
D
D
D
′′ 0 log log D1
5µ (t )
− log
− log[(s − 1)(N + 1 − s)] + O
,
16µ0
log D1
N
1 X ∗
t = t0 .
N k=1 k
(5.7)
By (5.6) and (5.7), we have t∗ ∈ Ωη if D is small enough.
We need to find tε ∈ Ωη such that Wε (tε ) = 0.
Setting e = (1, 1 . . . , 1)T , we have
c
C
∗
√
√
≤
kDW
(t
)ek
≤
0
D log D1
D log D1
and
c
C
√ ≤ kDW0 (t∗ )vk ≤ √ kvk
D
D
(5.6)
if v · e = 0.
For t ∈ Ωε , we expand
Wε (t) = Wε (t) − W0 (t) + W0 (t) − W0 (t∗) + W0 (t∗ )
ε
=O √
by (5.4)
D
20
JUNCHENG WEI AND MATTHIAS WINTER
+DW0 (t∗ ) · (t − t∗ ) + R0 (t − t∗ )
+W0 (t∗ ),
where R0 (τ ) = D 2 W0 (t∗ )(τ, τ ). Decomposing τ = v + αe, where v · e = 0, we estimate
c5
c6
c4
|R0 (τ )| ≤ |v|2 + √
α|v| + √
α2 .
1
D
D log D
D log D1
√
√
Noting that for t∗ + τ ∈ Ωη , we have |v| ≤ η D and α ≤ η D log D1 , we get
√
√
1
2
|R0 (τ )| ≤ η c4 + c5 D + c6 D log
.
D
This implies
√
|(DW0 (t∗ ))−1 R0 (τ )v| ≤ c7 η 2 D
and
√
1
|(DW0 (t∗ ))−1 R0 (τ )αe| ≤ c7 η 2 D log .
D
Setting τ = t − t∗ , we have to determine τ such that
−(DW0 (t∗ ))−1 [Wε (t∗ + τ ) − W0 (t∗ + τ ) + R0 (τ )] = τ
and so τ must be a fixed point of the mapping
τ → Mε,D (τ ) := −(DW0 (t∗ ))−1 [Wε (t∗ + τ ) − W0 (t∗ + τ ) + R0 (τ )],
B1 → B1 ,
where B1 = Ωη − t∗ (pointwise). We estimate
kMε,D (τ )k = k − (DW0 (t∗ ))−1 [Wε (t∗ + τ ) − W0 (t∗ + τ ) + R0 (τ )]k
√
ε √
1
1 √
1
2
≤ C √ · D log + η D log
D log .
D
D
D
D
Using projections, we have
√
ε
2
kMε,D (τ ) · vk ≤ C( √ + η
D if v · e = 0
D
and
√
ε
1
2
kMε,D (τ ) · (αe)k ≤ C √ + η
D log .
D
D
We now determine when the mapping Mε,D maps from B1 into B1 for max √εD , D small
enough. We need to have
ε
2
C √ + η ≤ η.
(5.8)
D
Now (5.8) is satisfied if we choose
ε
η = 2C √
(5.9)
D
and we assume
ε2
ε
Cη 2 = 4C 3 ≤ C √ .
(5.10)
D
D
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
21
Note that (5.10) is satisfied if
ε
1
√ ≤
4C 2
D
which holds if √εD is small enough since 4C1 2 is a constant which is independent of ε and D.
By Brouwer’s fixed point theorem, the mapping Mε,D possesses a fixed point τ ε ∈ B1 . Then
tε = t∗ + τ ε ∈ Ωη is the desired solution which satisfies Wε (tε ) = 0.
Thus we have proved the following proposition.
Proposition 9. For max √ǫD , D small enough, there exist points tǫ ∈ Ωη with tǫ → t0 such
that Wǫ (tε ) = 0.
Finally, we complete the proof of Theorem 1.
Proof. By Proposition 9, there exists tε → t0 such that Wε (tε ) = 0. Written differently, we
have Sε [wε,tε +φε,tε ] = 0. Let Aε = ξε (wε,tε +φε,tε ), Hε = ξε T [wε,tε +φε,tε ]. By the Maximum
Principle, Aε > 0, Hε > 0. Moreover (Aε , Hε ) satisfies all the properties of Theorem 1.
6. S TABILITY
0.
PROOF
I:
LARGE EIGENVALUES
In this section, we study the large eigenvalues which satisfy λε → λ0 6= 0 as max
√ε , D
D
→
First we consider the special case τ = 0. Then we need to analyze the eigenvalue problem
2Aε φε
A2ε
L̃ φε = ε ∆φε − µ(x)φε +
−
(T ′ [Aε ]φε ) = λε φǫ ,
(6.1)
2
T [Aε ]
(T [Aε ])
where λε is some complex number, Aε = wε,tε + φε,tε with tε ∈ Ωη determined in the previous
ε,tε
2
section,
φε ∈ HN2 (Ω).
and for φ ∈ L2 (Ω) the function T ′ [A]φ is defined as the unique solution of
(
D∆(T ′ [A]φ) − (T ′ [A]φ) + 2ξε Aφ = 0, −1 < x < 1,
(T ′ [A]φ)′ (−1) = (T ′ [A]φ)′ (1) = 0.
(6.2)
(6.3)
Because we study
the large eigenvalues, there exists some small c > 0 such that |λε | ≥ c > 0
√ǫ , D
D
small enough. We are looking for a condition under which Re(λε ) ≤ c < 0
for all eigenvalues λε of (6.1), (6.2) if max √εD , D is small enough, where c is independent
for max
of ε and D. If Re(λε ) ≤ −c, then λε is a stable large eigenvalue. Therefore, for the rest of this
section, we assume that Re(λε ) ≥ −c and study the stability properties of such eigenvalues.
22
JUNCHENG WEI AND MATTHIAS WINTER
We first derive the limiting problem of (6.1), (6.2) as max
by a system of NLEPs. Let us assume that
√ε , D
D
→ 0 which will be given
kφε kH 2 (Ωε ) = 1.
We cut off φε as follows: Introduce
εy φε,j (y) = φε (y)χ ,
δε
(6.4)
where y = (x − tj )/ε for x ∈ Ω, the cut-off function χ was introduced in (4.7) and δε satisfies
(4.8) .
√ε
D
From (6.1), (6.2), using Re(λε ) ≥ −c and kφε,tε kH 2 (Ωε ) = O
φε =
N
X
φε,j + O
j=1
ε
√
D
, it follows that
in H 2 (Ωε ).
(6.5)
Then, by a standard procedure, we extend φε,j to a function defined on R such that
kφε,j kH 2 (R) ≤ Ckφε,j kH 2 (Ωε ) ,
j = 1, . . . , N.
Since kφε kH 2 (Ωε ) =
kH 2 (Ωε ) ≤ C. By taking a subsequence, we may also assume that
1, kφε,j
φε,j → φj as max √εD , D → 0 in H 1 (R) for j = 1, . . . , N.
Taking the limit max √εD , D → 0 with λε → λ0 in (6.1) we get
−2 lim
D→0
∆y φi − µi φi + 2w̃i φi
! N
!−1
ˆ
ˆ N
2
X
X
ξˆk0 w̃k dy
w̃i2 = λ0 φi .
ĜD (t0i , t0k ) ξˆk0 w̃k φk dy
ĜD (t0i , t0k )
R
k=1
Using the transformation ỹ =
this implies that
k=1
√
R
µ y and the relations
1
0 0
ĜD (ti , tj ) = δik + O D log
,
D
1
1
0
ξˆk = 0 3/2 1 + O D log
(µ )
D
´
wφi dy 2
∆y φi − φi + 2wφi − ´ R 2
w = λ0 φ i ,
w
dy
R
2
i = 1, . . . , N,
(6.6)
where φi ∈ H 2 (RN ).
Then we have
Theorem 10. Let λε be an eigenvalue of (6.1) and(6.2) suchthat Re(λε ) > −c for some c > 0.
(1) Suppose that (for suitable sequences max √εDn n , Dn → 0) we have λεn → λ0 6= 0.
Then λ0 is an eigenvalue of the problem (NLEP) given in (6.6).
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
23
(2) Letλ0 6= 0with Re(λ0 ) > 0 be an eigenvalue of the problem (NLEP) given in (6.6). Then
for max √εD , D small enough, there is an eigenvalue λε of (6.1) and (6.2) with λε → λ0 as
ε
√
max D , D → 0.
Proof. (1) of Theorem 10 follows by asymptotic analysis similar to Appendix B.
To prove (2) of Theorem 10, we follow a compactness argument of Dancer [4]. The main idea
of his approach is as follows: Let λ0 6= 0 be an eigenvalue of problem (6.6) with Re(λ0 ) > 0.
Then can we rewrite (6.1) as follows:
"
#
2
A
2Aφε
φε = −Rε (λε )
−
T ′ [A]φε ,
(6.7)
T [A]
T [A]
where Rε (λε ) is the inverse of −∆+(µ(x)+λε ) in H 2 (R) (which exists if Re(λε ) > − minx∈R µ(x)
or Im(λε ) 6= 0) and the nonlocal operators have been defined in (4.21)
respectively.
and (6.3),
The main property is that Rε (λε ) is a compact operator if max
The rest of the argument follows in the same way as in [4].
√ε , D
D
is small enough.
We now study the stability of (6.1), (6.2) for large eigenvalues explicitly and prove Theorem
2.
By Lemma 8, for any nonzero eigenvalue λ0 in (6.6) we have
Re(λ0 ) ≤ c0 < 0
for some c0 > 0.
Thus by Theorem 10 (1), for max √εD , D small enough, all nonzero large eigenvalues of
(6.1), (6.2) have strictly negative real parts. More precisely, all eigenvalues λε of (6.1), (6.2),
for which λε → λ0 6= 0 holds, satisfy Re(λε ) ≤ −c < 0.
When studying the case τ > 0, we have to deal with nonlocal eigenvalue problems as in
(3.11), for which the coefficient γ of the nonlocal term is a function of τ α. Let γ = γ(τ α) be a
complex function of τ α. Let us suppose that
γ(0) ∈ R,
|γ(τ α)| ≤ C for Re(α) = αR ≥ 0, τ ≥ 0,
(6.8)
where C is a generic constant which is independent of τ and α. In our case the following simple
example of a function γ(τ α) satisfying (6.8) is relevant:
γ(α) = √
where
√
2
,
1 + τα
1 + τ α denotes the principal branch of the square root function, compare [35].
Now we have
24
JUNCHENG WEI AND MATTHIAS WINTER
Lemma 11. [43] Consider the nonlocal eigenvalue problem
´
wφ dy
φ′′ − φ + 2wφ − γ(τ α) ´R 2 w 2 = αφ,
w dy
R
(6.9)
where γ(τ α) satisfies (6.8). Then there is a small number τ0 > 0 such that for τ < τ0 ,
(1) if γ(0) < 1, then there is a positive eigenvalue to (3.11);
(2) if γ(0) > 1, then for any nonzero eigenvalue α of (6.9), we have
Re(α) ≤ −c0 < 0.
Proof. Lemma 11 follows from Theorem 5 by a regular perturbation argument. To make sure
that the perturbation argument works, we have to show that if αR ≥ 0 and 0 ≤ τ < 1, then
|α| ≤ C, where C is a generic constant which is independent of τ . In fact, multiplying (6.9) by
the conjugate φ̄ of φ and integration by parts, we obtain that
´
ˆ
ˆ
ˆ
wφ dy
′ 2
2
2
2
R
´
(|φ | + |φ| − 2w|φ| ) dy = −α |φ| dy − γ(τ α)
w 2φ̄ dy.
(6.10)
2
w dy R
R
R
R
From the imaginary part of (6.10), we obtain that
|αI | ≤ C1 |γ(τ α)|,
√
where α = αR + −1αI and C1 is a positive constant (independent of τ ). By assumption (6.8),
|γ(τ α)| ≤ C and so |αI | ≤ C. Taking the real part of (6.10) and noting that
ˆ
l.h.s. of (6.10) ≥ C |φ|2 for some C ∈ R,
R
we obtain that αR ≤ C2 , where C2 is a positive constant (independent of τ > 0). Therefore, |α|
is uniformly bounded and hence a perturbation argument gives the desired conclusion.
Now Theorem 10 can be extended to the case τ > 0 for eigenvalues such that Re(τ λε ) ≥ − 21 .
Then by Lemma 11 it follows that for 0 ≤ τ < τ0 all eigenvalues λε of (6.1), (6.2), for which
λε → λ0 6= 0 holds, satisfy Re(λε ) ≤ −c < 0.
For τ ≥ 0, the large eigenvalues in the limit are determined explicitly by the following result
from [47]:
where
√
−1λI be an eigenvalue of the problem
´
wφ 2
2
´R
∆φ − φ + 2wφ − √
w = λφ, φ ∈ H 1 (R),
2
1 + τλ R w
Lemma 12. Let λ =
τ ≥ 0, λ ∈ C, λ = λR + iλI , λR ≥ 0
(6.11)
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
25
√
and we take the principal branch of 1 + τ λ. Then λ is a solution of the algebraic equation


√
1, 3,
− 12 ,
2 ; 

1 + τλ
1
− 1 = −4 F3


2
5
2 + γ, 2 − γ, 2 ;


3
2
+
γ,
−
+
γ,
1
+
γ
;


5
2
2λ Γ(1 + γ)Γ( 2 )
1
F
+ b1
, (6.12)
3 2


3
Γ(γ + 32 )
3
1 + 2γ, 2 + γ ;
√
where γ = 1 + λ and b1 is given by
b1 =
9
(γ − 1)3 γ
π
.
24 (γ − 3/2)(γ − 1/2)22γ sin(π(γ − 1))
(6.13)
Here for two sequences a1 , a2 , . . . , aA and b1 , b2 , . . . , bB we let the series
1+
(a1 + 1)(a2 + 1) · · · (aA + 1) z 2
a1 a2 · · · aA z
+
+···
b1 b2 · · · bB 1!
(b1 + 1)(b2 + 1) · · · (bB + 1) 2!


 a1 , a2 , . . . , aA ; 
z
=: A FB


b1 , b2 , . . . , bB ;
(6.14)
be the generalized Gauss function or generalized hypergeometric function.
In conclusion, we have finished the study of the large eigenvalues (of order O(1)) and derived
results on their stability properties.
It remains to study the small eigenvalues (of order o(1)) which will be done in the next
section.
7. S TABILITY
PROOF
II:
CHARACTERIZATION OF SMALL EIGENVALUES
Now we study the eigenvalueproblem(6.1), (6.2) with respect to small eigenvalues. Namely,
we assume that λε → 0 as max
Let
√ε , D
D
→ 0.
w̄ε = wε,tε + φε,tε ,
H̄ε = T [wε,tε + φε,tε ],
(7.1)
where tε = (tε1 , . . . , tεN ) ∈ Ωη .
After re-scaling, the eigenvalue problem (6.1), (6.2) becomes

w̄ε
w̄ε2

2

ε
∆φ
−
µ(x)φ
+
2
φ
−
ψε = λε φǫ ,

ε
ε
ǫ

H̄ε
H̄ε2
D∆ψε − ψε + 2ξε w̄ε φǫ = λε τ ψε ,




φ′ε (−1) = φ′ε (1) = ψε′ (−1) = ψε′ (1) = 0.
Throughout this section, we denote
µj = µ(tεj ),
µ′j = µ′ (tεj ),
µ′′j = µ′′ (tεj ).
(7.2)
26
JUNCHENG WEI AND MATTHIAS WINTER
ˆ = (ξˆ1 (t), . . . , ξˆN (t))
By the implicit function theorem, there exists a (locally) unique solution ξ(t)
of the equation
N
X
3/2
ĜD (ti , tj )ξˆj2 µj = ξˆi , i = 1, . . . , N.
(7.3)
j=1
ˆ is C for t ∈ Ωη .
Moreover, ξ(t)
1
We have the estimates
ε
ˆ ) = O(1),
ξ(t
2 !
1
ξˆi (t ) − ξˆj (t ) = O D log
.
D
ε
ε
As a preparation, we first compute the derivatives of ξ̂(t).
Now from (7.3) we calculate
∇tj ξˆi = 2
N
X
l=1
∂
3/2
3/2
ĜD (ti , tl )ξˆl µl ∇tj ξˆl +
(ĜD (ti , tj ))ξˆj2 µj
∂tj
3
1/2
+ ĜD (ti , tj )ξˆj2 µj µ′j
2
∇ti ξˆi = 2
N
X
l=1
3/2
ĜD (ti , tl )ξˆl µl ∇ti ξˆl +
for i 6= j,
N
X
∂
3/2
(ĜD (ti , tl ))ξˆl2 µl
∂ti
l=1
3
1/2
+ ĜD (ti , ti )ξˆi2 µi µ′i
2
N
X
5 µ′ (ti )
3/2
3/2
= 2
ĜD (ti , tl )ξˆl µl ∇ti ξˆl + ∇ti ĜD (ti , ti )ξˆi2µi − ξˆi
4 µ(ti )
l=1
3
1/2
+ ĜD (ti , ti )ξˆi2 µi µ′i + Fi (t)
2
′
√
1
5
µ
(t
)
i
3/2
D log
= 2ĜD (ti , ti )ξˆi µi ∇ti ξˆi − ξˆi
+ Fi (t) + O
4 µ(ti )
D
(7.4)
i = 1, . . . , N.
Here F (t) is the vector field
F (t) = (F1 (t), . . . , FN (t)),
where
N
5 µ′ (ti ) X
3/2
Fi (t) = ξˆi
+
∇ti ĜD (ti , tl )ξˆl2 µl ,
4
µi
l=1
i = 1, . . . , N.
We compute
X
5 ˆ µ′ (ti )
1
3/2
2
3/2
Fi (t) = ξi
+
∇ti K̂D (ti , tl )ξˆl µl + O D log
,
4
µi
D
(7.5)
i = 1, . . . , N, (7.6)
l,|l−i|=1
by (3.5), (4.17).
Thus (7.4) implies that
ˆ =O
∇t ξ(t)
√
1
D log
D
.
(7.7)
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
Set
27
∂Fi (t)
.
(7.8)
(M(t))i,j =
∂tj
By the reduced problem (see Section 5), we have F (tε ) = 0 at tε = (tε1 , . . . , tεN ). In addition, if
M(tε )) is positive definite, then we will show that all small eigenvalues have negative real part
when 0 ≤ τ < τ0 for some τ0 > 0.
Next we compute M(t) using (7.4).
For i = j, we have
N
X
l=1
=
N
X
l=1
3/2
∇2ti ĜD (ti , tl )ξˆl2 µl
3/2
∇2ti K̂D (ti , tl )ξˆl2 µl
√
−2(d0 −η0 ) D
1 + O(e
)
h
i
1
3/2
3/2
2
2
2
2
= ∇ti K̂D (ti , ti−1 )ξˆi−1 µi−1 + ∇ti K̂D (ti , ti+1 )ξˆi+1µi+1 1 + O D log
D
"
2 !#
h
i
1
= ∇2ti K̂D (ti , ti−1 )(ξˆ0 )2 (µ0 )3/2 + ∇2ti K̂D (ti , ti+1 )(ξˆ0 )2 (µ0 )3/2 1 + O D log
.
D
(7.9)
For |i − j| = 1, we compute in case j = i − 1
N
X
l=1
=
N
X
l=1
3/2
∇ti−1 (∇ti ĜD (ti , tl ))ξˆl2 µl
∇ti−1 (∇ti K̂D (ti , tl ))ξˆl2 µl
3/2
h
i
√
1 + O(e−2(d0 −η0 ) D )
1
3/2
2
ˆ
= ∇ti−1 (∇ti K̂D (ti , ti−1 ))ξi−1 µi−1 1 + O D log
D
"
2 !#
1
0
2
0
3/2
= ∇ti−1 (∇ti K̂D (ti , ti−1 ))(ξˆ ) (µ )
1 + O D log
D
"
2 !#
1
= −∇2ti K̂D (ti , ti−1 )(ξˆ0)2 (µ0 )3/2 1 + O D log
D
(7.10)
and a similar result holds for j = i + 1. For |i − j| ≥ 2, we have
2 !
N
X
1
3/2
∇tj ∇ti ĜD (ti , tl ) ξˆl2 µl = O D log
.
D
l=1
(7.11)
This implies
0 N
M(tε ) = (mij (tε ))N
i,j=1 = (mij (t ))i,j=1
"
1
1 + O D log
D
2 !#
,
(7.12)
28
JUNCHENG WEI AND MATTHIAS WINTER
where
ˆ0 2
0 3/2
mij (t) = (ξ ) (µ )
−∇2ti K̂D (ti , ti−1 )δi,j+1
+2
N
X
j=1
∇2ti [K̂D (ti , ti−1 ) + K̂D (ti , ti+1 )]δi,j
−
∇2ti K̂D (ti , ti+1 )δi,j−1
3/2
∇ti K̂D (ti , ti−1 )ξˆi−1∇tj ξˆi−1 µi−1 + 2
N
X
j=1
5 µ′′
+ ξˆi i δi,j
4 µi
3/2
∇ti K̂D (ti , ti+1 )ξˆi+1 ∇tj ξˆi+1 µi+1
3
3
1/2
1/2
2
2
µi−1 µ′i−1 δi,j+1 + ∇ti K̂D (ti , ti+1 )ξˆi+1
µi+1 µ′i+1 δi,j−1
+ ∇ti K̂D (ti , ti−1 )ξˆi−1
2
2
′
µi ˆ (µ′i )2
5
ˆ
+ ∇ti ξi − ξi 2 δi,j .
4
µi
µi
Therefore, using (5.6), (5.7) and the estimate (7.7) we have
1
5
mij (tε ) = (ξˆ0 )2 (µ0 )1/2 µ′′ (tεi ) log
16
D
× − (i − 1)(N + 1 − i)δj,i−1 − i(N − i)δj,i+1 + [(i − 1)(N + 1 − i) + i(N − i)]δi,j
2 !
5 ˆ0 0 −1 ′′ ε
1
+ ξ (µ ) µ (ti )δi,j + O D log
4
D
=
5 ˆ0 2
(ξ ) (µ0 )1/2 µ′′ (t0 )
16
1
× log [−(i − 1)(N + 1 − i)δi,j−1 − i(N − i)δi,j+1 + [(i − 1)(N + 1 − i) + i(N − i)]δi,j ]+4δi,j
D
√
1
+O
D log
.
(7.13)
D
The matrix M(tε) will be the leading-order contribution to the small eigenvalues (compare
Lemma 20 and the comments following it). Thus we study the spectrum of the symmetric
N × N-matrix A defined by
as,s = (s − 1)(N − s + 1) + s(N − s),
as,s+1 = as+1,s = −s(N − s),
as,t = 0,
s = 1, . . . , N,
(7.14)
s = 1, . . . , N − 1,
|s − t| > 1.
We will show
Lemma 13. The eigenvalues of the matrix A are given by
λn = n(n + 1),
n = 0, 1, . . . , N − 1.
The corresponding eigenvectors are computed recursively from (7.17).
(7.15)
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
29
The matrix A has eigenvalue λ1 = 0 with eigenvector v1 = e. To compute the other eigen-
values and eigenvectors of A, we remark that this problem is equivalent to finding a suitable
finite-difference approximation ũ of the differential equation
h2 x(1 − x)u′′ + λu = 0,
in the interval (0, 1) for uniform stepsize h =
More precisely, we identify
u′ (0) = u′ (1) = 0
(7.16)
1
.
N
k
k − 1/2
and xk−1/2 =
for k = 1, . . . , N,
N
N
where in (7.16) we replace x(1 − x)u′′ (x) by
1
xk−1 (1 − xk−1 )ũ(xk−3/2 ) + xk (1 − xk )ũ(xk+1/2 )
h2
−[xk (1 − xk ) + xk−1 (1 − xk−1 )]ũ(xk−1/2 )
vik = ũ(xk−1/2 ) with xk =
= (k − 1)(N − k + 1)ũ(tk−3/2 ) + k(N − k)ũ(tk+1/2 )
−[(k − 1)(N − k + 1) + k(N − k)]ũ(tk−1/2 ).
To determine the eigenvectors vi , we have to solve this finite-difference problem exactly. We
assume that the solutions are given by polynomials of degree n (which will be shown later and
n will be specified). Using Taylor expansion around x = xk−1/2 and the identities
xk−1 (1 − xk−1 ) − xk (1 − xk ) = −h(1 − 2xk−1/2 )
and
xk−1 (1 − xk−1 ) + xk (1 − xk ) = 2xk−1/2 (1 − xk−1/2 ) −
h2
,
2
the finite-difference problem is equivalent to
[n/2]
h2 X h2l−2 (2l)
2x(1 − x) −
ũ (x)
4 l=1 (2l)!
[n/2]
+(1 − 2x)
X
Substituting the ansatz
l=1
h2l−2 (2l−1)
ũ
(x) + λn ũ(x) = 0,
(2l − 1)!
ũ(x) =
n
X
n = 0, . . . , N − 1.
ak xk
k=0
into this equation, considering the coefficient of the power xk , k = 0, . . . , n, implies that
(λn − k(k + 1))ak + (k + 1)2 ak+1
[n/2]+1
X h2l−2 (k + 2l − 1)! k + l
+
2
ak+2l−1 − ak+2l−2
(2l)!
(k
−
1)!
k
l=2
30
JUNCHENG WEI AND MATTHIAS WINTER
[n/2]
−
X h2l (k + 2l)!
ak+2l = 0,
2(2l)!
k!
l=1
(7.17)
where for k = 0 we put (0 − 1)! = 1 in the second line of (7.17).
For k = n, n = 0, 1, . . . , N − 1, this gives
(λn − n(n + 1))an + (n + 1)2 an+1 = 0.
Thus, if λn = n(n+1), we have an+1 = 0 and the solution ũ(x) is indeed a polynomial with degree n. After choosing the leading coefficient an 6= 0 arbitrarily, from (7.17) we compute an−1 ,
an−2 , . . . , a0 recursively in a unique way. Then we set vn = (ũ(t1−1/2 ), ũ(t2−1/2 ), . . . , ũ(tN −1/2 )).
There are two cases. Case 1. n < N: Then vn 6= 0 since otherwise we would have ũ ≡ 0,
in contradiction to the fact that we have chosen ũ to be a nontrivial eigenfunction with an 6= 0.
Thus (λn , vn ) is an eigenpair for A. The eigenvectors vn , n = 1, . . . , N are linearly indepen-
dent. From Case 1, we get N eigenpairs with eigenvalues λn = n(n + 1) for n = 0, . . . , N − 1.
Case 2. n ≥ N: Then vn = 0 although ũ 6≡ 0. The resulting eigenfunctions for A are trivial
and so in this case there are no new eigenpairs.
Thus we have found N eigenpairs with linearly independent eigenvectors.
Remark 14. The eigenvector v0 with eigenvalue λ0 = 0 corresponds to a rigid translation of
all N spikes.
The leading eigenpair for mutual movement of spikes is (λ1 , v1 ).
The eigenvector for λ1 = 2 can be computed as follows:
ũ(x) = 1 − 2x, 0 < x < 1
v1,k = ũ(tk−1/2 ), k = 1, . . . , N,
v1,k = 1 −
2(k − 1/2)
N − 2k + 1
=
.
N
N
The components of v1,k are linearly increasing and have odd symmetry around the center of the
spike cluster which corresponds to k =
N +1
2
or x = 21 .
Remark 15. The stability of the small eigenvalues follows from the results in [29] but the eigenvalues have not been determined explicitly.
The technical analysis for the small eigenvalues has been postponed to Appendix C.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
31
8. D ISCUSSION
We conclude this paper with a discussion of our results. We have considered a particular
biological reaction-diffusion system with two small diffusivities, the Gierer-Meinhardt system
with precursor. We have proved the existence and stability of cluster solutions which have three
different length scales: a scale of order O(1) coming from the precursor inhomogeneity and
two small scales which are of the same size as the square roots of the small diffusivities. In
particular, the cluster solution can be stable for a suitable choice of parameter values.
Such systems and their solutions play an important role in biological modeling to account
for the bridging of length scales, e.g. between genetic, nuclear, intra-cellular, cellular and
tissue levels. Our solutions incorporate and combine multiple scales in a robust and stable
manner. A particular example of biological multi-scale patterns concerns the pattern formation
of hypostome, tentacles, and foot in hydra. Meinhardt’s model [18] correctly describes the
following experimental observation: with tentacle-specific antibodies, Bode et al. [3] have
shown that after head removal tentacle activation first reappears at the very tip of the gastric
column. Then this activation becomes shifted away from the tip to a new location, where the
tentacles eventually appear. There are different lengthscales involved for this tentacle pattern:
diameter of the gastric column, distance between tentacles, and diameter of tentacles.
Systems of the type considered in this paper are also a key to understanding the hierarchy
of multi-stage biological processes such as in signalling pathways, where typically first largescale structures appear which induce patterns on successively smaller scales. In our example,
the multi-spike cluster is a typical small-scale pattern which is established near a pre-existing
large-scale precursor inhomogeneity. The precursor can represent previous information from
an earlier stage of development leading to the formation of fine structure at the present time.
An example of hierarchical pattern formation is seen in the determination of cell states for
segmentation in Drosophila wings. In this case three different hierarchy levels are involved in
the process: maternal positional information, gap genes, and pair rule genes [17].
9. A PPENDIX A: E XISTENCE
PROOF
II –
ERROR TERMS
In this section, we compute the error terms caused by the approximate solutions in Section
4. We begin by considering the spatial dependence of the inhibitor near the spikes which is
given by the difference T [wε,t ](xs ) − T [wε,t ](ts ) for x = (x1 , . . . , xN ) ∈ Ωη and t ∈ Ωη , where
the nonlocal operator T [A] has been defined in (4.21) and the approximate solution has been
introduced in (4.23).
To simplify our notation, we let
Hε,t (xs ) = T [wε,t](xs ).
(9.1)
32
JUNCHENG WEI AND MATTHIAS WINTER
Let xs = ts + εy. We calculate
Hε,t (ts + εy) − Hε,t (ts )
= ξε
ˆ
Ω
[GD (ts + εy, x) − GD (ts , x)]
N
X
ξˆk2 w̃k2 (x) +
k=1
= J1 ,
X
ξˆk ξˆl w̃k (x)w̃l (x)
k6=l
!
dx
(9.2)
where ξε has been introduced in (2.8) and J1 is defined by the last equality. For J1 , we have
!
ˆ
N
X
X
J1 = ξε [GD (ts + εy, x) − GD (ts , x)]
ξˆk2 w̃k2 (x) +
ξˆk ξˆl w̃k (x)w̃l (x) dx
Ω
k=1
= ξε
N
X
k=1
ξˆk2
ˆ
Ω
k6=l
[GD (ts + εy, x) − GD (ts , x)] w̃k2 dx.
(9.3)
by (4.6). We further compute
ˆ
ξε [GD (ts + εy, x) − GD (ts , x)] w̃k2 dx
Ω
ˆ √
√ 1
−|ts −x|/ D
−|ts +εy−x|/ D
√
e
−e
− (HD (ts + εy, x) − HD (ts , x)) w̃k2 dx
= ξε
Ω 2 D
ˆ √
√ √ 1
−|ts +εy−x|/ D
−|ts −x|/ D
2
−2(d0 −η0 )/ D
e
= ξε √
−e
w̃k dx 1 + O e
.
(9.4)
2 D Ω
using (4.14). Let x = tk + εz̃. For k = s, we have
ˆ √
√ 1
−|ts +εy−x|/ D
−|ts −x|/ D
e
ξε √
−e
w̃s2 (x) dx
2 D Ω
ˆ √
√ ε
= ξε √
e−ε|y−z̃|/ D − e−ε|z̃|/ D wµ2 s (z̃) dz̃ 1 + O(ε10)
2 D R
2 ˆ
ε
ε
ε 2
2
√
y
1 + O(ε10 )
= ξε √
(|z̃| − |y − z̃|) wµs (z̃) dz̃ + O
D
2 D
D R
!!
2 2
ˆ
ε
ε
ε
1
√
= ξε √
(|z̃| − |y − z̃|) wµ2 0 (z̃) dz̃ + O
y2
1 + O D log
+ ε10
D
D
2 D
D R
!!
2 2
1
ε
ε 2
1
√ T0 (y) + O
=´ 2
y
1 + O D log
+ ε10
,
(9.5)
D
D
w
(y)
dy
D
R
where wµs has been defined in (3.1) and
ˆ
T0 (y) =
(|z̃| − |y − z̃|) wµ2 0 dz̃
(9.6)
R
is an even function, using (2.8) and (4.9). For k 6= s, we have
ˆ √
√ 1
e−|ts +εy−x|/ D − e−|ts −x|/ D w̃k2 (x) dx
ξε √
2 D Ω
ˆ √
√ ε
= ξε √
e−|ts −tk +ε(y−z̃)|/ D − e−|ts −tk −εz̃|/ D wµ2 k (z̃) dz̃ 1 + O(ε10)
2 D R
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
33
√
εy
ε
ts − tk
1 ε2 2
3/2
−|ts −tk |/ D
√ + O D log
= ξε √ (µk )
e
−
y
|ts − tk |
DD
2 D
D
ˆ
× w 2 (z̃) dz̃ 1 + O(ε10 )
R
√
ts − tk
εy
1 2
0 3/2
−|ts −tk |/ D
2
√ + O ε log y
= (µ )
e
−
|ts − tk |
D
D
!!
2
1
× 1 + O D log
+ ε10
,
D
(9.7)
using (4.9) and (2.8). Combining (9.5) and (9.7), we have
Hε,t (ts + εy) − Hε,t (ts )
2 ˆ
ε
ε 2
1
2
2
√
(|z̃| − |y − z̃|) wµs (z̃) dz̃ + O
= ξˆs ´ 2
y
D
w
(y)
dy
D
R
R
!
√
X
t
−
t
εy
1
s
k
√ + O ε2 log y 2
+
ξˆk2 (µk )3/2 e−|ts −tk |/ D −
|t
−
t
|
D
D
s
k
k6=s
!!
2
1
+ ε10
.
× 1 + O D log
D
(9.8)
Remark 16. (i) The second line in (9.8) is an even function in the inner variable y which will
drop out in many subsequent computations due to symmetry.
(ii) The third line in (9.8) is an odd function in the inner variable y. For t ∈ Ωη , we have
√
1
−|ts −tk |/ D
e
, |k − s| = 1,
= O D log
D
2 !
√
1
−|ts −tk |/ D
e
, |k − s| ≥ 2.
=O
D log
D
√
Thus the third line in (9.8) is of exact order O ε D log D1 y .
Next we compute and estimate the error terms of the Gierer-Meinhardt system (4.20) for the
approximate solution wε,t . We recall that a steady state for (4.20) is given by Sε [A] = 0, where
Sε [A] := ε2 A′′ − A +
A2
T [A]
(9.9)
and T [A] is defined by (4.21), combined with Neumann boundary conditions A′ (−1) = A′ (1) =
0. We now compute the error term
Sε [wε,t] = Sε
" N
X
s=1
ξˆs w̃s
#
34
JUNCHENG WEI AND MATTHIAS WINTER
N
X
= ε2 ∆
ξˆs w̃s
s=1
=
−
N
X
s=1
"
!
− µ(x)
N
X
ξˆs w̃s +
s=1
P
N ˆ
s=1 ξs w̃s
Hε,t
2
2 
ˆ
N
ξs w̃s  X
 2 ˆ
ˆ
(µ(x) − µ(ts ))ξˆs w̃s
ε ∆(ξs w̃s ) − µs ξs w̃s +
−
H
(t
)
ε,t s
s=1
s=1
N
X
ξˆs w̃s

2
# √ ε
−2(d0 −η0 )/ D
√
[H
(x)
−
H
(t
)]
1
+
O
|y|
1
+
O
e
ε,t
ε,t s
(Hε,t(ts ))2
D
"
N
X
=−
(µ′ (ts )εy) ξˆs wµs
s=1
−
N
X
+
s=1
1
ε
√
2
w (y) dy D
R
ξˆs2 wµ2 s ´
N
X
wµ2 s
s=1
X
ˆ
R
(|z̃| − |y − z̃|) wµ2 s (z̃) dz̃
#
ts − tk
εy
√
−
|ts − tk |
D
!!
√
ξˆk2 (µk )3/2 e−|ts −tk |/ D
k6=s
2
1
× 1 + O D log
+ ε10
D
"
N
X
= −
µ′′ (t0 )(ts − t0 )εy ξˆs wµs
s=1
−
+
N
X
ξˆs2 wµ2 s ´
R
s=1
N
X
s=1
wµ2 s
1
ε
√
2
w (y) dy D
ˆ
R
(|z̃| − |y −
z̃|) wµ2 s
(z̃) dz̃ + O
ε2 2
y
D
#
t
−
t
εy
1
s
k
√ + O ε2 log y 2
ξˆk2(µk ) e
−
|ts − tk |
D
D
k6=s
!!
2
1
× 1 + O D log
+ ε10
.
(9.10)
D
X
√
3/2 −|ts −tk |/ D
Now we readily have the estimate
kSε [wε,t ]kL2 (− 1 , 1 ) = O
ε ε
ε
√
D
.
(9.11)
Remark 17. The estimates derived in this section will be needed to conclude the existence
proof using Liapunov-Schmidt reduction in Appendix B. In particular, they will imply an explicit
formula for the positions of the spikes in Section 5.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
10. A PPENDIX B: E XISTENCE
PROOF
III – L IAPUNOV-S CHMIDT
35
REDUCTION
In this section, we study the linear operator defined by
L̃ε,t :=
Sǫ′ [A]φ
A2
2Aφ
= ε ∆φ − µ(x)φ +
−
(T ′ [A]φ),
2
T [A] (T [A])
2
L̃ε,t : H 2 (Ω) → L2 (Ω),
where A = wε,t and T ′ [A] has been defined in (6.3).
We will prove results on its invertibility after suitable projections. This will have important
implications on the existence of solutions of the nonlinear problem including bounds in suitable
norms. The proof uses the method of Liapunov-Schmidt reduction which was also considered
in [10], [11], [8], [25], [26] and [37] and other works.
We define the approximate kernel and co-kernel of the operator L̃ε,t , respectively, as follows:
dw̃i Kε,t := span
i = 1, . . . , N ⊂ H 2 (Ω),
dx dw̃i Cε,t := span
i = 1, . . . , N ⊂ L2 (Ω).
dx Recall that the vectorial linear operator L has been introduced in (3.12) as follows:
´
wΦ
LΦ := ∆Φ − Φ + 2wΦ − 2 ´R 2 w 2 ,
(10.1)
w
R
where

φ1
 φ2 
2
N

Φ=
 ...  ∈ (H (R)) .

φN
By Lemma 7, we know that
L : (X0 ⊕ · · · ⊕ X0 )⊥ ∩ (H 2 (R))N → (X0 ⊕ · · · ⊕ X0 )⊥ ∩ (L2 (R))N
n o
with X0 = span dw
is invertible and possesses a bounded inverse.
dy
⊥
⊥
We also introduce the orthogonal projection πε,t
: L2 (Ω) → Cε,t
and study the operator
⊥
⊥
⊥
Lε,t := πε,t
◦ L̃ε,t . We will show that Lε,t : Kε,t
→ Cε,t
is invertible with a bounded inverse
ε
provided max √D , D is small enough. In proving this, we will use the fact that this system
is the limit of the operator Lε,t as max √εD , D → 0. This statement is contained in the
following proposition.
ε
√
Proposition 18. There exist positive constants δ̄, λ such that for max D , D ∈ (0, δ̄) and all
t ∈ Ωη we have
kLε,t φε kL2 (Ωε ) ≥ λkφε kH 2 (Ωε ) .
(10.2)
36
JUNCHENG WEI AND MATTHIAS WINTER
Further, the map
⊥
⊥
Lε,t = πε,t ◦ L̃ε,t : Kε,t
→ Cε,t
is surjective.
k
k
Proof.
Suppose(10.2) is false. Then there exist sequences {εk }, {Dk }, {t }, {φ } such that
max √εDk , Dk → 0, tk ∈ Ωη , φk = φεk ∈ Kε⊥k ,tk , k = 1, 2, . . . and
k
kLǫk ,tk φk kL2 (Ωεk ) → 0
kφk kH 2 (Ωεk ) = 1,
as k → ∞,
(10.3)
k = 1, 2, . . . .
(10.4)
We define φε,i , i = 1, 2, . . . , N and φε,N +1 as follows:
x − ti
φε,i (x) = φε (x)χ
,
δε
φε,N +1(x) = φε (x) −
N
X
x ∈ Ω,
φε,i (x),
i=1
(10.5)
x ∈ Ω.
At first (after rescaling), the functions φε,i are only defined on Ωε . However, by a standard result
2
they can be extended to
R such that their norm in H (R) is bounded by a constant independent
of ε, D and t for max √εD , D small enough. In the following we will study this extension. For
simplicity of notation we keep the same notation for the extension. Since for i = 1, 2, . . . , N
2
(R) it has a weak limit in
each sequence {φki } := {φεk ,i } (k = 1, 2, . . .) is bounded in Hloc
2
2
∞
Hloc
(R),
 andtherefore also a strong limit in Lloc (R) and Lloc (R). Call these limits φi . Then
φ1


 φ2 



φ=
 ..  solves the system Lφ = 0. By Lemma 6, φ ∈ Ker(L) = X0 ⊕ · · · ⊕ X0 . Since
 . 


φN
φεk ∈ Kε⊥k ,tk by taking k → ∞ we get φ ∈ Ker(L)⊥ . Therefore, φ = 0.
By elliptic estimates we have kφεk ,i kH 2 (R) → 0 as k → ∞ for i = 1, 2, . . . , N.
Furthermore, φεk ,N +1 → φN +1 in H 2 (R), where ΦN +1 satisfies
∆φN +1 − φN +1 = 0
in R.
Therefore we conclude φN +1 = 0 and kφkN +1 kH 2 (R) → 0 as k → ∞. This contradicts
kφk kH 2 (Ωεk ) = 1.
To complete the proof of Proposition 18 we just need to show that the conjugate operator to
⊥
⊥
Lε,t (denoted by L∗ε,t ) is injective from Kε,t
to Cε,t
.
The proof for L∗ε,t follows along the same lines as for Lε,t and is omitted.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
37
Now we are in the position to solve the equation
⊥
πǫ,t
◦ Sǫ [wε,t + φ] = 0.
(10.6)
−1
Since Lǫ,t |Kǫ,t
⊥ is invertible (call the inverse Lǫ,t ) we can rewrite this as
⊥
−1
⊥
φ = −(L−1
ǫ,t ◦ πǫ,t ◦ Sǫ [wε,t )] − (Lǫ,t ◦ πǫ,t ◦ Nǫ,t [φ]) ≡ Mǫ,t [φ],
(10.7)
Nε,t [φ] = Sε [wε,t + φ] − Sε [wε,t ] − Sε′ [wε,t]φ
(10.8)
where
and the operator Mǫ,t is defined by (10.7) for φ ∈ H 2 (Ωǫ ). We are going to show that the
operator Mǫ,t is a contraction on
Bǫ,δ ≡ {φ ∈ H 2 (Ωǫ ) : kφkH 2 (Ωǫ ) < δ}
ε
√
for suitably chosen δ if max D , D is small enough. By (9.11) and Proposition 18 we have
⊥
⊥
kMǫ,t[φ]kH 2 (Ωǫ ) ≤ λ−1 kπǫ,t
◦ Nǫ,t [φ]kL2 (Ωε ) + πǫ,t
◦ Sε [wε,t ]L2 (Ω )
ε
ε
−1
≤ λ C c(δ)δ + √
kφkH 2 (Ωǫ ) ,
D
where λ > 0 is independent of δ > 0, ε > 0, D > 0 and c(δ) → 0 as δ → 0. Similarly, we
show that
kMǫ,t[φ] − Mǫ,t [φ′ ]kH 2 (Ωε ) ≤ λ−1 C(c(δ)δ)kφ − φ′ kH 2 (Ωε ) ,
where c(δ) → 0 as δ → 0. If we choose
⊥
δ = 2λ−1 πǫ,t
◦ Sε [wε,t ]L2 (Ω
ǫ)
then, for max √ǫD , D small enough, the operator Mǫ,t is a contraction on Bε,δ . The existence
of a fixed point φǫ,t now follows from the standard contraction mapping principle and φǫ,t is a
solution of (10.7).
We have thus proved
Lemma 19. There exists δ > 0 such that for every pair of ǫ, t with 0 < ǫ < δ and t ∈ Ωη , there
⊥
exists a unique φǫ,t ∈ Kǫ,t
satisfying Sǫ [wε,t + φε,t ] ∈ Cε,t . Furthermore, we have the estimate
ǫ
kφǫ,t kH 2 (Ωǫ ) ≤ C √
.
(10.9)
D
Using the symmetry discussed in Remark 16, we can decompose
φε,t = φε,t,1 + φε,t,2,
(10.10)
38
JUNCHENG WEI AND MATTHIAS WINTER
where φε,t,1 is an even function in the inner variable y which can be estimated as
ε
kφε,t,1kH 2 (− 1 , 1 ) = O √
ε ε
D
and φε,t,2 is an odd function in the inner variable y which can be estimated as
√
1
.
kφε,t,2kH 2 (− 1 , 1 ) = O ε D log
ε ε
D
11. A PPENDIX C: S TABILITY P ROOF III –
TECHNICAL ANALYSIS OF SMALL
EIGENVALUES
In this section we perform a technical analysis of the small eigenvalues and conclude the
proof of Theorem 2.
First let us define
w̃ε,j (x) = χ
x − tεj
δε
w̄ε (x),
j = 1, . . . , N,
(11.11)
where χ(x) is given in (4.7) and δε satisfies (4.8). We define similar to Section 5
2
new
′
Kε,t
ε := span {εw̃ε,j : j = 1, . . . , N} ⊂ H (Ωε ),
new
′
2
Cε,t
ε := span {εw̃ε,j : j = 1, . . . , N} ⊂ L (Ωε ).
Then it is easy to see that
w̄ε (x) =
N
X
w̃ε,j (x) + O(ε10 ).
(11.12)
j=1
Note that w̃ε,j satisfies
(w̃ε,j )2
+ O(ε10 ) = 0.
H̄ε
x−tε j
ε
1 2
ˆ
√
Further, we have w̃ε,j (x) = ξj wj
+ O D + D log D
in H 2 (Ωε ), where wj has
ε
ε2 ∆w̃ε,j − µ(x)w̃ε,j +
been defined in(4.5).
dw̃
′
Thus w̃ε,j
:= dxε,j satisfies
′
′
ε2 ∆w̃ε,j
− µ(x)w̃ε,j
+
2
w̃ε,j
2w̃ε,j ′
w̃ε,j −
H̄ε′ − µ′ (x)w̃ε,j + O(ε9) = 0.
2
H̄ε
(H̄ε )
(11.13)
Let us now decompose
φε = ε
N
X
′
aεj w̃ε,j
+ φ⊥
ε,
(11.14)
j=1
new
where aεj are complex numbers and φ⊥
ε ⊥ Kε,tε . Note that the scaling factor ε has been introduced to ensure that φε = O(1) in H 2 (Ωε ).
Suppose that kφε kH 2 (Ωε ) = 1. Then |aεj | ≤ C.
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
39
The decomposition of φε given in (11.14) implies
ψε = ε
N
X
aεj ψε,j + ψε⊥ ,
(11.15)
j=1
where ψε,j satisfies
′
D∆ψε,j − ψε,j + 2ξε w̄ε w̃ε,j
= 0,
′
′
ψε,j
(−1) = ψε,j
(1) = 0
(11.16)
and ψε⊥ is given by
D∆ψε⊥ − ψε⊥ + 2ξε w̄ε φ⊥
ε = 0,
(ψε⊥ )′ (−1) = (ψε⊥ )′ (1) = 0.
(11.17)
Substituting the decompositions of φε and ψε into (7.2) we have, using (11.13),
N
N
X
X
(w̃ε,j )2 ′ (w̄ε )2
ε
ε
aj
H̄ε −
ψε,j + ε
aεj µ′ (x)w̃ε,j
2
2
H̄
H̄
ε
ε
j=1
j=1
+ε
2
∆φ⊥
ε
−
µ(x)φ⊥
ε
w̄ε ⊥ w̄ε2 ⊥
9
+ 2 φε − 2 ψε − λε φ⊥
ε + O(ε )
H̄ε
H̄ε
!
N
X
′
= λε ε
aεj w̃ε,j
.
(11.18)
j=1
We first compute
I2 := ε
N
X
aεj
j=1
= ε
= ε
(w̃ε,j )2 ′ (w̄ε )2
H̄ε −
ψε,j
H̄ε2
H̄ε2
N
X
(w̃ε,j )
aεj
H̄ε2
j=1
N
X
aεj
j=1
We estimate I2 as follows
= −ε
Let us also put
j=1 |k−j|=1
[−ψε,j +
H̄ε′ ]
−ε
N
X
aεj ψε,j
j=1
(w̃ε,j )2 ′
−ψ
+
H̄
ε,j
ε −ε
H̄ε2
j=1
aεk
N X
N
X
j=1 k=1
aεk
X (w̃ε,k )2
k6=j
N X
X
I2 = −ε
N
X
X
2
aεk ψε,k
k6=j
H̄ε2
+ O(ε9 )
(w̃ε,j )2
+ O(ε9).
H̄ε2
(w̃ε,j )2 ′
ψ
−
H̄
δ
+ O(ε9 )
ε,k
jk
ε
2
H̄ε
2
(w̃ε,j )
1
ψε,k + O(ε9 ) + O εD 3/2 log
2
D
H̄ε
2
⊥
⊥
L̃ε φ⊥
ε := ε ∆φε − µ(x)φε +
2w̄ε ⊥ w̄ε2 ⊥
φ − 2 ψε
H̄ε ε
H̄ε
2 !
.
(11.19)
(11.20)
and
aε := (aε1 , . . . , aεN )T .
(11.21)
40
JUNCHENG WEI AND MATTHIAS WINTER
′
Multiplying both sides of (11.18) by w̃ε,l
and integrating over (−1, 1), we obtain, using (3.3),
r.h.s. = ελε
N
X
aεj
j=1
ˆ
1
−1
′
′
w̃ε,j
w̃ε,l
dx
2 !!
ε
1
= λε aεl ξˆl2 (wl′(y))2 dy 1 + O √ + D log
D
D
R
2 !!
ˆ
ε
1
5/2
= λε aεl ξˆl2µl
(w ′ (z))2 dz 1 + O √ + D log
D
D
R
ˆ
(11.22)
(11.23)
and, using (11.19),
l.h.s. = −ε
+ε
N
X
ˆ
1
aεk
ˆ
1
aεj
k=1
N
X
j=1
−1
−1
2 ′
w̃ε,l
′
ψ
−
H̄
δ
w̃ε,l dx
ε,k
lk
ε
H̄ε2
′
µ′ w̃ε,j w̃ε,l
dx
2
w̃ε,l
(H̄ε′ φ⊥
ε ) dx
2
H̄
−1
ε
ˆ 1 2
ˆ 1
w̃ε,l ⊥ ′
−
(ψε wε,l ) dx +
µ′ φ⊥
ε wε,l dx
2
−1 H̄ε
−1
= (J1,l + J2,l + J3,l + J4,l + J5,l ),
+
ˆ
1
where Ji,l , i = 1, 2, 3, 4, 5 are defined by the last equality.
The following is the key lemma.
Lemma 20. We have
J1,l
ˆ
1
5/2
3/2
3
2
ˆ
w dy ξl µl
− ∇2tεl K̂D (tεl , tεl−1 )ξˆl−1
µl−1 aεl−1
= −ε
3 R
3/2
2
−∇2tεl K̂D (tεl , tεl+1 )ξˆl+1
µl+1 aεl+1
3/2
3/2 ε
2
ε ε
2
2
ε ε
2
ˆ
ˆ
+∇tεl K̂D (tl , tl+1 )ξl+1µl+1 + ∇tεl K̂D (tl , tl−1 )ξl−1 µl−1 ]al
2 !!
ε
1
+O ε2 √ + D log
,
D
D
2
(11.24)
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
J2,l = −ε2
5
12
J3,l = O ε2
J4,l = O ε2
J5,l = O ε2
ˆ
R
3/2
w 3 dy ξˆl2 µl µ′′l aεl + O ε2
41
2 !!
1
ε
√ + D log
, (11.25)
D
D
2 !!
1
ε
√ + D log
,
D
D
2 !!
ε
1
√ + D log
,
D
D
2 !!
ε
1
√ + D log
,
D
D
(11.26)
(11.27)
(11.28)
where aεl has been defined in (11.21) and
a0l = lim aεl ,
ε→0
a0 = (a01 , . . . , a0N ).
(11.29)
Proof. We prove Theorem 1 by using Lemma 20. We compute
2 !!
ε
1
l.h.s. = J1,l + J2,l + O ε2 √ + D log
D
D
ˆ
1
5/2
3/2
2
= −ε2
w 3 dy ξˆl µl
− ∇2tεl K̂D (tεl , tεl−1 )ξˆl−1
µl−1 aεl−1
3 R
3/2
2
−∇2tεl K̂D (tεl , tεl+1 )ξˆl+1
µl+1aεl+1
3/2
3/2 ε
2
ε ε
2
2
ε ε
2
ˆ
ˆ
+∇tεl K̂D (tl , tl+1 )ξl+1 µl+1 + ∇tεl K̂D (tl , tl−1 )ξl−1 µl−1 ]al
−ε2
5
12
ˆ
R
3/2
w 3 dy ξˆl2 µl µ′′l aεl + O ε2
2 !!
ε
1
√ + D log
.
D
D
Comparing with r.h.s. and recalling the computation of M(t0 ) at (7.12), we obtain
√
ε
1
2 ˆ0
0 5/2
0
−2.4ε ξ (µ ) M(t )aε 1 + O √ + D log
D
D
ˆ
√
1
ε
0 5/2 ˆ0 2
′
2
= λε (µ ) (ξ ) aε (w (y)) dy 1 + O √ + D log
,
(11.30)
D
D
R
using (2.3). Equation (11.30) shows that the small eigenvalues λε of (7.2) are given by
λε ∼ −2ε2 ξˆ0 σ M(t0) ,
using (2.3).
Arguing as in Theorem 10, this shows that if all the eigenvalues of M(t0 ) have positive real
part, then the small eigenvalues are stable. On the other hand, if M(t0 ) has an eigenvalue with
negative real part, then there are eigenfunctions and eigenvalues to make the system unstable.
42
JUNCHENG WEI AND MATTHIAS WINTER
This proves Theorem 2.
Next we prove Lemma 20.
Proof. We first study the asymptotic behavior of ψε,j .
Lemma 21. We have
(ψε,k − H̄ε′ δkl )(tεl ) = −∇tεk K̂D (tεl , tεk )ξˆk2 µk [δk,l−1 + δk,l+1 ] − δkl
3/2
+O
√
1
D log
D
X
m,|m−l|=1
2 3/2
∇tεl K̂D (tεl , tεm )ξˆm
µm
2 !!
1
ε
√ + D log
.
D
D
(11.31)
Proof. Note that for l 6= k, we have
ˆ 1
ε
′
ψε,k (tl ) = 2ξε
GD (tεl , z)w̄ε w̃ε,k
dz
−1
3/2
= −∇tεk ĜD (tεk , tεl )ξˆk2 µk + O
3/2
= −∇tεk K̂D (tεk , tεl )ξˆk2 µk + O
√
D log
√
D log
2 !!
ε
1
√ + D log
(11.32)
D
D
2 !!
1
ε
√ + D log
.
D
D
1
D
1
D
Next we compute ψε,l − H̄ε′ near tεl :
ˆ 1
H̄ε (x) = ξε
GD (x, z)w̄ε2 dz
−1
+∞
= ξε
ˆ
−∞
+ξε
2
KD (|z|)w̃ε,l
(x
Xˆ
k6=l
+ z)dz − ξε
1
−1
2
GD (x, z)w̃ε,k
dz + O
ˆ
1
−1
2
HD (x, z)w̃ε,l
dz
2 !
ε
1
√ + D log
.
D
D
So
H̄ε′ (x)
= ξε
ˆ
+∞
KD (|z|)(2w̃ε,l(x +
−∞
+ξε
Xˆ
k6=l
=
X
k,|k−l|=1
+O
′
z)w̃ε,l
(x
1
−1
2
∇x GD (x, z)w̃ε,k
dz + O
+ z)) dz − ξε
√
D log
1
D
ˆ
1
2
∇x HD (x, z)w̃ε,l
dz
2 !!
ε
1
√ + D log
D
D
−1
3/2
∇x K̂D (x, tεk )ξˆk2 µk
√
1
D log
D
2 !!
ε
1
√ + D log
.
D
D
(11.33)
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
43
Thus
H̄ε′ (x)
− ψε,l (x) = −ξε
ˆ
1
2
∇x HD (x, z)w̃ε,l
−1
+2ξε
ˆ
√
X
=
k,|k−l|=1
+O
1
−1
k6=l
1
−1
+O
dz + ξε
Xˆ
2
∇x GD (x, z)w̃ε,k
dz
′
HD (x, z)w̃ε,l w̃ε,l
dz
1
D log
D
2 !!
ε
1
√ + D log
D
D
3/2
∇x K̂D (x, tεk )ξˆk2µk
√
D log
1
D
2 !!
ε
1
√ + D log
.
D
D
(11.34)
Therefore, we have,
H̄ε′ (tεl )
−
ψε,l (tεl )
= −ξε
ˆ
1
2
∇tεl HD (tεl , z)w̃ε,l
−1
dz + ξε
3/2
−∇tεl HD (tεl , tεl )ξˆl2 µl + O
=
N
X
k=1
X
√
−1
1
D log
D
2
∇tεl GD (tεl , z)w̃ε,k
dz
2 !!
ε
1
√ + D log
D
D
3/2
√
k,|k−l|=1
+O
k6=l
1
3/2
∇tεl ĜD (tεl , tεk )ξˆk2 µk − ∇tεl HD (tεl , tεl )ξˆl2 µl
+O
=
Xˆ
D log
1
D
ε
1
√ + D log
D
D
2 !!
3/2
∇tεl K̂D (tεl , tεk )ξˆk2 µk
√
1
D log
D
2 !!
ε
1
√ + D log
.
D
D
(11.35)
Combining (11.33) and (11.35), we have shown (11.31).
Similar to the proof of Lemma 21, the following result is derived.
Lemma 22. We have
ψε,k (tεl + εy) − ψε,k (tεl )
(11.36)
"
2 !#
ε
1
3/2
= −εy∇tεl ∇tεk ĜD (tεl , tεk )ξˆk2 µk 1 + O √ y + O D log
D
D
"
2 !#
ε
1
3/2
= −εy∇tεl ∇tεk K̂D (tεl , tεk )ξˆk2µk [δl,k−1 + δl,k+1] 1 + O √ y + O D log
D
D
44
JUNCHENG WEI AND MATTHIAS WINTER
for l 6= k and
(ψε,l − H̄ε′ )(tεl + εy) − (ψε,l − H̄ε′ )(tεl )
= −εy
= −εy
N
X
m=1
"
2 3/2
∇2tεl ĜD (tεl , tεm )ξˆm
µm 1 + O
X
m,|m−l|=1
"
2 3/2
∇2tεl K̂D (tεl , tεm )ξˆm
µm 1 + O
(11.37)
2 !#
1
ε
√ y + O D log
D
D
2 !#
ε
1
√ y + O D log
.
D
D
For J1,l , we compute
J1,l = −ε
= −ε
−ε
N
X
aεk
ˆ
aεk
ˆ
aεk
ˆ
k=1
N
X
k=1
N
X
k=1
1
−1
1
−1
1
−1
= J6,l + J7,l .
2 ′
w̃ε,l
′
ψ
−
H̄
δ
w̃ε,l dx
ε,k
lk
ε
H̄ε2
2 ′
w̃ε,l
ψε,k (tεl ) − H̄ε′ (tεl )δlk w̃ε,l
dx
2
H̄ε
2
′
w̃ε,l
′
ε
′ ε
ψ
(x)
−
H̄
(x)δ
−
ψ
(t
)
−
H̄
(t
)δ
w̃ε,l dx
ε,k
lk
ε,k
lk
ε
l
ε
l
H̄ε2
For J6,l , we use (11.33) and Lemma 21 to obtain
J6,l
ˆ
N
3
2 X ε 1 w̃ε,l
′
ε
′ ε
= − ε
ak
H̄
ψ
(t
)
−
H̄
(t
)δ
dx
ε,k
lk
ε
l
ε
l
3
3 k=1
−1 H̄ε
ˆ
N
2 2X ε
3
= − ε
ak
wl dy H̄ε′ (tεl ) ψε,k (tεl ) − H̄ε′ (tεl )δlk
3 k=1
R
"
2 !#
ε
1
× 1 + O √ + D log
D
D
ˆ
N
X
2
5/2
2
ε
3
= ε
ak
w dy µl
3
R
k=1
"
#
N
X
3/2
3/2
∇tε ĜD (tε , tε )ξˆ2µ −
∇tε ĜD (tε , tε )ξˆ2 µ
k
l
k
k k
l
l
k
k k
k=1
×
" N
X
j=1
3/2
∇tεl ĜD (tεl , tεj )ξˆj2µj
#"
1+O
2 !#
ε
1
√ + D log
D
D
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
= ε
2
N
X
ˆ
2
5/2
3
w dy µl
3 R
aεk
k=1

∇tε K̂D (tεl , tεk )ξˆk2µ3/2
k [δl,k−1 + δl,k+1 ] −
k

×
= ε
2
X
j,|j−l|=1
N
X
aεk
k=1


k,|k−l|=1
3/2
∇tεl K̂D (tεl , tεk )ξˆk2 µk 
2 !!
1
ε
√ + D log
D
D
2
ˆ
2
5/2
3
w dy µl
3 R
X
×
X

1
3/2
∇tεl K̂D (tεl , tεj )ξˆj2 µj  + O ε2 D log
D
∇tε K̂D (tεl , tεk )ξˆk2 µ3/2
k [δl,k−1 + δl,k+1 ] −
k

45
j,|j−l|=1

3/2
∇tεl K̂D (tεl , tεj )ξˆj2 µj 
X
k,|k−l|=1

3/2
∇tεl K̂D (tεl , tεk )ξˆk2 µk 
2 !!
ε
1
√ + D log
D
D
2 !!
ε
1
√ + D log
D
D
2
1
2
+ O ε D log
D
2 !
2
1
1
2
2
= O ε D log
+ O ε D log
D
D
!
2
1
2
= O ε D log
.
D
(11.38)
Similarly, we compute, using Lemma 22, (7.10) and (7.11),
J7,l = ε2 ξˆl
ˆ
ywl2wl′(y) dy
R
N
X
k=1
3/2
∇tεl ∇tεk ĜD (tεl , tεk )ξˆk2 µk +
N
X
m=1
2 3/2
∇2tεl ĜD (tεl , tεm )ξˆm
µm δk,l
2 !!
√
1
ε
1
2
√ + D log
+O ε D log
D
D
D
ˆ
X
1
5/2
3/2
= −ε2
w 3 dy µl
∇tεl ∇tεk K̂D (tεl , tεk )ξˆk2 µk aεk
3 R
k,|k−l|=1
N
X
+
m,|m−l|=1
2 3/2 ε
∇2tεl K̂D (tεl , tεm )ξˆm
µm al + O ε
ˆ
1
5/2
3
= −ε
w dy µl
−
3 R
2
+
N
X
k,|k−l|=1
X
k,|k−l|=1
√
2
2 !!
ε
1
√ + D log
D
D
1
D log
D
3/2
∇2tεl K̂D (tεl , tεk )ξˆk2 µk aεk
3/2
∇2tεl K̂D (tεl , tεk )ξˆk2 µk aεl + O ε
√
2
D log
1
D
ε
1
√ + D log
D
D
2 !!
!
aεk
46
JUNCHENG WEI AND MATTHIAS WINTER
ˆ
1
5/2
3
= −ε
w dy µl
3 R
2
+O ε
√
2
X
k,|k−l|=1
3/2
∇2tεl K̂D (tεl , tεk )ξˆk2 µk (aεl − aεk )
2 !!
ε
1
√ + D log
D
D
1
D log
D
ˆ
1
5/2
3/2
3
2
= −ε
w dy µl
− ∇2tεl−1 K̂D (tεl , tεl−1 )ξˆl−1
µl−1 aεl−1
3 R
3/2
2
−∇2tεl+1 K̂D (tεl , tεl+1 )ξˆl+1
µl+1 aεl+1
2
ε ε
2 3/2 ε
2
ε ε
2 3/2
ˆ
ˆ
+∇tεl K̂D (tl , tl+1 )ξl µl+1 + ∇tεl K̂D (tl , tl−1 )ξl µl−1]al
2 !!
√
1
ε
1
√ + D log
+O ε2 D log
.
D
D
D
2
(11.39)
Combining (11.38) and (11.39), we obtain (11.24).
For J2,l , integration by parts gives
ˆ 1
N
X
ε
′
J2,l = ε
aj
µ′ w̃ε,j w̃ε,l
dx
−1
j=1
2 !#
1
ε
2
µ′′ w̃ε,l
dx 1 + O √ + D log
D
D
−1
"
2 !#
ˆ
ε2 aεl ˆ2 3/2 ′′
ε
1
ξ µ µ
= −
w 2 dy 1 + O √ + D log
2 l l l R
D
D
εaε
= − l
2
ˆ
"
1
and (11.25) follows.
These arethemain terms. The remaining
terms are small and we will show that they are of
1 2
2 √ε
+ D log D
.
the order O ε
D
Similar to the proof of Proposition 18, it can beshown that L̃ε is invertible from (Kεnew )⊥ to
(Cεnew )⊥ with uniformly bounded inverse for max
√ε , D
D
small enough. By (11.18), (11.19),
Lemma 21 and the fact that L̃ε is uniformly invertible, we deduce that
√
1
⊥
kφε kH 2 (Ωε ) = O ε D log
.
D
Then we have by the equation for ψε⊥
ˆ 1
√
1
ε
⊥
⊥ ε
ψε (tj ) = 2ξε
GD (tj , z)w̄ε φε dz = O ε D log
.
D
−1
Further, we estimate
ψε⊥ (tεj
+ εy) −
ψε⊥ (tεj )
= 2ξε
ˆ
1
−1
[GD (tεj + εy, z) − GD (tεj + εy, z)]w̄εφ⊥
ε dz
(11.40)
STABLE SPIKE CLUSTERS FOR THE ONE-DIMENSIONAL GIERER-MEINHARDT SYSTEM
"
2 !#
ε
1
√ + D log
= 2εyξε
∇tεj GD (tεj y, z)w̄ε φ⊥
ε dz 1 + O
D
D
−1
2 !
√
1 √
1
1
= O ε D log ε D log y = O ε2 D log
y .
D
D
D
ˆ
47
1
(11.41)
These estimates of ψε⊥ and φ⊥
ε are important for the rest of the proof.
For J3,l , we have by (11.33), (11.40)
"
ˆ 1
J3,l = H̄ε′ (tεl )
wl2 φ⊥
ε dx 1 + O
2 !#
ε
1
√ + D log
D
D
−1
2 !
√
1
1
=O
D log εkφ⊥
= O ε2 D log
ε kH 2 (Ωε )
D
D
which proves (11.26).
For J4,l , we decompose
J4,l = J8,l + J9,l ,
where
J8,l = −
J9,l =
ˆ
1
2
w̃ε,l
(ψ ⊥ (tε )w̃ ′ ) dx,
H̄ε2 ε l ε,l
−1
ˆ 1 2
w̃ε,l ⊥
(ψε (x)
−
2
−1 H̄ε
′
− ψε⊥ (tεl ))w̃ε,l
dx.
(11.42)
(11.43)
For J8,l , we have using (11.33), (11.41)
ˆ 1 2
w̃ε,l ′
⊥ ε
J8,l = −ψε (tl )
w̃ dx
2 ε,l
−1 H̄ε
"
2 !#
ˆ 1 3
w̃ε,l ′
2 ⊥ ε
ε
1
=
ψ (t )
H̄ dx 1 + O √ + D log
3 ε l −1 H̄ε3 ε
D
D
"
ˆ
2 !#
ε
2 ′ ε ⊥ ε 5/2
1
1 + O √ + D log
w 3 dy
= −ε H̄ε (tl )ψε (tl )µl
3
D
D
R
!
2
√
1 √
1
1
2
= O ε D log ε D log
.
(11.44)
= O ε D log
D
D
D
For J9,l , we have using (11.41)
2 !
ˆ 1 2
w̃ε,l ⊥
1
′
J9,l = −
(ψε (x) − ψε⊥ (tεl ))w̃ε,l
dx = O ε2 D log
.
2
D
−1 H̄ε
(11.45)
Now (11.27) follows from (11.44), (11.45).
√
1
′
Finally, we estimate using (11.40) and µ (ti ) = O
D log D that
J5,l
√
1 √
1
=
µ′ φ⊥
D log
ε wε,l dx = O ε ε D log
D
D
−1
ˆ
1
2 !
1
= O ε2 D log
D
(11.46)
48
JUNCHENG WEI AND MATTHIAS WINTER
and (11.28) follows.
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