Advances in Mathematical GakkotoshoSciences and Applications Tokyo, JapanVol.xx, No.x (xxxx), pp.xx-xx

TRANSITION LAYERS AND SPIKESFOR A BISTABLE REACTION-DIFFUSION EQUATION

Michio UranoDepartment of Mathematics, Waseda University,3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555 Japan

(michio u@akane.waseda.jp)

Kimie NakashimaDepartment of Ocean Science, Tokyo University of Marine Sciences and Technology,

4-5-7 Konan, Minato-ku, Tokyo, 108-8477 Japan(nkimie@s.kaiyodai.ac.jp)

Yoshio YamadaDepartment of Mathematics, Waseda University,3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555 Japan

(yamada@waseda.jp)

Abstract. This paper is concerned with a steady-state problem forut = ε2uxx + u(1− u)(u− a(x)), (x, t) ∈ (0, 1)× (0,∞),

with ux(0, t) = ux(1, t) = 0, where a is a C2-function satisfying 0 < a(x) < 1. When ε isvery small, the problem has various solutions. Among them, we are interested in solutionswith transition layers and spikes. Our main purpose is to study profiles of such solutionsand determine the location of transition layers and spikes. Moreover, we will show someconditions under which one can observe multi-layers and multi-spikes.

————————————————————Communicated by Editors; Received xxxxxx xx, xxxx.

This work was partially supported by Grant-in-Aid for Scientific Research (C) No. 15540216, Ministry of

Education, Sciences, Sports and Culture, Japan.

AMS Subject Classification 35K57, 35B25

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1 Introduction

In this paper we consider the following reaction-diffusion equation :ut = ε2uxx + f(x, u), 0 < x < 1, t > 0,ux(0, t) = ux(1, t) = 0, t > 0,u(x, 0) = u0(x), 0 < x < 1.

(1.1)

Here ε is a positive parameter and f(x, u) is given by

f(x, u) = u(1− u)(u− a(x)), (1.2)

where a is a C2[0, 1]-function with the following properties :(A.1) 0 < a(x) < 1 in [0, 1],(A.2) if

Σ := {x ∈ (0, 1) ; a(x) = 1/2}, (1.3)

then Σ is a finite set and a′(x) = 0 at any x ∈ Σ,(A.3) if

Λ := {x ∈ (0, 1) ; a′(x) = 0}, (1.4)

then Λ is a finite set,(A.4) a′(0) = a′(1) = 0.The above problem appears as a model which describes a phase transition phenomenonin various fields. See the monograph of Fife [5] and the references therein.

We will mainly discuss the steady state problem associated with (1.1) :{ε2u′′ + f(x, u) = 0, 0 < x < 1,u′(0) = u′(1) = 0,

(1.5)

where ‘ ′ ’ denotes the derivative with respect to x. Angenent, Mallet-Paret and Peletier [3]proved that, for sufficiently small ε > 0, (1.5) admits a stable solution uε which possessesa single transition layer near each x0 ∈ Σ with a′(x0) = 0 and that uε(x) is sufficientlyclose to 0 (resp. 1) for x in any compact subset of {x ∈ (0, 1) ; a(x) > 1/2} (resp.{x ∈ (0, 1) ; a(x) < 1/2}).

The appearance of such a solution with transition layers is closely related to thebistable property of reaction-term f(x, u). As an energy functional associated with (1.1),one can find

E(u) =

∫ 1

0

{1

2ε2ux(x)

2 +W (x, u(x))

}dx,

where

W (x, u) = −∫ u

ϕ0(x)

f(x, s)ds with ϕ0(x) =

{0 if a(x) ≤ 1/2,1 if a(x) > 1/2.

(1.6)

Here W is called a bistable potential because W takes its local minimums at u = 0and u = 1. It is well known that every solution of (1.1) converges to a solution of (1.5)as t → ∞ and that E(u(·, t)) is decreasing with respect to t. Therefore, a minimizer

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of E will be a stable solution of (1.5). We should note that the minimum of W (x, ·)is attained at u = 1 (resp. u = 0) when a(x) < 1/2 (resp. a(x) > 1/2). Intuitively,this fact assures that E has a minimizer uε with a transition layer near an x0 ∈ Σ withu′ε(x0)a

′(x0) < 0. We also refer to a work of Hale and Sakamoto [6], who proved that(1.5) admits an unstable solution uε with a single transition layer near x0 ∈ Σ and thatit satisfies u′

ε(x0)a′(x0) > 0. Moreover, Dancer and Yan [4] have shown the existence of

a solution uε with multi-layers to (1.5). Here a multi-layer means a part of uε wheremultiple transition layers appear as a cluster in a neighborhood of a certain point. Moreprecisely, it is proved that there exists a solution which possesses a prescribed number oftransition layers near a designated point x0 ∈ Σ. (They have discussed such solutions ina ball of RN .) See also Nakashima [7, 8], where a solution with multi-layers is studiedin a balanced case with f(x, u) = A(x)u(1 − u)(u − 1/2). See also the work of Ai andHastings [2].

Recently, Ai, Chen and Hastings [1] have obtained remarkable results on the structureof solutions uε of (1.5) with transition layers and spikes. They give interesting informa-tion on complicated patterns of transition layers and spikes. The existence and stability(Morse index) of such solutions are also discussed. In order to discuss patterns, they de-rived asymptotic results which describes how close uε(ζ) approaches to 0 or 1 at its localminimum or maximum point when ε is sufficiently small. Here, ζ denotes a local maxi-mum or minimum point of uε. Using these results, they reduce the pattern determinationproblem to a certain kind of an algebraic system ; they have solved it and determinedthe patterns. This paper is greatly motivated by their work. Our main purpose is toderive more precise results on the profiles of solutions with transition layers and spikes.We will develop more general results on the asymptotic behavior of uε(x) (Theorems 3.3and 3.6). Furthermore, we will discuss patterns by using different approach based on ourasymptotic results.

When we concentrate ourselves on a solution uε of (1.5) with oscillatory profiles suchas transition layers and spikes, it is useful to take account of the number of intersectingpoints of the graphs of uε and a in (0, 1). We introduce the notion of n-mode solution ;uε is called an n-mode solution if the graph of uε has n intersecting points with thatof a in (0, 1). Roughly speaking, for any n-mode solution of (1.5), its graph is classifiedinto the following three groups (see Lemmas 2.2 and 2.4) :

(i) uε(x) is close to 0 or 1,(ii) uε(x) forms a transition layer connecting 0 and 1,(iii) uε(x) forms a spike based on 0 or 1.Here it should be noted that, if uε has a spike, then its peak is distant away from u = 0 andu = 1. In order to study patterns of solutions with transition layers, we note that uε(x)is very close to 0 or 1 at one of end-points of any transition layer, when ε is sufficientlysmall. The situation is similar when we discuss a spike ; if uε has a spike based on 1,then uε(x) is very close to 1 at both end-points of the spike. Therefore, as is stated inthe preceeding paragraph, it will be important to study the asymptotic rate of 1− uε(x)(resp. uε(x)) as ε → 0 in a certain interval containing one local maximum point (resp.local minimum point) of uε. The analysis to get the asymptotic rate will be carried outby a kind of barrier method.

The content of this paper is as follows. In Section 2, we will give some fundamental

3

properties of n-mode solutions of (1.5). In Section 3, the asymptotic rates of 1 − uε(x)and uε(x) as ε → 0 for x in a suitable interval will be discussed. The asymptotic resultsare given by Theorems 3.3 and 3.6. These results enable us to show that any transitionlayer (resp. spike) appears only in a neighborhood of a point of Σ (resp. Λ) in Section 4.Finally, Section 5 is devoted to the study of multi-layers and multi-spikes. It will beshown that each multi-layer consists of an odd number of transition layers. Furthermore,we will show that multi-layers (resp. multi-spikes) can appear only in a neighborhood ofa point in a suitable subset of Σ (resp. Λ).

2 Transition layers and spikes for n-mode solutions

In this section we will give some basic properties of solutions of (1.5).

Lemma 2.1. Let uε be a solution of (1.5). Then

0 ≤ uε(x) ≤ 1 for all x ∈ (0, 1).

Furthermore, if uε ≡ 0 or 1, then

0 < uε(x) < 1 for all x ∈ (0, 1).

Proof. Assume thatuε(x0) = max{uε(x) ; x ∈ [0, 1]} > 1 (2.1)

for some x0 ∈ [0, 1]. It follows from u′′ε(x0) ≤ 0 that f(x0, uε(x0)) ≥ 0. On the other

hand, (2.1) together with (1.2) implies f(x0, uε(x0)) < 0, which is a contradiction. Henceuε(x) ≤ 1. Similarly, it is easy to show uε(x) ≥ 0.

To give the proof of the last assertion, assume uε(x1) = max{uε(x) ; x ∈ [0, 1]} = 1at some x1 ∈ [0, 1]. Since u′

ε(x0) = 0, we immediately get uε ≡ 1 by the uniqueness ofsolutions for the initial value problem of the second order differential equation. Therefore,uε(x) < 1 in [0, 1] unless uε ≡ 1. Similarly, one can see that, if uε ≡ 0, then uε(x) > 0 in[0, 1]. This completes the proof.

Let uε be a solution of (1.5). Recall that uε is called an n-mode solution of (1.5) ifuε − a has exactly n zero-points in (0, 1). Denote by Sn,ε the set of all n-mode solutionsand we fix arbitrary n ∈ N. For uε ∈ Sn,ε, define

Ξ = {x ∈ [0, 1] ; uε(x) = a(x)}. (2.2)

In what follows, we sometimes extend uε to a function over R by the standard reflec-tion. This is possible because uε satisfies u′

ε(0) = u′ε(1) = 0 ; so that uε is regarded as a

periodic function with period 2. Similarly, by virtue of (A.4), f(x, u) can be extended for(x, u) ∈ R × R by the reflection with respect to x-variable. So we may consider that uε

satisfies (1.5) for all x ∈ R.

4

Lemma 2.2. For n ∈ N, it holds that

limε→0

supuε∈Sn,ε

maxx∈[0,1]

∣∣∣∣uε(x)(1− uε(x))

[1

2ε2(u′

ε(x))2 −W (x, uε(x))

]∣∣∣∣ = 0, (2.3)

where W (x, u) is defined by (1.6).

Proof. Although this lemma is given in [1, Lemma 2.1] we will give a proof for the sakeof completeness. Suppose that (2.3) is not true, then there exist {(εk, uk, xk)} such thatuk ∈ Sn,εk , xk ∈ [0, 1] and∣∣∣∣uk(xk)(1− uk(xk))

[1

2εk

2(u′k(xk))

2 −W (xk, uk(xk))

]∣∣∣∣ ≥ δ (2.4)

with some δ > 0.We use a change of variable x = xk + εkt and introduce a new function Uk by Uk(t) =

uk(xk + εkt). Clearly, Uk satisfies

U ′′k + f(xk + εkt, Uk) = 0 in R, (2.5)

where ‘ ′ ’ denotes the derivative of t.We first prove the uniform boundedness of {Uk}, {U ′

k} and {U ′′k }. By Lemma 2.1,

sup{|Uk(t)| ; t ∈ R} < 1 ; so that it follows from (2.5) that sup{|U ′′k (t)| ; t ∈ R} = m1 < ∞.

To study U ′k, we take any t ∈ R. The mean value theorem assures that there exists a

number t0 ∈ (t, t+ 1) such that

U ′k(t0) = Uk(t+ 1)− Uk(t) ;

then −1 ≤ U ′k(t0) ≤ 1 from Lemma 2.1. Hence it holds that

|U ′k(t)| =

∣∣∣∣U ′k(t0) +

∫ t

t0

U ′′k (s) ds

∣∣∣∣ ≤ 1 +m1.

These estimates implies that {Uk}, {U ′k}, {U ′′

k } are uniformly bounded in R. Therefore,it is easy to see that {Uk} and {U ′

k} are equi-continuous. Moreover, it also follows from(2.5) that {U ′′

k } is also equi-continuous.On account of the above results, one can apply Ascoli-Arzela’s theorem and use a

diagonal argument to show that {Uk} has a subsequence, which is still denoted by {Uk},such that

limk→∞

Uk = U in C2loc(R)

with a suitable function U ∈ C2(R). Here we recall that {xk} is bounded. Since one canchoose a convergent subsequence from {xk}, we may assume

limk→∞

xk = x∗ ∈ [0, 1].

Then it is seen in the standard manner that U satisfies

U ′′ + f(x∗, U) = 0 in R. (2.6)

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Multiplying (2.6) by U ′ and integrating the resulting expression with respect to t weget

1

2U ′(t)2 −W (x∗, U(t)) = C in R (2.7)

with some constant C. If U ≡ 0 or U ≡ 1, then it is easy to derive a contradiction to(2.4) from (2.7).

We will show C = 0 in (2.7) in the case that U ≡ 0 and U ≡ 1. If C > 0, thenwe see from the phase plane analysis that U is unbounded. This is impossible because{Uk} is bounded . If C < 0, then the phase plane analysis tells us that U is a periodicfunction. So the graph of U(t) has infinitely many intersecting points with that of a(x∗)and, therefore, the graph of Uk(t) also has infinitely many intersecting points providedthat k is sufficiently large. This fact implies that, if k is sufficiently large, then uk(x)−a(x)has many zero-points near x = x∗. This result contradicts to the definition of n-modesolutions. Thus we have proved C = 0 in (2.7).

Hence

limk→∞

∣∣∣∣12εk2(u′k(xk))

2 −W (xk, uk(xk))

∣∣∣∣ = limk→∞

∣∣∣∣12(U ′k(0))

2 −W (xk, Uk(0))

∣∣∣∣=

∣∣∣∣12(U ′(0))2 −W (x∗, U(0))

∣∣∣∣ = 0,

which contradicts to (2.4). Thus the proof is complete.

Lemma 2.3. For uε ∈ Sn,ε, set Ξ = {ξ1, ξ2, . . . , ξn} with 0 < ξ1 < ξ2 < · · · < ξn < 1. If εis sufficiently small, then u′

ε has exactly (n− 1) zero points {ζk}n−1k=1 in (0, 1) satisfying

0 < ξ1 < ζ1 < ξ2 < ζ2 < · · · < ξn−1 < ζn−1 < ξn < 1.

Proof. Let ξ ∈ Ξ and take any small η > 0. Lemma 2.2 implies that, if ε is sufficientlysmall, then∣∣∣∣uε(ξ)(1− uε(ξ))

[1

2ε2(u′

ε(ξ))2 −W (ξ, uε(ξ))

]∣∣∣∣=

∣∣∣∣a(ξ)(1− a(ξ))

[1

2ε2(u′

ε(ξ))2 −W (ξ, a(ξ))

]∣∣∣∣ < η.

Since a(ξ)(1− a(ξ)) > M with some ε-independent M > 0, we get

− η

M<

1

2ε2(u′

ε(ξ))2 −W (ξ, a(ξ)) <

η

M.

Observe that W (ξ, a(ξ)) ≥ c1 > 0, where c1 is a positive constant independent of ε.Hence, taking a sufficiently small ε > 0 one can conclude

ε2(u′ε(ξ))

2 ≥ c22 > 0

with some c2 > 0. Thus

|u′(ξ)| > c2ε

(2.8)

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for sufficiently small ε.We study the case uε(x) > a(x) in (ξk, ξk+1). By (2.8) and the boundedness of a′(x),

it is easy to see u′ε(ξk) > 0 and u′

ε(ξk+1) < 0. On the other hand, since (1.5) impliesu′′(x) < 0 in (ξk, ξk+1), u

′ε has a unique zero-point in (ξk, ξk+1), which is denoted by ζk.

Clearly uε attains its local maximum at x = ζk.Since the proof is analogous for the case uε(x) < a(x) in (ξk, ξk+1), it remains to show

the nonexistence of zero point of u′ε in (0, ξ1) ∪ (ξn, 1). Assume uε(x) > a(x) in (0, ξ1).

Since u′ε(0) = 0 and u′′

ε(x) < 0 in (0, ξ1), it is clear that u′ε(x) < 0 in (0, ξ1). The other

cases can be discussed in the same way.

Lemma 2.4. For uε ∈ Sn,ε, let ξε be any point in Ξ and define Uε by Uε(t) = uε(ξ

ε+ εt).Then there exists a subsequence {εk} ↓ 0 such that ξk = ξεk and Uk = Uεk satisfy

limk→∞

ξk = ξ∗ and limk→∞

Uk = ϕ in C2loc(R),

where ϕ ∈ C2(R) is a function satisfying one of the following properties.(i) In the case a(ξ∗) = 1/2, ϕ is a unique solution to the following problem :

ϕ′′ + f(ξ∗, ϕ) = 0 in R,ϕ(−∞) = 0, ϕ(+∞) = 1 (resp. ϕ(−∞) = 1, ϕ(+∞) = 0),ϕ(0) = 1/2,

if ϕ′(0) > 0 (resp. ϕ′(0) < 0). Moreover, ϕ′(t) > 0 for t ∈ R if ϕ′(0) > 0, while ϕ′(t) < 0for t ∈ R if ϕ′(0) < 0.(ii) In the case a(ξ∗) < 1/2, ϕ is a unique solution to the following problem :{

ϕ′′ + f(ξ∗, ϕ) = 0 in R,ϕ(0) = a(ξ∗), ϕ(±∞) = 0.

Moreover, ϕ satisfies sup{ϕ(x) ; x ∈ R} > a(ξ∗).(iii) In the case a(ξ∗) > 1/2, ϕ is a unique solution to the following problem :{

ϕ′′ + f(ξ∗, ϕ) = 0 in R,ϕ(0) = a(ξ∗), ϕ(±∞) = 1.

Moreover, ϕ satisfies inf{ϕ(x) ; x ∈ R} < a(ξ∗).

Proof. Clearly, Uε(t) satisfies

U ′′ε + f(ξε + εt, Uε) = 0 and Uε(0) = a(ξε).

As in the proof of Lemma 2.2, one can prove that {Uε} is bounded in C2(R) ; so thatthere exists a subsequence {εk} ↓ 0 such that Uk = Uεk is convergent in C2

loc(R) ; i.e.,

limk→∞

Uk = ϕ in C2loc(R) (2.9)

with some ϕ ∈ C2(R). Moreover, since {ξk} (ξk = ξεk) is also bounded, we may assumelimk→∞

ξk = ξ∗ ∈ [0, 1]. Therefore, the limiting procedure yields

ϕ′′(t) + f(ξ∗, ϕ(t)) = 0 with ϕ(0) = a(ξ∗).

7

The same argument as in the proof of (2.7) with C = 0 also shows

1

2ϕ′(t)2 −W (ξ∗, ϕ(t)) = 0

for t ∈ R. Hence the phase plane analysis enables us to conclude that ϕ satisfies one of(i)-(iii).

Lemma 2.5. For uε ∈ Sn,ε, let ξε1, ξε2 be two successive points in Ξ. Then one of the

following properties holds true :(i) (ξε2 − ξε1)/ε is unbounded as ε → 0,(ii) For sufficiently small ε > 0, it holds that

M1 <ξε2 − ξε1

ε< M2,

where M1 and M2 are positive constants independent of ε.

Proof. We denotes the derivative with respect to t by ‘ ˙ ’ and the derivative with respectto x by ‘ ′ ’. Put Uε(t) = uε(ξ

ε1 + εt) ; then Uε(t) = εu′

ε(ξε1 + εt). Therefore,{

Uε(0) = εu′ε(ξ

ε1),

Uε((ξε2 − ξε1)/ε) = εu′

ε(ξε2).

In view of (2.8) we see

Uε(0)Uε((ξε2 − ξε1)/ε) = εu′

ε(ξε1)εu

′ε(ξ

ε2) < −c22. (2.10)

Suppose that {(ξε2 − ξε1)/ε} is bounded. Then one can choose a subsequence {εk} suchthat

0 ≤ M = limk→∞

ξεk2 − ξεk1εk

< +∞.

Recalling the proof of Lemma 2.4 we may regard {Uεk} as a convergent sequence satisfying(2.9). Setting ε = εk in (2.10) and letting k → ∞ we get

ϕ(0)ϕ(M) ≤ −c22.

Hence it follows from Lemma 2.4 that M must be positive. Thus we have shown (ii) when(i) does not hold.

3 Asymptotic profiles of n-mode solutions

In this section we will derive some asymptotic behavior of uε or 1 − uε as ε → 0 in acertain interval containing a local minimum or local maximum of uε. For this purpose,we first prepare the following lemma.

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Lemma 3.1. Let g(v) = v(1 − v)(v − a0) with a0 ∈ (0, 1). Then for any σ ∈ (0, 1)satisfying σ > max{a0, (a0 + 1)/3} and M > 0, there exists a unique solution of

vzz + g(v) = 0 in (−M, 0),

v(−M) = σ, vz(0) = 0,

v > σ in (−M, 0).

(3.1)

Moreover, there exists a constant σ∗ ∈ ((a0+1+√a20 − a0 + 1)/3, 1) such that, if σ > σ∗,

thenc1 exp(−RM) < 1− v(0) < c2 exp(−rM), (3.2)

where r =√

−g′(σ), R =√

−g′(1) and c1, c2 (0 < c1 < c2) are positive constants depend-ing only on σ.

Proof. In order to solve (3.1), we employ the time-map method (see, e.g., Smoller andWasserman [9]). Take σ ∈ (a0, 1) with σ > max{a0, (a0+1)/3} and consider the followinginitial value problem : {

vzz + g(v) = 0 for z > −M,

v(−M) = σ, vz(−M) = p,(3.3)

where p is a positive parameter. Let v(z; p) the solution of (3.3). Multiplying (3.3) byvz(z; p) and integrating the resulting expression over (−M, z) we get

1

2v2z(z; p)−G(v(z; p)) =

1

2p2, (3.4)

where

G(v) = −∫ v

σ

g(s)ds.

Since we look for p satisfying vz(0; p) = 0 and vz(z; p) > 0 for z ∈ (−M, 0), we haveto restrict the range of p. By the phase plane analysis, 0 < p <

√−2G(1) = p∗ (note

−G(1) > 0 because of σ > a0).For such p, define α(p) ∈ (σ, 1) by p2/2 = −G(α(p)), and let T (p) be a time-map

defined byT (p) = inf { z > −M ; v(z) = α(p) }+M.

Then α(p) = max{v(z; p) ; z > −M} and T (p) denotes the distance from −M to the firstzero point of vz. If we can find a number pM satisfying T (pM) = M , then v(z; pM) givesa unique solutions of (3.1). Hence the study of T (p) is essential to show the existence ofa solution of (3.1).

As a first step, we will show that T (p) is strictly monotone increasing for 0 < p < p∗.It follows from (3.4) that

1√G(v)−G(α(p))

dv

dz=

√2

Integrating this equation over (−M,−M + T (p)) yields

√2T (p) =

∫ α(p)

σ

dv√G(v)−G(α(p))

. (3.5)

9

From the definition, α(p) is a strictly increasing function of p satisfying α(p) → σ asp → 0 and α(p) → 1 as p → p∗. So it is convenient to treat T (p) in (3.5) as a function ofα in place of p. Set

S(α) =

∫ α

σ

dv√G(v)−G(α)

=

∫ 1

0

α− σ√G(s(α− σ) + σ)−G(α)

ds.

We will prove that S(α) is strictly monotone increasing for α ∈ (σ, 1). Differentiation ofS(α) with respect to α gives

S ′(α) =

∫ 1

0

2(∆G) + (α− σ)sg(s(α− σ) + σ)− (α− σ)g(α)

2(∆G)3/2ds

=1

α− σ

∫ α

σ

θ(v)− θ(α)

2(∆G)3/2dv,

(3.6)

where∆G = G(v)−G(α) and θ(v) = 2G(v) + (v − σ)g(v).

Note ∆G > 0 for σ < v < α. We will investigate θ to show S ′(α) > 0 for α ∈ (σ, 1). It iseasy to see

θ′(v) = −g(v) + (v − σ)g′(v) and θ′′(v) = (v − σ)g′′(v).

Observe θ′(σ) = −g(σ) < 0 for a0 < σ < 1. Moreover, θ′′(v) < 0 in (σ, α) by the concavityof g(v). Therefore, θ′(v) < 0 in (σ, α). Since θ is monotone decreasing in (σ, α), we seefrom (3.6) that S ′(α) > 0 in (σ, α). Therefore, S(α) is monotone increasing in (σ, α) andso is T (p) in (0, p∗).

Furthermore, we will showlimp→0

T (p) = 0 (3.7)

andlimp→p∗

T (p) = +∞. (3.8)

We use

G(v)−G(α) =

∫ α

v

g(s)ds

≥ min{g(α), g(σ)}(α− v) for v ∈ (σ, α)

to prove (3.7). Hence it is easy to see limα→σ

S(α) = 0, which implies (3.7). To prove (3.8),

we note α(p) → 1 when p → p∗. For α → 1, we see

G(v)−G(α) → −1

2g′(1)(v − 1)2 + o((v − 1)2) as v → 1.

Therefore, limα→1

S(α) = +∞, whence follows (3.8).

We have shown that T (p) is a strictly increasing function satisfying (3.7) and (3.8).Hence it is easy to see that, for each M > 0, there exists a unique pM ∈ (0, p∗) suchthat T (pM) = M . Clearly, pM is strictly increasing and continuous with respect to M

10

and limM→∞

pM = p∗. Set vM = v(0; pM) ; vM is also strictly increasing and continuous with

respect to M and satisfies limM→∞

vM = 1.

We will prove that vM satisfies (3.2). Recall

√2M =

∫ vM

σ

dv√G(v)−G(vM)

, (3.9)

from (3.5). By the mean value theorem, there exists a constant θ1 ∈ (σ, vM) satisfying

G(v)−G(vM)

(1− v)2 − (1− vM)2= − g(θ1)

2(θ1 − 1)= −g(θ1)− g(1)

2(θ1 − 1). (3.10)

Using the mean value theorem again, we see that the right-hand side of (3.10) is equal to−g′(θ2)/2 with some θ2 ∈ (θ1, 1). Take σ∗ ∈ ((a0 + 1 +

√a20 − a0 + 1)/3, 1). It should be

noted that g′(s) is decreasing and negative for s ∈ (σ∗, 1). Then for σ ∈ (σ∗, 1)

r2

2< −g′(θ2)

2<

R2

2(3.11)

with r =√

−g′(σ) and R =√−g′(1). With use of (3.10) and (3.11), it follows from (3.9)

that1

RBM < M <

1

rBM , (3.12)

where

BM =

∫ vM

σ

dv√(1− v)2 − (1− vM)2

= log

(bM +

√b2M − 1

),

with bM = (1− σ)/(1− vM). Since BM ∈ [log bM , log 2bM ], (3.12) yields

(1− σ) exp(−RM) < 1− vM < 2(1− σ) exp(−rM).

Thus the proof is complete.

Replacing z by −z in the proof of Lemma 3.1, we can show the following lemma.

Lemma 3.2. Let g be the same function as in Lemma 3.1. Then for any σ ∈ (0, 1)satisfying σ > max{a0, (a0 + 1)/3} and M > 0, there exists a unique solution v of

vzz + g(v) = 0 in (0,M),

vz(0) = 0, v(M) = σ,

v > σ in (0,M).

Furthermore, there exists a constant σ∗ ∈ ((a0 + 1 +√a20 − a0 + 1)/3, 1) such that, if

σ > σ∗, then v satisfies (3.2).

In what follows, let ξ1, ξ2 be two successive points in Ξ and let (ξ1, ξ2) be an intervalsuch that

uε(x)− a(x) > 0 in (ξ1, ξ2). (3.13)

Let ζ ∈ (ξ1, ξ2) be a unique point satisfying u′ε(ζ) = 0 and u′

ε(x) > 0 in (ξ1, ζ). Theexistence of such ζ is assured by Lemma 2.3.

We will study asymptotic behavior of uε in (ξ1, ξ2) as ε ↓ 0.

11

Theorem 3.3. For uε ∈ Sn,ε, assume (3.13) and let ζ ∈ (ξ1, ξ2) satisfy u′(ζ) = 0. If(ζ − ξ1)/ε → +∞ as ε → 0, then there exist positive constants C1, C2, r, R with C1 < C2

and r < R such that

C1 exp

(−R(ζ − ξ1)

ε

)< 1− uε(x) < C2 exp

(−r(x− ξ1)

ε

)(3.14)

for x ∈ [ξ1, ζ] and sufficiently small ε > 0.

Proof of Theorem 3.3. We begin with the proof of the right-hand side inequality of (3.14).Let a∗ be a constant which satisfies a∗ > max{a(x) ; x ∈ [ξ1, ζ]} and take δ∗ ∈ (a∗, 1)which is close to 1. By assumptions and Lemma 2.4 we can find a point ξ1 ∈ (ξ1, ζ) suchthat uε(ξ1) = δ∗ and uε(x) > δ∗ in (ξ1, ζ) provided that ε is sufficiently small. Clearly,ξ1 − ξ1 = O(ε) as ε → 0 ; so ζ − ξ1 > ε.

Now take any x∗ ∈ (ξ1 + ε, ζ) and apply Lemma 3.1. Let v(z) be a solution of (1.5)with a0 = a∗, σ = δ∗ and M = (x∗− ξ1−ε)/ε. We use a change of variable z = (x−x∗)/εand define V1 by V1(x) = v((x− x∗)/ε) ; then

ε2V ′′1 + V1(1− V1)(V1 − a∗) = 0 in (ξ1 + ε, x∗),

V1(ξ1 + ε) = δ∗, V ′1(x

∗) = 0,

V1 > δ∗ in (ξ1 + ε, x∗).

(3.15)

By virtue of Lemma 3.1, V1 satisfies

c1eR exp

(−R(x∗ − ξ1)

ε

)< 1− V1(x

∗) < c2er exp

(−r(x∗ − ξ1)

ε

), (3.16)

where c1, c2, r and R are positive constants depending only on a∗ and δ∗.We will show

V1(x) ≤ uε(x) in (ξ1 + ε, x∗). (3.17)

For this purpose, it is convenient to introduce the following auxiliary function

h1(x) =V1(x)− a∗

uε(x)− a∗in [ξ1 + ε, x∗],

and show h1(x) ≤ 1 in [ξ1 + ε, x∗] by contradiction. Suppose that there exists an x1 ∈[ξ1 + ε, x∗] such that

h1(x1) = max{h1(x) ; x ∈ [ξ1 + ε, x∗]} =1

η> 1.

Then {Vη(x) ≤ uε(x) in [ξ1 + ε, x∗],

Vη(x1) = uε(x1),

whereVη(x) = η(V1(x)− a∗) + a∗ (< V1(x)).

12

We will proveV ′′η (x1) ≤ u′′

ε(x1). (3.18)

Clearly, h1(ξ1 + ε) < 1. Moreover, since u′ε(x

∗) > 0 and V ′1(x

∗) = 0 (by (3.15)), it is easyto see h′

1(x∗) < 0. Therefore, x1 must be an interior point in (ξ1 + ε, x∗). So

h′1(x1) = 0 and h′′

1(x1) ≤ 0. (3.19)

From the definition of h1,

h1(x)(uε(x)− a∗) = V1(x)− a∗.

Differentiating the above identity two times with respect to x and setting x = x1 we get

u′′ε(x1) + 2ηu′

ε(x1)h′1(x1) + η(uε(x1)− a∗)h′′

1(x1) = ηV ′′1 (x1) = V ′′

η (x1). (3.20)

Then (3.19) and (3.20) imply (3.18).We next use f(x, Vη) > ηV1(1 − V1)(V1 − a∗). Indeed, since V1(x) > a∗ > 1/2 in

(ξ1 + ε, x∗), a simple calculation yields

f(x, Vη) = Vη(1− Vη)(Vη − a(x))

= η(V1 − a∗)Vη(1− Vη) + (a∗ − a(x))Vη(1− Vη)

> η(V1 − a∗)Vη(1− Vη)

> ηV1(1− V1)(V1 − a∗)

provided that δ∗ is sufficiently close to 1. Hence it follows from (3.15) that

ε2V ′′η + f(x, Vη) = ηε2V ′′

1 + f(x, Vη) > η{ε2V ′′1 + V1(1− V1)(V1 − a∗)} = 0.

Therefore, using (3.18) we have

0 = ε2u′′ε(x1) + f(x1, uε(x1)) ≥ ε2V ′′

η (x1) + f(x1, Vη(x1)) > 0,

which is a contradiction. Thus we have shown (3.17).Now (3.16) and (3.17) imply

1− uε(x∗) ≤ 1− V1(x

∗) < c2er exp

(−r(x∗ − ξ1)

ε

).

Here we should note that c2 and r can be chosen independently of x∗. Recalling that x isan arbitrary point in (ξ1 + ε, ζ), one can conclude that

1− uε(x) < c2er exp

(−r(x− ξ1)

ε

)(3.21)

is valid for x ∈ (ξ1 + ε, ζ).

13

Moreover, since ξ1 − ξ1 < Kε with some K > 0, it follows from (3.21) that

1− uε(x) < c2er exp

(−r(x− ξ1)

ε

)exp

(r(ξ1 − ξ1)

ε

)

< c2er(K+1) exp

(−r(x− ξ1)

ε

) (3.22)

for x ∈ (ξ1 + ε, ζ). On the other hand, we note that

exp(−r(K + 1)) < exp

(−r(x− ξ1)

ε

)for x ∈ (ξ1, ξ1 + ε). Hence, we can choose a sufficiently large constant L > 0 such that

1− uε(x) ≤ 1− uε(ξ1) = 1− a(ξ1)

< L exp(−r(K + 1)) < L exp

(−r(x− ξ1)

ε

)(3.23)

for x ∈ (ξ1, ξ1 + ε). Thus (3.22) and (3.23) enable us to extend (3.21) for all x ∈ [ξ1, ζ]with ξ1 replaced by ξ1 (for x = ζ, it is sufficient to use the continuity of uε with respectto x).

We will prove the left-hand side inequality of (3.14). Let a∗ be a constant satisfyinga∗ < min{a(x) ; x ∈ [ξ1, ζ]} and take δ∗ ∈ (a∗, 1) which is close to 1. In particular, weassume that δ∗ > max{1/2, max{a(x) ; x ∈ [ξ1, ζ]}}. Then there exists a point ξ ∈ (ξ1, ζ)such that uε(ξ1) = δ∗ and ξ1 − ξ1 = O(ε).

If ε is sufficiently small, then ζ − ξ1 > ε. We apply Lemma 3.1 by setting σ = δ∗,a0 = a∗ and M = (ζ − ξ1 + ε)/ε and define v as the solution of (3.1). With use of thechange of variable z = (x− ζ)/ε, we see that V2(x) = v((x− ζ)/ε) satisfies

ε2V ′′2 + V2(1− V2)(V2 − a∗) = 0 in (ξ1 − ε, ζ),

V2(ξ1 − ε) = δ∗, V′2(ζ) = 0,

V2 > δ∗ in (ξ1 − ε, ζ).

Lemma 3.1 gives

c1eR exp

(−R(ζ − ξ1)

ε

)< 1− V2(ζ). (3.24)

We will proveV2(x) ≥ uε(x) in [ξ1 − ε, ζ], (3.25)

which, together with (3.24), yields the assertion because uε(ζ) is the maximum of uε in[ξ1, ζ] and ξ1 < ξ1 < ζ. To prove (3.25), we introduce the following function

h2(x) =uε(x)− a∗V2(x)− a∗

in [ξ1, ζ]

and will show h2(x) ≤ 1 by contradiction. Assume that there exists x2 ∈ [ξ1, ζ] such that

h2(x2) = max{h2(x) ; x ∈ [ξ1, ζ]} = η > 1. (3.26)

14

By (3.26) {uε(x) ≤ Wη(x) in [ξ1, ζ],

uε(x2) = Wη(x2),

where Wη(x) = η(V2(x) − a∗) + a∗. Since h2(ξ1) < 1, x2 must satisfy ξ1 < x2 ≤ ζ. If x2

lies in (ξ1, ζ), then it is easy to see

u′′ε(x2) ≤ W ′′

η (x2). (3.27)

For the case x2 = ζ, note h′2(x2) = h′

2(ζ) = 0. Therefore, (3.27) is also valid for x2 = ζ.As the next step, we will prove

f(x,Wη) < ηV2(1− V2)(V2 − a∗). (3.28)

As a function of η, set P (η) = ηV2(1− V2)(V2 − a∗)− f(x,Wη). Then

P ′(η) = V2(1− V2)(V2 − a∗)− (V2 − a∗)fu(x,Wη) = (V2 − a∗)Q(η),

whereQ(η) = V2(1− V2)− fu(x,Wη).

Observe that

Q′(η) = −fuu(x,Wη)(V2 − a∗) = 2(V2 − a∗){(Wη − a(x)) + (2Wη − 1)}.

Recalling the definition of δ∗ and η > 1, we can see that Wη(x) ≥ V2(x) > δ∗ >max{1/2,max{a(x) ; x ∈ [ξ1, ζ]}} in (ξ1, ζ) ; this implies Q′(η) > 0. Therefore,

Q(η) ≥ Q(1) = (V2 − a(x))(2V2 − 1) > 0,

which leads to P ′(η) > 0 for η ≥ 1. Hence we get

P (η) ≥ P (1) = V2(1− V2)(a(x)− a∗) > 0

and (3.28) is proved.We finally combine (3.27) and (3.28) to get

0= ε2u′′ε(x2)+ f(x2, uε(x2))≤ ε2W ′′

η (x2) + f(x2,Wη(x2))< η{(V ′′

2 (x2)+ V2(x2)(1− V (x2))(V (x2)− a∗)} = 0.

Since this is a contradiction, we have shown (3.25) ; thus the proof is complete.

Remark 3.4. We should note that (3.14) depends on the position x ; for any x ∈ [ξ1, ζ],1−uε(x) is estimated in terms of the distance between x and ξ1 when ζ is a local maximumpoint. Although similar results as Theorem 3.3 have been obtained by Ai, Chen andHastings [1, Lemma 2.3], their results are only concerned with the order of 1− uε(ζ). Inthis point of view, we believe that (3.14) gives us more precise information on the profileof uε. Indeed, (3.14) helps us to study the ε-dependence of the width of each transitionlayer, spike, multi-layer and multi-spike in Sections 4 and 5.

15

Remark 3.5. In (3.14), we can choose r =√1− A∗ + O(1) and R =

√1− A∗ + O(1)

where A∗ = min{a(x) ; x ∈ [ξ1, ζ]} and A∗ = max{a(x) ; x ∈ [ξ1, ζ]}. These facts can beshown from the proof of Theorem 3.3 by taking account of the definition of r and R inLemma 3.1.

Using the same method as the proof of Theorem 3.3 one can prove the following resultfrom Lemma 3.2 :

Theorem 3.6. For u ∈ Sn,ε, assume (3.13) and let ζ ∈ (ξ1, ξ2) satisfy u′(ζ) = 0. If (ξ2 −ζ)/ε → +∞, then for sufficiently small ε > 0, there exist positive constants C ′

1, C′2, r

′, R′

with C ′1 < C ′

2 and r′ < R′ such that

C ′1 exp

(−R′(ξ2 − ζ)

ε

)< 1− uε(x) < C ′

2 exp

(−r′(ξ2 − x)

ε

)(3.29)

for x ∈ [ζ, ξ2].

Remark 3.7. Theorems 3.3 and 3.6 deal with the case that ζ ∈ (ξ1, ξ2) is a local maximumpoint of uε ; i.e., the case that uε(ζ) is very close to 1. On the contrary, assume that ζ isa local minimum point of uε and (ζ − ξ1)/ε → ∞ or (ξ2 − ζ)/ε → ∞ as ε → 0. Then wecan derive similar estimates as (3.14) and (3.29) with 1− uε(x) replaced by uε(x).

4 Location of transition layers and spikes

We will study the location of transition layers and spikes of n-mode solution uε with useof (1.3) and (1.4).

Theorem 4.1. Let ξ be any point in Ξ. Then ξ lies in a neighborhood of a point in Σ∪Λwhen ε is sufficiently small. Moreover, if uε has a transition layer near a point x0 ∈ Σ∪Λ,then x0 belongs to Σ, and if uε has a spike near a point x0 ∈ Σ∪Λ, then x0 belongs to Λ.

Remark 4.2. Theorem 4.1 has been obtained by Ai, Chen and Hastings [1, Theorem 1].In the proof, they have reduced the location problem to a certain kind of algebraic system.We give a different proof ; we will derive a contradiction to the finiteness of Ξ for uε bymeans of asymptotic properties developed in Section 3.

Proof. Define {ξk}nk=1, {ζk}n−1k=1 as in Lemma 2.3 and set ζ0 = 0, ζn = 1. Let Σ =

{z1, z2, . . . , zm} with 0 < z1 < z2 < · · · < zm < 1. By Lemma 2.4 it can be shown that, ifuε ∈ Sn,ε has a transition layer in a neighborhood of ξε ∈ Ξ, then ξε must be very closeto one of zj when ε is sufficiently small.

It is sufficient to show that if uε has a spike near ξε, then ξε lies in a vicinity of apoint in Λ. For this purpose, let a(x) − 1/2 > 0 in (zj, zj+1) and denote all points ofΛ ∩ (zj, zj+1) by y1, y2, . . . , yl with zj < y1 < y2 < · · · < yl < zj+1.

We will prove by contradiction that every spike lies near a point in Λ. Take any smallδ > 0 and fix it. Assume that uε has spikes in an interval (zj + δ, y1 − δ). Note a′(x) > 0in this interval. By (iii) of Lemma 2.4, then there exist ξk and ξk+1 such that

zj + δ < ξk < ξk+1 < y1 − δ, u′ε(ξk) < 0 and u′

ε(ξk+1) > 0,

16

if ε is sufficiently small. By Lemma 2.3 there exist ζk−1, ζk, ζk+1 satisfying ζk−1 < ξk <ζk < ξk+1 < ζk+1.

We will show1− uε(ζk−1) > κ

√ε (4.1)

with some κ > 0, in the case that neither ζk−1 nor ζk+1 belongs to (zj, y1). The othercases can be discussed in the same way and the proof is easier.

We rewrite (1.5) as

ε2u′′ε + f(ζk, uε) = uε(1− uε)(a(x)− a(ζk)). (4.2)

Multiplying (4.2) by u′ε and integrating the resulting expression over (ζk−1, ζk+1) with

respect to x we get

W (ζk, uε(ζk−1))−W (ζk, uε(ζk+1))

=

∫ ζk+1

ζk−1

uε(x)(1− uε(x))(a(x)− a(ζk))u′ε(x)dx

=

(∫ zj

ζk−1

+

∫ y1

zj

+

∫ ζk+1

y1

)uε(x)(1− uε(x))(a(x)− a(ζk))u

′ε(x)dx

= : I + II + III.

(4.3)

We will estimate I, II and III.We begin with the study of II. Since a is monotone increasing in (ζk, y1),

II >

∫ y1

ζk+ε

uε(x)(1− uε(x))(a(x)− a(ζk))u′ε(x)dx

> (a(ζk + ε)− a(ζk))

∫ y1

ζk+ε

uε(x)(1− uε(x))u′ε(x)dx

= (a(ζk + ε)− a(ζk))

∫ uε(y1)

uε(ζk+ε)

s(1− s)ds

> Kε

∫ uε(y1)

uε(ζk+ε)

s(1− s)ds

with a positive constant K. Moreover, Theorem 3.3 gives

1− uε(y1) < C exp

(−r(y1 − ξk)

ε

)< C exp

(−rδ

ε

),

and Lemma 2.4 implies uε(ζk+ ε) < A with some A ∈ (0, 1) provided that ε is sufficientlysmall. Hence ∫ uε(y1)

uε(ζk+ε)

s(1− s)ds >

∫ uε(y1)

A

s(1− s)ds > C∗ (4.4)

with a positive constant C∗ independent of ε ; so

II > C∗Kε.

17

We next estimate I ;

|I| ≤∫ zj

ζk−1

|uε(1− uε)(a(x)− a(ζk))u′ε|dx

≤∫ zj

ζk−1

uε(1− uε)|u′ε|dx =

∫ uε(ζk−1)

uε(zj)

s(1− s)ds ≤ 1− uε(zj).

Theorem 3.6 implies

1− uε(zj) ≤ C2 exp

(−r(ξk − zj)

ε

)≤ C2 exp

(−rδ

ε

).

Therefore, we get |I|=O(exp(−1/ε))). Similarly, one can also derive |III|=O(exp(−1/ε)).Thus we get

W (ζk, uε(ζk−1))−W (ζk, uε(ζk+1)) = I + II + III > K∗ε (4.5)

with some K∗ > 0.On the other hand, we will estimate the left-hand side of (4.3). In the same way as

the proof of (3.10), one can see

W (ζk, uε(ζk−1))−W (ζk, uε(ζk+1)) = −1

2fu(ζk, θ){(1− uε(ζk−1))

2 − (1− uε(ζk+1))2}

with some θ ∈ (uε(ζk−1), 1). Since θ is very close to 1, there exists a positive constant M ,which is independent of ε, such that

W (ζk, uε(ζk−1))−W (ζk, uε(ζk+1)) < M(1− uε(ζk−1))2. (4.6)

Hence (4.1) follows from (4.5) and (4.6).We use (4.1) and Theorem 3.6 with x = ζk−1 and ξ2 = ξk+1 to get

κ√ε < c′2 exp

(−r′(ξk − ζk−1)

ε

)(4.7)

with some c′2 > 0 and r′ > 0. Here recall that uε is periodic with period 2. So we seethat there exists ξk−1 such that uε(x) > a(x) for x ∈ (ξk−1, ξk). Therefore, Theorem 3.3together with (4.1) implies

κ√ε < 1− uε(ζk−1) < C exp

(−r(ζk−1 − ξk−1)

ε

). (4.8)

Hence (4.7) and (4.8) implyξk − ξk−1 < Kε| log ε| (4.9)

with some positive constant K. This fact implies that ξk−1 belongs to the interval (zj +δ, y1 − δ) if ε is sufficiently small.

When ξk−1 lies in (zj + δ, y1− δ), Lemma 2.4 tells us that there must be another spikesuch that ξk−2, ξk−1 ∈ Ξ with zj+δ < ξk−2 < ξk−1 < y1−δ and u′

ε(ξk−2) < 0, u′ε(ξk−1) > 0.

Note that uε has a peak at x = ζk−1 ∈ (ξk−2, ξk−1). Repeating this procedure, we see thatthe number of points of Ξ∩(zj+δ, y1−δ) increases in each process. This is a contradictionto the definition of n-mode solutions; so that uε has no spikes in (zj + δ, y1 − δ).

The same argument is valid to show that uε has no spikes in (yi + δ, yi+1 − δ) fori = 1, 2, . . . , l − 1 and (yl, zj+1). Thus the proof is complete.

18

We will discuss the location of each single transition layer more carefully.

Theorem 4.3. Let uε ∈ Sn,ε possess a single transition layer near z ∈ Σ for sufficientlysmall ε > 0. If Ξ ∩ (z − δ, z + δ) = {ξ} with some δ > 0, then ξ − z = O(ε).

Proof. We only consider the case where a′(z) > 0, z < ξ and u′ε(ξ) > 0 for the sake of

simplicity. The other case can be shown in the same way as follows.Choose critical points ζ0 and ζ1 of uε such that u′

ε(x) > 0 for x ∈ (ζ0, ζ1). Since uε hasa single transition layer in (ζ0, ζ1), there exists ξ

∗ ∈ Ξ such that ξ∗ > ξ+σ with a positiveconstant σ independent of ε and uε(x) > a(x) for x ∈ (ξ, ξ∗). (Regarding uε as a functiondefined for all x ∈ R by reflectionm we can take such ξ∗.) Since ζ1 is distant from ξ orξ∗ independently of ε, Theorem 3.3 or 3.6 enables us to get 1 − uε(ζ1) = O(exp(−1/ε)).Similarly, we can also show uε(ζ0) = O(exp(−1/ε)).

We introduce

W (x, u) := −∫ u

ϕ0(x)

f(x, s)ds with ϕ0(x) :=

{0 in (ζ0, ξ),1 in (ξ, ζ1).

We use the following identity for x ∈ (ζ0, ξ) :

d

dx

{1

2ε2u′

ε(x)2 − W (x, uε(x))

}= {ε2u′′

ε(x) + f(x, uε(x))}u′ε(x)− Wx(x, uε(x))

= a′(x)G(u(x)),

(4.10)

where,

G(u(x)) :=

{−u(x)2/2 + u(x)3/3 in (ζ0, ξ),

(1− u(x)2)/2− (1− u(x)3)/3 in (ξ, ζ1).

By Remark 3.7, we see that

|a′(x)G(u(x))| < C2 exp

(−r(ξ − x)

ε

)in (ζ0, ξ)

with some positive constants C2 and r. Therefore, there exists a positive constant K suchthat ∣∣∣∣∫ ξ

ζ0

a′(x)G(u(x))dx

∣∣∣∣ < Kε. (4.11)

On the other hand, integrating the left-hand side of (4.10) over (ζ0, ξ) yields that∫ ξ

ζ0

d

dx

{1

2ε2u′

ε(x)2 − W (x, uε(x))

}dx =

1

2ε2u′

ε(ξ)2 +

∫ u(ξ)

0

f(ξ, s)ds+ W (ζ0, uε(ζ0)).

Hence it follows from (4.10) and (4.11) that

1

2ε2u′

ε(ξ)2 +

∫ u(ξ)

0

f(ξ, s)ds+ W (ζ0, uε(ζ0)) ≤ Kε. (4.12)

19

Repeating the same argument as above with (ζ0, ξ) replaced by (ξ, ζ1), one can obtainthat

−1

2ε2u′

ε(ξ)2 − W (ζ1, uε(ζ1)) +

∫ 1

u(ξ)

f(ξ, s)ds ≤ K1ε. (4.13)

with some K1 > 0. Therefore, (4.12) and (4.13) imply that

W (ζ0, uε(ζ0))− W (ζ1, uε(ζ1)) +

∫ 1

0

f(ξ, s)ds = O(ε).

It follows from W (ζ0, uε(ζ0)) = O(exp(−1/ε)) and W (ζ1, uε(ζ1)) = O(exp(−1/ε)) that∫ 1

0

f(ξ, s)ds = O(ε).

Taking account of∫ 1

0

f(ξ, s)ds = −1

6a(ξ) +

1

12and a(ξ) =

1

2+ a′(z)(ξ − z) +O((ξ − x)2),

we can conclude that ξ − z = O(ε).

5 Multiplicity of transition layers and spikes

In this section we will discuss a cluster of multiple transition layers and spikes. ByTheorem 4.1, such a cluster of multiple transition layers appears in a neighborhood of apoint in Σ if it exists, while a cluster of multiple spikes appears in a neighborhood a pointin Λ if it exists.

Definition 5.1 (multi-layer). Let uε be a solution of (1.5). If uε has a cluster ofmultiple transition layers in a neighborhood of a point in Σ, then such a cluster is calleda multi-layer.

Definition 5.2 (multi-spike). Let uε be a solution of (1.5). If uε has a cluster of multiplespikes in a neighborhood of a point in Λ, then such a cluster is called a multi-spike.

We introduce some notations to study multi-layers and multi-spikes.

Σ+ = {x∗ ∈ Σ ; a′(x∗) > 0}, Σ− = {x∗ ∈ Σ ; a′(x∗) < 0},Λ+ = {x∗ ∈ Λ ; a(x∗) < 1/2 and a attains its local maximum at x = x∗},Λ− = {x∗ ∈ Λ ; a(x∗) > 1/2 and a attains its local minimum at x = x∗}.

We begin with the study of multi-layers. We only discuss the case where uε has amulti-layer in a neighborhood of z ∈ Σ+ because the analysis for the case z ∈ Σ− isalmost the same.

By virtue of Lemma 2.4, there exists a one-to-one correspondence between a transitionlayer and a point in Ξ defined by (2.2).

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Lemma 5.1. For z ∈ Σ+, let ξ1, ξ2 ∈ (z − δ, z + δ) be successive points in Ξ satisfyingu′ε(ξ1) < 0 and u′

ε(ξ2) > 0 (resp. u′ε(ξ1) > 0 and u′

ε(ξ2) < 0) with some δ > 0. Then thereexitst another ξ ∈ Ξ such that z − δ < ξ < ξ1 and u′

ε(ξ) > 0 (resp. ξ2 < ξ < z + δ andu′ε(ξ) > 0) provided that ε is sufficiently small.

Proof. We give the proof in the case u′ε(ξ1) < 0 and u′

ε(ξ2) > 0. By Lemma 2.3, thereexist critical points ζ0, ζ1 and ζ2 of uε with ζ0 < ξ1 < ζ1 < ξ2 < ζ2. Since a′(x) > 0 in(z − δ, z + δ), the argument used in the proof of (4.1) is valid to show

1− uε(ζ0) > κ√ε (5.1)

with some κ > 0 independent of ε. Theorem 3.3 implies the existence of the othersuccessive point ξ ∈ Ξ to ξ1 (with ξ < ξ1) satisfying

1− uε(ζ0) < C exp

(−r(ζ0 − ξ)

ε

)(5.2)

with some C, r > 0. As in the proof of (4.9), it follows from (5.1) and (5.2) that ξ ∈ Ξsatisfies ξ < ξ1, ξ1 − ξ < Kε| log ε| and u′

ε(ξ) > 0. Hence ξ lies in (z − δ, z + δ) if ε issufficiently small.

Lemma 5.2. Let z ∈ Σ+ and assume that uε has a multi-layer in (z − δ, z + δ) withsome δ > 0. If ε is sufficiently small, then Ξ∩ (z − δ, z + δ) consists of an odd number ofelements. Moreover, if

Ξ ∩ (z − δ, z + δ) = {ξl, . . . , ξm} (5.3)

with some l,m ∈ N such that m− l is even, then u′ε(ξl) > 0 and u′

ε(ξm) > 0.

Proof. Define ξi, i = l, · · · ,m, by (5.3). We will show this lemma by contradiction.Assume that m− l is odd. Then one of the following properties holds true :

u′ε(ξl) < 0, u′

ε(ξl+1) > 0 and u′ε(ξm−1) < 0, u′

ε(ξm) > 0, (5.4)

u′ε(ξl) > 0, u′

ε(ξl+1) < 0 and u′ε(ξm−1) > 0, u′

ε(ξm) < 0. (5.5)

Lemma 5.1 implies that there exists ξl−1 ∈ Ξ (resp. ξm+1 ∈ Ξ) such that z− δ < ξl−1 < ξl(resp. ξm < ξm+1 < z + δ) when (5.4) (resp. (5.5)) is satisfied. This is a contradiction to(5.3). Hence m− l is even.

It is clear that either u′ε(ξl) > 0 and u′

ε(ξm) > 0, or u′ε(ξl) < 0 and u′

ε(ξm) < 0.However, in the latter case, Lemma 5.1 enables us to derive a contradiction in the sameway as above. So the proof is complete.

Let uε possess a multi-layer in a neighborhood of z ∈ Σ+. Set Ξ ∩ (z − δ, z + δ) ={ξl, ξl+1, . . . , ξm} with some δ > 0. By Lemma 2.3 uε has critical points ζl−1, ζl, . . . , ζmsuch that ζl−1 < ξl < ζl < · · · < ξm < ζm. Here we should note that uε(ζl−1) is close to0 and that uε(ζm) is close to 1. Such a multi-layer is called a multi-layer from 0 to 1. Amulti-layer from 1 to 0 is defined in a similar manner.

We can also show that, if there exists a multi-layer in a neighborhood of a point inΣ−, it must be a multi-layer from 1 to 0.

Summarizing these facts we have the following theorem.

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Theorem 5.3. A multi-layer from 0 to 1 (resp. from 1 to 0) appears only in a neighbor-hood of a point in Σ+ (resp. Σ−).

Next we will study multi-spikes. Note that for each spike there exist exactly twopoints in Ξ. So if uε has a multi-spike in a neighborhood of some y ∈ Λ, we can denoteΞ ∩ (y − δ, y + δ) = {ξl, ξl+1, . . . , ξm} with some δ > 0 and some l,m ∈ N such thatm− l is odd. Moreover, by Lemmas 2.3 and 2.4, there exist critical points of uε denotedby {ζk}mk=l−1 such that ζl−1 < ξl < ζl, · · · , ξm < ζm and both uε(ζl−1) and uε(ζm) aresufficiently close to 0 or 1. If uε(ζl−1) and uε(ζm) are close to 1 (resp. 0), then such amulti-spike is called a multi-spike based on 1 (resp. 0).

Theorem 5.4. A multi-spike based on 1 (resp. 0) appears only in a neighborhood of apoint in Λ− (resp. Λ+).

Proof. We only show that a multi-spike based on 1 appears in a neighborhood of a pointof Λ−. Since any spike based on 1 appears only in a neighborhood of a critical point y ofa with a(y) > 1/2, it sufffices to show that, if a takes its local maximum at y, then anymulti-spike based on 1 cannot appear in a neighborhood of such y in order to completethe proof.

We take a contradiction method. Let y be a local maximum point of a satisfyinga(y) > 1/2. Assume that uε has a multi-spike based on 1 in (y−δ, y+δ) with some δ > 0.

Observe that, if there is a multi-spike in (y − δ, y + δ), then

Ξ ∩ (y − δ, y + δ) = {ξl, ξl+1, . . . , ξm}

with some l,m ∈ N such that m− l is odd. By Lemma 2.3, we can choose ζl−1, ζl, . . . , ζmsuch that u′(ζk) = 0 for k = l− 1, l, . . . ,m and ζl−1 < ξl < ζl < · · · < ξm < ζm. Moreover,Lemma 2.4 implies that ξk+1 − ξk = O(ε) for k = l, l + 1, . . . ,m− 1 ; so that at least twopoints in Ξ belong to either (y − δ, y) or (y, y + δ).

We will consider the case when ξl, ξl+1 ∈ (y − δ, y). Note that a′(x) > 0 in (y − δ, y).For the sake of simplicity, we assume that ζl−1 lies in (y− δ, y). (If not, see the argumentdeveloped in the proof of Theorem 4.1.) Similarly to the proof of (4.2) and (4.3) we have

W (ζk, uε(ζl−1))−W (ζk, uε(ζl+1)) =

∫ ζl+1

ζl−1

uε(x)(1− uε(x))(a(x)− a(ζl))u′ε(x)dx. (5.6)

For the left-hand side of (5.6), observe that (4.6) is valid with k replaced by l. So it issufficient to consider the right-hand side of (5.6). Since a′′(x) < 0 in (y − δ, y) by (A.3),the right-hand side of (5.6) is bounded from below by∫ ζl−ε

ζl−1

uε(x)(1− uε(x))(a(ζl)− a(x))(−u′ε(x))dx

>

∫ ζl−ε

ζl−1

uε(x)(1− uε(x))(a(ζl)− a(ζl − ε))(−u′ε(x))dx

=(a(ζl)− a(ζl − ε))

∫ uε(ζl−1)

uε(ζl−ε)

s(1− s)ds,

22

when ε is sufficiently small. By the Taylor expansion, we see that

a(ζl)− a(ζl − ε) > Cε2

with a positive constant C independent of ε. Moreover, the same argument as in theproof of (4.4) leads to ∫ uε(ζl−1)

uε(ζl−ε)

s(1− s)ds > C∗

with some positive constant C∗. Thus one can deduce

1− uε(ζl−1) > κε

with some κ > 0 (cf. (4.1)). We repeat the argument developed in the proof of Theo-rem 4.1 with use of Theorems 3.3 and 3.6. It is seen that there exists another spike in(y− δ, y) when ε is sufficiently small. This is a contradiction to the definition of ξl. Thuswe complete the proof.

Finally, we will discuss ε-dependence of the width and the location of multi-layer andmulti-spike. For this purpose, we will collect important properties of multi-layers andmulti-spikes.

By Lemma 5.2 any multi-layer consists of an odd number of transition layers. If uε

has a multi-layer in δ-neighborhood of z ∈ Σ = Σ+ ∪ Σ− with small δ > 0, then thereexist m ∈ N \ {1} and {ξk}2m−1

k=1 ⊂ Ξ satisfying

(z − δ, z + δ) ∩ Ξ = {ξk}2m−1k=1 (5.7)

when ε is sufficiently small. Then from Lemma 2.3 we can choose a set of critical pointsof uε, which is denoted by {ζk}2m−1

k=0 , satisfying ζ0 < ξ1 < ζ1 < · · · < ξ2m−1 < ζ2m−1. Weshould note that ξk − ξk−1 = O(ε| log ε|) for any k = 1, 2, . . . , 2m − 1 by (4.9). It alsoshould be noted that u(ζ0) = O(exp(−1/ε)) and 1− u(ζ2m−1) = O(exp(−1/ε)) if z ∈ Σ+,while 1 − u(ζ0) = O(exp(−1/ε)) and u(ζ2m−1) = O(exp(−1/ε)) if z ∈ Σ− by the samereasoning as in the proof of Theorem 4.3.

Similarly, if uε has a multi-spike in a neighborhood of y ∈ Λ+ ∪ Λ− ⊂ Λ, then thereexist l ∈ N \ {1}, {ξk}2lk=1 ⊂ Ξ and critical points {ζk}2lk=0 of uε which satisfy

(z − δ, z + δ) ∩ Ξ = {ξk}2lk=1 (5.8)

and ζ0 < ξ1 < ζ1 < · · · < ξ2l < ζ2l. Observe that Lemma 2.4 implies that ξ2k−ξ2k−1 = O(ε)for any k = 1, 2, . . . , l. Furthermore, by the same argument as in the proof of Theorem 5.4,we obtain that ξ2k+1 − ξ2k = O(ε| log ε|) for any k = 1, 2, . . . , l − 1. We also note that, ify ∈ Λ+, then uε(ζ0) = O(exp(−1/ε)) and uε(ζ2l) = O(exp(−1/ε)), while if y ∈ Λ+, then1− uε(ζ0) = O(exp(−1/ε)) and 1− uε(ζ2l) = O(exp(−1/ε)).

Theorem 5.5. Let uε ∈ Sn,ε posses a multi-layer satisfying (5.7) for sufficiently smallε > 0. Then ξk − z = O(ε| log ε|) for k = 1, 2, . . . , 2m− 1.

23

Proof. For the sake of simplicity, we only consider the case that m = 2 and z ∈ Σ+. Inthis case, there exist a set of critical points {ζk}3k=0 of uε satisfying ζ0 < ξ1 < ζ1 < ξ2 <ζ2 < ξ3 < ζ3, and a constant C > 0 such that ξ3 − ξ1 < Cε| log ε|. Therefore, it suffices toconsider the case z < ξ1 or ξ3 < z in order to complete the proof.

We will give the proof in the case ξ3 < z. It also should be noted that a multi-layernear z ∈ Σ+ must be a multi-layer from 0 to 1. Rewrite (1.5) as

ε2u′′ε + f(z, uε) = uε(1− uε)(a(x)− 1/2). (5.9)

Multiplying (5.9) by u′ε and integrating the resulting expression over (ζ2, z) we get

1

2ε2u′

ε(z)2−W (z, uε(z))+W (z, uε(ζ2)) =

∫ z

ζ2

uε(x)(1−uε(x))(a(x)−1/2)u′ε(x)dx. (5.10)

We should note that both a and uε are monotone increasing in (ζ2, z). Hence the right-hand side of (5.10) is negative ; so that W (z, uε(z)) > W (z, uε(ζ2)). Taking account ofthe profile of the graph of W (z, u), we get

uε(ζ2) < 1− uε(z). (5.11)

Applying Theorems 3.3, 3.6 and Remark 3.7 to (5.11), we can obtain

C1 exp

(−R(ξ3 − ζ2)

ε

)< C2 exp

(−r(z − ξ3)

ε

)with some positive constants C1, C2, r and R. This implies that there is a constant K > 0such that

0 < z − ξ3 < K(ξ3 − ζ2) < K(ξ3 − ξ1) < KCε| log ε|

when ε is sufficiently small. Thus the proof is complete.

Theorem 5.6. Let uε ∈ Sn,ε posses a multi-spike satisfying (5.8) for sufficiently smallε > 0. Then ξk − z = O(ε| log ε|) for k = 1, 2, . . . , 2l.

Proof. For the sake of simplicity, we only consider the case m = 2 and y ∈ Λ−. Thenthere exists a set of critical points {ζk}4k=0 of uε satisfying ζ0 < ξ1 < ζ1 < · · · < ξ4 < ζ4.We should note that this multi-spike is based on 1. Then ξ2 − ξ1 = O(ε), ξ4 − ξ3 = O(ε)and ξ3 − ξ2 = O(ε| log ε|) ; so ξ4 − ξ1 = O(ε| log ε|). Therefore, it is sufficient to discussthe case y < ξ1 or ξ4 < y in order to complete the proof. We only consider the latter case.

We rewrite (1.5) as

ε2u′′ε + f(ζ3, uε) = uε(1− uε)(a(x)− a(ζ3)). (5.12)

Multiplying (5.12) by u′ε and integrating the resulting expression over (ζ2, y) with respect

to x, we obtain

1

2ε2u′

ε(y)2 −W (ζ3, uε(y)) +W (ζ3, uε(ζ2)) =

∫ y

ζ2

uε(x)(1− uε(x))(a(x)− a(ζ3))u′ε(x)dx.

(5.13)

24

Since a′(x) < 0 in (ζ2, y), u′ε(x) > 0 in (ζ3, y) and uε(x) < 0 in (ζ2, ζ3), the right-hand

side of (5.13) is negative. This fact implies W (ζ3, uε(ζ2)) < W (ζ3, uε(y)). Therefore,1− uε(ζ2) < 1− uε(y). Applying Theorems 3.3 and 3.6 we can obtain

C1 exp

(−R(ξ3 − ζ2)

ε

)< C2 exp

(−r(y − ξ4)

ε

)with some positive constants C1, C2, r and R. Thus we can conclude that y−ξ4 < ξ3−ζ2 <ξ4 − ξ1 = O(ε| log ε|) .

References

[1] S. Ai, X. Chen, and S. P. Hastings, Layers and spikes in non-homogeneous bistablereaction-diffusion equations, to appear in Trans. Amer. Math. Soc.

[2] S. Ai and S. P. Hastings, A shooting approach to layers and chaos in a forced Duffingequation, J. Differential Equations, 185(2002), 389–436.

[3] S. B. Angenent, J. Mallet-Paret, and L. A. Peletier, Stable transition layers in asemilinear boundary value problem, J. Differential Equations, 67(1987), 212–242.

[4] E. N. Dancer and S. Yan, Multi-layer solutions for an elliptic problem, J. DifferentialEquations, 194(2003), 382-405.

[5] P. C, Fife, Mathematical Aspect of Reacting and Diffusing System, Lecture Notes inBiomath., Vol. 28, Springer-Verlag, Berlin-Heidelberg-New York, 1979.

[6] J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J.Appl. Math., 5(1988), 367–405.

[7] K. Nakashima, Multi-layered stationary solutions for a spatially inhomogeneousAllen-Cahn equation, J. Differential Equations, 191(2003), 234–276.

[8] K. Nakashima, Stable transition layers in a balanced bistable equation, DifferentialIntegral Equations, 13(2000), 1025–1238.

[9] J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differ-ential Equations, 39(1981), 269–290.

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