turing patterns and wavefronts for reaction-diffusion systems in an infinite channel

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. APPL. MATH. c© 2010 Society for Industrial and Applied MathematicsVol. 70, No. 8, pp. 2822–2843

TURING PATTERNS AND WAVEFRONTS FORREACTION-DIFFUSION SYSTEMS IN AN INFINITE CHANNEL∗

CHAO-NIEN CHEN† , SHIN-ICHIRO EI‡ , AND YA-PING LIN†

Dedicated to the memory of Chung-Wei Ha

Abstract. This paper deals with reaction-diffusion systems on an infinitely long strip in R2.Through a pitchfork bifurcation, spatially heterogeneous patterns exist in a neighborhood of Turinginstability. Motivated by the works of Kondo and Asai, we study wavefront solution heteroclinic toTuring patterns. It will be seen that the dynamics of a wavefront can be approximated by a fourthorder equation of buckling type.

Key words. Turing pattern, wavefront, reaction-diffusion system

AMS subject classifications. 35J50, 35K55, 35K57, 37L65

DOI. 10.1137/090747348

1. Introduction. In 1995, an article by Kondo and Asai [17] showed that somechemical waves have been observed in the skin of angelfish. By using reaction-diffusionsystems in simulations, their results seem to fit into the formation of patterns verywell. A typical pattern structure is the rearrangement of the stripe patterns. Forexample, when the width of body varies in locations, the number of stripes thatappear on the fish skin become different. Moreover, patterns with different numbersof stripes seem to be joined with a heteroclinic-like solution and some defects appearin between. According to the observation on the growth of skin, the locations ofthe defects change. After the work [17], although there have been many simulationsrelated to the investigation of such phenomena (e.g., [1, 20, 30, 31]), there have beenno theoretical results yet.

Fig. 1.1. Turing patterns observed on angelfish [17].

∗Received by the editors January 21, 2009; accepted for publication (in revised form) June 6, 2010;published electronically September 16, 2010. This research was supported in part by the NationalScience Council, Taiwan, Republic of China, and Grants-in-Aid for Scientific Research from JSPS.

http://www.siam.org/journals/siap/70-8/74734.html†Department of Mathematics, National Changhua University of Education, Changhua, 500,

Taiwan ([email protected], [email protected]).‡Faculty of Mathematics, Kyushu University, Motooka Nishi-ku, Fukuoka 819-0395, Japan

([email protected]).

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TURING PATTERNS AND WAVEFRONTS 2823

To seek a theoretical framework to the above phenomena (see Figure 1.1), webegin with studying a mathematical problem on a cylindrical domain and constructinga solution heteroclinic to two stripe patterns; that is, consider a reaction-diffusionsystem

(1.1) Wt = DΔW + F (W ), t > 0, x = (x, y) ∈ Ω,

under homogeneous Neumann boundary conditions, where Ω := (−∞,∞) × (0, l) ∈R2, W = t(w1, . . . , wN ) ∈ RN , F is a smooth function on RN and D =diag(d1, . . . , dN ), a diagonal matrix with positive entries dj . We look for a solutionW (t,x) of (1.1) satisfying

(1.2) W (t,±∞, y) =W±(y).

A heteroclinic solution of this type is a standing or traveling front of (1.1). Of particu-lar interest is the situation where W+(y) and W−(y) are two nonconstant stationarysolutions of

(1.3) Wt = DWyy + F (W ), t > 0, y ∈ (0, l),

with the boundary conditions ∂W∂y = 0 at y = 0, l.

The aim of this paper is to start with the investigation of the problems (1.1)–(1.2) in the neighborhood of Turing instability [33]. Through a pitchfork bifurcation,we obtain two nonconstant stationary solutions for (1.3), say W±(y), which will bereferred to as Turing patterns. In addition, we study standing wave solutions of (1.1)joining withW±(y) as x→ ±∞. It will be seen that the dynamics of such a wavefrontare essentially governed by

(1.4)∂R

∂T= −γ1 ∂

4R

∂ζ4+R(M2 −M1R

2), T > 0, ζ ∈ R,

where all the coefficients γ1, M1, and M2 are positive numbers.In the past, a number of authors [2, 21, 22, 34] studied wavefront solutions in

cylindrical domains. If a solution of (1.1) has the form W (t, x, y) = U(z, y) withz = x− ct, then U satisfies

(1.5) D(Uzz + Uyy) + cUz + F (U) = 0, U(±∞, y) =W±(y).

In case (1.1) is a scalar equation. Vega [34] considered the solutions of (1.5) satisfying

(1.6) W−(y) < U(z, y) < W+(y) for all z ∈ R.

Under certain stability conditions onW+ andW−, Vega proved existence and unique-ness results [34] for the solutions of (1.5). As a consequence of the maximum principle,such a wave is monotone in the z-direction. Interesting examples including the KPPequation, bistable reaction-diffusion equation, and combustion model have been in-vestigated in [2, 22, 34].

The flow generated by a scalar reaction-diffusion equation is order-preserving. Ina convex domain, it is known [5, 19] that a stable solution satisfying homogeneousNeumann boundary condition must be constant. This type of result has been extendedby Jimbo and Morita [14] and Lopes [18] to the case of minimizers in a gradient system.On the other hand, many interesting patterns [6, 8, 9, 10, 11, 15, 16, 24, 25, 26, 27,

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2824 CHAO-NIEN CHEN, SHIN-ICHIRO EI, AND YA-PING LIN

28, 29, 32, 37] have been found in reaction-diffusion systems of activator-inhibitortype. Particular examples including FitzHugh–Nagumo type equations [9, 23] and theGierer–Mainhardt system [11] have received a great deal of attention. Based on a skew-gradient structure proposed in [35, 36], a number of stability criteria [6, 35, 36] forthe stationary solutions of FitzHugh–Nagumo type equations have been established.Although in the absence of order-preserving, calculus of variations provides a tool inanalyzing Turing patterns of skew-gradient systems. However, further work is neededto use this approach to study wavefront joining with two Turing patterns. In contrastwith the results [2, 23] obtained for the scalar reaction-diffusion equation, numericalsimulation (see Figures 5.3 and 5.4) indicates that in a FitzHugh–Nagumo type modela standing wavefront W (x, y) joining with W+(y) and W−(y) may not be monotonein the x-direction.

This paper is organized as follows: In section 2, a bifurcation analysis under thecondition of Turing instability will be derived. Through a pitchfork bifurcation, weobtain two nonconstant stationary solutions W±(y) of (1.3). Furthermore, on thecylindrical domain Ω, by taking as the x-independent solutions of (1.1), the stabil-ity of W±(y) will be studied in section 3. For a standing wave of (1.1) joining withW+(y) and W−(y), an approximate solution will be constructed in section 4. Forthe FitzHugh–Nagumo type model, it is convinced by numerical study that the ap-proximate solution fits in quite well. In certain situations, according to numericalsimulation, defect can be found out between Turing patterns with a different numberof stripes. Moreover, the variational method provides an additional tool for studyingwavefront in the FitzHugh–Nagumo type model. Detailed analysis will be given insection 5. Lastly, in section 6 we end up with some discussions on future works.

2. Bifurcation analysis of Turing patterns. In this section the existenceof nonconstant stationary solutions of (1.3) in a neighborhood of Turing instabilitywill be investigated. Let 0 be a linearly stable equilibrium of Wt = F (W ); that is,F (0) = 0 and all the eigenvalues of the linearized matrix B := F ′(0) are of negativereal parts. The function F treated here is in a general setting. To find out non-constant stationary solutions, we use bifurcation analysis. We introduce a parameterη and assume a bifurcation occurs at η = 0. Then the problem near the bifurcationpoint η = 0 is

(2.1) Wt = DWyy + F (W ) + ηG(W ), t > 0, 0 < y < l.

Here the function G satisfies G(0) = 0. Let L be the linearized operator with respectto the zero solution of

(2.2) DWyy + F (W ) = 0, y ∈ (0, l).

Substituting the Fourier series∑∞n=0 cosCnyan (an ∈ RN ) for the function W into

the eigenvalue problem LW = λW gives

{−C2nD +B}an = λan,

where Cn := nπl . This leads to the consideration of matrix Ξ(τ) := −τD+B parame-

terized by τ ≥ 0. Let λj(τ) (j = 1, . . . , N) be the eigenvalues of Ξ(τ). We assume thatthere exist positive numbers γ0 and τ0 such that Re(λj(τ)) < −γ0 for j = 2, . . . , Nand λ1(τ) ≤ 0 for any τ ≥ 0, and λ1(τ) = 0 if and only if τ = τ0 (Figure 2.1). It isalso assumed that λ1(τ) is a simple eigenvalue of Ξ(τ) with eigenvector α(τ) ∈ RN

for all τ ≥ 0.

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Fig. 2.1. Eigenvalue λ1(τ) and others.

Let φn(y) := cosCnyan. If the width l of Ω satisfies(πl

)2= τ0 and α(τ0) = a1,

then Lφ1 = 0. In this situation, if unstable solutions in the neighborhood of Turinginstability exist, they must be 1-mod solutions and close to a multiple of φ1(y).

To obtain the existence of nonconstant stationary solutions of (2.1), we studythe flow on an invariant manifold: Let α0 := α(τ0) and α∗

0 be the vector satisfyingtΞ(τ0)α

∗0 = 0 and 〈 α0,α

∗0 〉 = 1. Define

M1 := −{〈 F ′′(0)b0 ·α0,α

∗0 〉+ 1

8〈 F ′′′(0)α3

0,α∗0 〉+ 1

2〈 F ′′(0)b2 ·α0,α

∗0 〉}

and

M2 := 〈 G′(0)α0,α∗0 〉 ,

where bn := − 14 (−C2

nD+B)−1F ′′(0)α20 (n = 0, 2, 3, . . .). Applying the center manifold

theory (e.g., [4, 13]) yields the following properties for the solution W (t, y) of (2.1)on the invariant manifold.

Proposition 2.1. If η is sufficiently small, there exists a function σ = σ(r; η)(y)with ‖σ‖ = O(|η| + r2), σ(0; η) = 0, and σ(r; η)(y) = r2ψ2 +O(|η| + r3) such that

W (t, y) = r(t)ϕ1(y) + σ(r(t); η)(y)

with r(t) being a solution of

(2.3) r = H(r; η) := −M1r3 +M2rη +O(|η|2 + r4),

where ψ2 := b0 + cosC2yb2.The proof of Proposition 2.1 is standard. We omit it.Remark 1. The matrix (−C2

1D+B) is not invertible due to the assumptions ofΞ(τ) and C2

1 = τ0.Since λ′1(τ0) = 0, the eigenvalue λ1(τ) of the matrix Ξ(τ) has the following ex-

pansion:

(2.4) λ1(τ0 + δ) = −γ1δ2 +O(δ3).

If γ1 > 0, the stationary solutions of (2.1) in a small neighborhood of Turing instabilityare stable (Figure 2.1). The constant γ1 can be determined through the followingderivation.

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Substituting α(τ0 + δ) = α0 + δα1 + · · · and (2.4) into the eigenvalue problemΞ(τ)α(τ) = λ1(τ)α(τ), we see from the coefficients of the terms with order of δ1 that

(2.5) (−τ0D +B)α1 −Dα0 = 0.

Taking the inner product with α∗0 in (2.5) yields 〈 Dα0,α

∗0 〉 = 0.

Next, look at the coefficients for the terms of order δ2:

(−τ0D +B)α2 −Dα1 = −γ1α0,

from which we know

(2.6) 〈 Dα1,α∗0 〉 = γ1,

by taking the inner product with α∗0. To determine γ1, we calculate α1 from (2.5)

first. Since the matrix (−τ0D+B) has 0 eigenvalue with the eigenvector α0, it followsthat

α1 = e1 + aα0,

where e1 is a uniquely determined vector obtained by solving (−τ0D+B)e1−Dα0 = 0and using 〈 e1,α∗

0 〉 = 0. We remark that the matrix (−τ0D +B) is invertible in the

subspace {v ∈ RN ; 〈 v,α∗0 〉 = 0}. Then we see from (2.6) that

γ1 = 〈 Dα1,α∗0 〉

= 〈 D(e1 + aα0),α∗0 〉

= 〈 De1,α∗0 〉+ a 〈 Dα0,α

∗0 〉

= 〈 De1,α∗0 〉 ,

by making use of 〈 Dα0,α∗0 〉 = 0.

In summary, we have the following lemma.Lemma 2.2. If α(τ0 + δ) = α0 + δα1 + · · · and λ1(τ0 + δ) = −γ1δ2 +O(δ3), then

〈 Dα0,α∗0 〉 = 0 and γ1 = 〈 De1,α

∗0 〉.

If both M1 and M2 are positive, a supercritical pitchfork bifurcation diagramoccurs. This case holds for the FitzHugh–Nagumo equations, as will be shown insection 5. If M1 is negative, the bifurcation diagram is subcritical, which can occur in(5.1) if γ is negative and the nonlinearity is different.

In what follows, we consider the case where both M1 and M2 are positive. Then

if η > 0 and sufficiently small, r± := ±√

M2ηM1

+O(|η|) are the stable equilibria of the

ODE r = H(r; η) as stated in Proposition 2.1. Furthermore,

W±(y) := r±φ1(y) + σ(r±; η)(y) = ±√M2η

M1cosC1yα0 +O(|η|)

are the stable stationary solutions of (2.1).

3. Stability of Turing patterns as x-independent solutions in Ω. In thissection, the stability of W+(y) and W−(y) on the cylinder Ω will be investigated.Since W+(y) and W−(y) are x-independent solutions of

(3.1) Wt = DΔW + F (W ) + ηG(W ), t > 0, x = (x, y) ∈ Ω,

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they will be referred as planar stationary solutions.Let α∗(τ) be the eigenvector satisfying tΞ(τ)α∗(τ) = λ1(τ)α

∗(τ) and〈 α(τ),α∗(τ) 〉 = 1. Recall that α∗

0 = α∗(τ0). It will be seen that the stability ofW±(y) depends on the sign of ω0 defined by

ω0 :=1

8〈 F ′′(0)α0 · (v1 − b2),α

∗0 〉+ 1

8〈 F ′′′(0)α3

0,α∗0 〉 ,

where vn := −(−(C2n + τ0)D +B)−1F ′′(0)α2

0 (n = 1, 2, . . .).Theorem 3.1. If ω0 < 0 (> 0), then W±(y) are stable (respectively, unstable)

planar stationary solutions of (3.1).Proof. Since the solution under consideration is of order O(

√η), setting ε :=

√η

and W = εZ in (3.1), we arrive at

(3.2) Zt = DΔZ + F (Z) + ε2G(Z), t > 0, x = (x, y) ∈ Ω,

where F = F (Z; ε) := F (εZ)/ε and G = G(Z; ε) := G(εZ)/ε. Notice that

F (Z; ε) = F ′(0)Z +1

2εF ′′(0)Z2 + ε2

1

6F ′′′(0)Z3 + · · · .

A similar expansion holds for G(Z; ε).Let Z±(y) := 1

εW±(y). Consider the linearized operator L+ defined by

(3.3) L+Z = DΔZ + F ′(Z+(y))Z + ε2G′(Z+(y))Z.

We remark that Z+(y) = R+ cosC1yα0+εR2+ψ2+O(ε

2), where ψ2 := b0+cosC2yb2

and R+ := 1εr+ =

√M2

M1+O(ε), as noted in section 2. From (3.3), we obtain

L+Z = DΔZ +BZ + εR+ cosC1yF′′(0)α0 · Z

+ ε2{R2

+F′′(0)ψ2 · Z +

1

2R2

+ cos2 C1yF′′′(0)α2

0 · Z +G′(0)Z}+O(ε3)Z

=: L2Z +O(ε3)Z.

Taking Fourier transformation of this equation with respect to the x-variable, weobtain

L+Z = L+Z = −ξ2DZ + LyZ +O(ε3)Z,

where ξ is the Fourier transformation variable and

LyU := DUyy +BU + εR+ cosC1yF′′(0)α0 · U

+ ε2{R2

+F′′(0)ψ2 · U +

1

2R2

+ cos2 C1yF′′′(0)α2

0 · U +G′(0)U}.

Now, the small eigenvalues of L+ will be under investigation. Since L+U = Ξ(τ)U +

DUyy + O(ε)U , we denote L+ by L(ε, ξ2). Note that there are only two situations

that L(ε, ξ2) could have eigenvalues close to 0, namely, ξ2 ∼ 0 or ξ2 ∼ τ0.Let us first look at the case in the neighborhood of ξ2 = 0; that is, ξ2 = δ �

1. Let L0(δ)U := Ξ(δ)U + DUyy and λ0(δ) be the eigenvalue of L0(δ). SinceC2

1 = τ0, it follows that L0(δ)(cosC1yα(δ + τ0)) = cosC1yΞ(δ + τ0)α(δ + τ0) =

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λ1(δ+ τ0) cosC1yα(δ+ τ0), from which we know λ0(δ) = λ1(τ0 + δ), where λ1(τ0 + δ)is the first eigenvalue of Ξ(τ0 + δ) as stated in section 2. From the assumptions of

λ1(τ) (see Figure 2.1), we know λ0(δ) < −K1ε if δ > K2√ε, where K1,K2 are pos-

itive constants. Since L(ε, δ) = L0(δ) + O(ε), the eigenvalue of L(ε, δ), say λ(ε, δ),

satisfies λ(ε, δ) < −K3ε for some positive constant K3. Thus, it suffices to consider

δ ∈ [0,K2√ε]. Substituting λ(ε, δ) = 0+ελ1,0+δλ0,1+εδλ1,1+· · · with the correspond-

ing eigenfunction Z = cosC1yα0+εZ1,0+δZ0,1+εδZ1,1+· · · into L(ε, δ)Z = λ(ε, δ)Z,we see from the coefficients of the terms with order of ε1 that

(3.4) L0(0)Z1,0 +R+ cosC1yF′′(0)α0 · φ1 = λ1,0φ1,

where φ1 = cosC1yα0 as mentioned in section 2. Taking the inner product with

φ∗1 := cosC1yα

∗0 yields λ1,0 = 0. Next the terms with an order of δ1 imply

(3.5) L0(0)Z0,1 −Dφ1 = λ0,1φ1.

Taking the inner product with φ∗1 yields λ0,1 = 0. Furthermore, from (3.4) and (3.5),

we get Z1,0 = b1,0 + cos 2C1yb′1,0 + a1,0 cosC1yα0 and Z0,1 = cosC1y{b0,1 + a0,1α0}

for some constants a1,0, a0,1 and vectors b1,0, b′1,0, b0,1. Since λ1,0 = λ0,1 = 0, the

terms with order of ε1δ1 lead to

(3.6) L0(0)Z1,1 −DZ1,0 +R+ cosC1yF′′(0)α0 · Z0,1 = λ1,1φ1.

Substituting Z1,0 and Z0,1 into (3.6) and taking the inner product with φ∗1 =

cosC1yα0, we obtain λ1,1 = 0.

Now the expansion of λ(ε, δ) is reduced to ε2λ2,0 + δ2λ0,2 + O(ε3 + δ3). In the

next lemma, we will determine formulae for λ2,0 and λ0,2.

Lemma 3.2. In the expansion of λ(ε, δ), λ2,0 = −2M2 + O(ε) and λ0,2 = −γ1hold.

Proof. Since λ(0, δ) = λ0(δ) = λ1(τ0 + δ) = −γ1δ2 +O(δ3), it follows that λ0,2 =−γ1.

On the other hand, the terms with an order of ε2 in the eigenvalue problem ofL(ε, δ) give

L0(0)Z2,0 +R+ cosC1yF′′(0)α0 · Z1,0 + R2

+F′′(0)ψ2 · φ1(3.7)

+1

2R2

+ cos2 C1yF′′′(0)α2

0 · φ1 +G′(0)φ1 = λ2,0φ1.

Recall that R+ =√

M1

M2+O(ε). The precise form of Z1,0 is

Z1,0 = 2R+(b0 + cosC2yb2) + a1,0 cosC1yα0 = 2R+ψ2 + a1,0φ1.

Substituting Z1,0 into (3.7) and taking the inner product with φ∗1 = cosC1yα0, we

obtain

3R2+

⟨F ′′(0)α0 ·

(l

2b0 +

l

4b2

),α∗

0

⟩+3l

16R2

+ 〈 F ′′′(0)α30,α

∗0 〉+〈 G′(0)α0,α

∗0 〉= l

2λ2,0.

Consequently,

−3

2R2

+M1 +M2 =1

2λ2,0

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and

λ2,0 = −3R2+M1 + 2M2 = −3

M2

M1·M1 +M2 +O(ε) = −2M2 +O(ε).

As a consequence of Lemma 3.2, we know that λ(ε, δ) = −2ε2M2 − δ2γ1 +O(ε3 + δ3) ≤ −M2ε

2.

Next, we treat the case of ξ2 ∼ τ0 and consider the eigenvalue problem L(ε,

τ0 + δ)Z = μ(ε, δ)Z. Note that in the previous case, we suppress the dependence of ε

and δ from Z(ε, δ). In the following calculation, we use Z to denote Z(ε, τ0 + δ); the

terms in the expansion of Z will be treated in the same manner.The argument is similar to the previous case; we first obtain Z0,0 = α0, so

L0(τ0)α0 = 0 holds. The terms with an order of ε1 imply

L0(τ0)Z1,0 +R+ cosC1yF′′(0)α2

0 = μ1,0α0.

Taking the inner product with α∗0 yields μ1,0 = 0.

On the other hand, L(0, δ) = L0(δ) implies μ(0, δ) = λ1(τ0+δ). Hence, it is easilyseen that μ0,1 = 0 and μ0,2 = −γ1.

The terms with an order of ε1δ1 lead to

(3.8) L0(τ0)Z1,1 +R+ cosC1yF′′(0)α0 · Z0,1 −DZ1,0 = μ1,1α0.

Substituting Z1,0 = R+ cosC1yv1 + a1,0α0 and Z0,1 = e1 + a0,1α0 into (3.8) andtaking the inner product with α∗

0, we obtain μ1,1 = 0 by making use of Lemma 2.2.Finally, let us turn to the terms with order of ε2; that is,

L0(τ0)Z2,0 +R+ cosC1yF′′(0)α0 · Z1,0 + R2

+F′′(0)ψ2 ·α0(3.9)

+1

2R2

+ cos2 C1yF′′′(0)α3

0 +G′(0)α0 = μ2,0α0.

Taking the inner product with α∗0 in (3.9), we see that the right-hand side is lμ2,0.

The left-hand side becomes

R2+

∫ l

0

cos2 C1ydy 〈 F ′′(0)α0 · v1,α∗0 〉+R2

+l 〈 F ′′(0)b0 ·α0,α∗0 〉

+R2+

1

2

∫ l

0

cos2 C1ydy 〈 F ′′′(0)α30,α

∗0 〉+ l 〈 G′(0)α0,α

∗0 〉

=l

2R2

+ 〈 F ′′(0)α0 · v1,α∗0 〉+R2

+l 〈 F ′′(0)b0 · α0,α∗0 〉

+l

4R2

+ 〈 F ′′′(0)α30,α

∗0 〉+ l 〈 G′(0)α0,α

∗0 〉

= l

[R2

+

{1

2〈 F ′′(0)α0 · v1,α

∗0 〉+ 〈 F ′′(0)b0 · α0,α

∗0 〉+ 1

4〈 F ′′′(0)α3

0,α∗0 〉}+M2

]= l(R2

+ω1 +M2),

where ω1 := 12 〈 F ′′(0)α0 · v1,α

∗0 〉 + 〈 F ′′(0)b0 · α0,α

∗0 〉 + 1

4 〈 F ′′′(0)α30,α

∗0 〉. Then

we have

μ2,0 = R2+ω1 +M2

=M2

M1ω1 +M2 +O(ε)

=M2

{1 +

ω1

M1

}+O(ε).

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Hence μ2,0 < 0 if ω1

M1< −1 and ε is sufficiently small. We remark that ω1

M1< −1 is

equivalent to ω0 = ω1 +M1 < 0.To summarize the above results, we know ω0 < 0 implies that μ(ε, δ) = μ2,0ε

2 −γ1δ

2 + O(ε3 + δ3) < 12 μ2,0ε

2 < 0. Since both λ(ε, δ) and μ(ε, δ) are negative, theplanar solution W+ is stable. On the other hand, if ω0 > 0 and ε is sufficiently small,then μ(ε, δ) is positive. Thus W+ is unstable.

Corollary 3.3. Let W = t(u, v) ∈ R2 and F (W ) + ηG(W ) = t(f(u) − v +ηk(u, v), 1

d4(u − γv)), where d4 > 0 and γ > 0. Suppose f(0) = k(0, 0) = 0 and

f ′′(0) = 0. Then W±(y) are stable (or unstable) if f ′′′(0) < 0 (or > 0) and η issufficiently small.

Typical examples of f are cubic-like functions such as f(u) = u(1 − u2). For theFitzHugh–Nagumo type model, it will be seen in section 5 that the hypotheses ofCorollary 3.3 are satisfied and the planar solutions W± are stable.

Remark 2. In the case of anisotropic media in the directions of x- and y-axis,i.e., diffusion rates are different along x- and y-directions, the model equation becomes

(3.10) Wt = DxWxx +DWyy + F (W ) + ηG(W ), t > 0, x = (x, y) ∈ Ω,

where Dx is a diagonal matrix with positive entries. We note that the stability ofplanar stationary solutions of (3.10) can be proved if Dx is sufficiently close to D.However, this is a specially limited case, we do not carry out the proof here. Thecase without such a limitation on the diffusion matrices will be investigated in afuture work.

4. Approximation for connecting orbits joining with Turing patterns.Consider a center manifold defined by Σ(r)(y) := rφ1(y) + σ(r; η)(y). As we knowΣ(r)(y) is invariant under the flow generated by (2.1), and there exists a functionH(r; η) such that

HΣr = A(Σ),

where Σr := dΣdr and A(W ) := DWyy + F (W ) + ηG(W ). Note that we sometimes

suppress the dependence of η from the notation. In view of Proposition 2.1, H(r; η) =−M1r

3 +M2rη +O(|η|2 + r4) and W±(y) are given by Σ(r±; η).Using the above information on the center manifold Σ(r), we define an approxi-

mate function

W ∗(t, x, y) := Σ(√ηR(T, ζ))(y)

through the rescaling T := ηt and ζ := 4√ηx, where the function R(T, ζ) will be

determined in Theorem 4.1. Set H(R; η) := 1η√ηH(

√ηR; η). Then H(R; η) = R(M2−

M1R2) +O(

√η).

Theorem 4.1. Let γ′1 := 〈 Dxe1,α∗0 〉. Then for any given t = T ∗,

W ∗t − {DxW

∗xx +DW ∗

yy + F (W ∗) + ηG(W ∗)} = O(|η|3/2)

holds. Moreover, R(T, ζ) satisfies

(4.1) RT = −γ′1Rζζζζ + H(R; η) +O(√η)

uniformly for 0 ≤ T ≤ T ∗ and (x, y) ∈ Ω.

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Proof. Let Σ(R) := 1εΣ(εR) and A(Z) := DZyy+ F (Z)+ ε2G(Z), where ε :=

√η.

Then Σ(R) is an invariant manifold for (3.2), that is, ε2HΣR = A(Σ) holds. Consider a

solution of (3.2) with the expansion Z = Z(T, ζ, y) = Σ(R(T, ζ))+εZ1(T, ζ, y)+O(ε2),

where T := ε2t and ζ :=√εx. Then Z satisfies

(4.2) ε2ZT = A(Z) + εDxZζζ .

Substituting the expansion of Z into (4.2) gives

ε2RT ΣR + ε3∂TZ1 +O(ε4) = A(Σ(R)) + εA′(Σ(R))(Z1 + εZ2)

+ ε21

2A′′(Σ(R))Z2

1 + εDx(Σ + εZ1)ζζ +O(ε3).

Set

(4.3) h(R) := RT − H(R).

Invoking ε2HΣR = A(Σ), we arrive at

(4.4) ε2hΣR + ε3∂TZ1 +O(ε4) = εA′(Σ(R))(Z1 + εZ2) + εDx(Σ + εZ1)ζζ +O(ε3).

Recall that LZ := DZyy +BZ. Using R = R0 + εR1 +O(ε2), h = h0 + εh1 +O(ε2),and equating the terms with order of ε in (4.4), we get

(4.5) 0 = LZ1 + ∂2ζR0 cosC1yDxα0,

by making use of Σ(R) = R cosC1yα0 + εR2ψ2 + O(ε2) and A′(Σ(R))Z = LZ +εR cosC1yF

′′(0)α0 · Z +O(ε2). Solving (4.5) yields

(4.6) Z1 = cosC1y(−∂2ζR0e1 + a0α0)

with a0 being a constant.Next we turn to the terms with an order of ε2 in (4.4):

h0 cosC1yα0(4.7)

= LZ2 +R0 cosC1yF′′(0)α0 · Z1

+(R20)ζζDxψ2 + ∂2ζR1 cosC1yDxα0 +

1

2F ′′(0)(Z1)

2 +Dx∂2ζZ1.

Substituting (4.6) into (4.7) and taking the inner product with φ∗1 = cosC1yα

∗0, we

get

(4.8)

∫ l

0

〈 h0 cosC1yα0,φ∗1 〉 dy =

l

2h0

from the left-hand side of (4.7). On the right-hand side of (4.7), it follows from straight-forward calculation that∫ l

0

〈 R0 cosC1yF′′(0)α0 · Z1,φ

∗1 〉 dy

=

∫ l

0

cos3 C1ydy 〈 R0F′′(0)α0 · (−∂2ζR0e1 + a0α0),α

∗0 〉 = 0

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and ∫ l

0

〈 (R20)ζζDxψ2,φ

∗1 〉 dy

= (R20)ζζ

∫ l

0

{cosC1y 〈 Dxb0,α0 〉+ cosC2y cosC1y 〈 Dxb2,α0 〉}dy = 0.

Moreover, as mentioned in the proof of Theorem 3.1, ψ2 := b0 + cosC2yb2. Thendirect calculation gives∫ l

0

〈 ∂2ζR1 cosC1yDxα0,φ∗1 〉 = ∂2ζR1

∫ l

0

cos2 C1ydy 〈 Dxα0,α∗0 〉 = 0,

∫ l

0

⟨1

2F ′′(0)(Z1)

2,φ∗1

⟩dy =

∫ l

0

cos3 C1ydy

⟨1

2F ′′(0)(−∂2ζR0e1 + a0α0)

2,α∗0

⟩= 0

and ∫ l

0

〈 Dx∂2ζZ1,φ

∗1 〉 dy =

∫ l

0

cos2 C1ydy 〈 (−∂2ζR0Dxe1 + a0Dxα0)ζζ ,α∗0 〉

= − l

2〈 Dxe1,α

∗0 〉 ∂4ζR0 = − l

2γ′1∂

4ζR0.

Therefore,

(4.9) h0 = −γ′1∂4ζR0,

and by (4.3) we know the dynamics of R is governed by the following equation:

RT = H(R) + h(R) = H(R)− γ′1Rζζζζ +O(ε).

The proof of Theorem 4.1 is complete.Remark 3. When Dx = D + κD1 and κ is sufficiently small, we know γ′1 =

γ1 +O(κ) > 0. On the other hand, for large κ the question remains open.Remark 4. Assume that R0(T, ζ) satisfies

(4.10) RT = −γ′1Rζζζζ + H(R; η)

and R0(T,±∞) = R± for 0 ≤ T ≤ T ∗. If W ∗0 (t, x, y) := Σ(

√ηR0(T, ζ))(y), then

W ∗0 (t,±∞, y) = Σ(

√ηR0(T,±∞))(y) = Σ(

√ηR±)(y) = Σ(r±)(y) =W±(y).

Thus W ∗0 (t, x, y) is an approximation for a connecting orbit joining with W±(y) at

x = ±∞.Remark 5. Let Φ(R) =

∫∞−∞[

γ′1

2 R2ζζ +

∫ R0 H(s; η)ds]dζ. If R(T, ζ) is a solution

of (4.10) satisfying R(T,±∞) = R±, then ddT Φ(R(T, ζ)) = − ∫∞

−∞R2Tdζ. It has been

observed in numerical simulation that R0(T, ζ) converges to a stationary heteroclinicsolution of (4.10), say R0 = R0(ζ).

Remark 6. In the above derivation, all the coefficients mentioned hitherto suchas Mj are given explicitly.

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5. FitzHugh–Nagumo type model. In [30, 31] the authors studied the direc-tionality of stripe patterns for tropical fishes. They proposed that anisotropic diffusion(directional difference in diffusion rate) might be responsible for the contrasting dif-ference in the directionality of stripes, because most scales are arranged parallel tothe anterior-posterior axis and the substances (for example, activators and inhibitors)controlling the pattern formation may diffuse along the anterior-posterior axis at aspeed different from that along the dorso-ventral axis.

In this section, we study Turing patterns and wavefronts in the FitzHugh–Nagumotype system with anisotropic diffusion:

(5.1)

{ut = d1uxx + d3uyy + f(u)− v + ηu,vt = d′2vxx + vyy +

1d4(u− γv),

where u can be viewed as an activator while v acts as inhibitor. System (5.1) canbe written in the form of (3.10); that is, W = t(u, v), D = diag(d3, 1), F (W ) =t(f(u) − v, 1

d4(u − γv)), and G(W ) = t(u, 0). A typical example for nonlinearities is

f(u) = ku(1 − u2) for a positive constant k. Assume that f(0) = 0 and γf ′(0) < 1.In case d1 = d4 = 1, d′2 = d3 = 0.002277, f(u) = 0.1u(1 − u2), η = 0.05, γ = 1, andl = 2, Figure 5.1 illustrates that W−(y) and W+(y) are two different stripe patternsfor (5.1), while in Figure 5.2 we see a wavefront joining with W±(y); that is, there isa defect appeared in between.

Fig. 5.1. Profiles of u-component of W±(y) on the interval [0, l].

Fig. 5.2. Defect (u-component of wavefront) in (5.1).

5.1. Calculation for the coefficients of (1.4). As mentioned in the intro-duction, the dynamics of standing wavefront is governed by (1.4). We now calculate

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that the coefficients of (1.4) in case (5.1) is the model taken into consideration. Forconvenience in notation, let f ′

0 := f ′(0), f ′′0 := f ′′(0), and so on. The linearized matrix

is given by B = F ′(0) =( f ′

0 −11d4

− γd4

). Set Ξ(τ) := −τD + B. Suppose Ξ(τ) has zero

eigenvalue at τ = τ0, then det Ξ(τ0) = 0, or, equivalently,

(5.2) d4d3τ20 − (d4f

′0 − d3γ)τ0 − γf ′

0 + 1 = 0.

Since the largest eigenvalue λ1(τ) of Ξ(τ) is tangent to the τ -axis at τ = τ0, we knowthat τ0 is a multiple root of (5.2). This implies that

(5.3) (d4f′0)

2 + (d3γ)2 + 2d4d3γf

′0 − 4d4d3 = 0.

Furthermore, p0 := d3d4

is a positive root of

(5.4) γ2p20 − 2(−γf ′0 + 2)p0 + (f ′

0)2 = 0

and τ0 = 12p0d4

(f ′0 − γp0), from which we see f ′

0 − γp0 > 0.Below is a number of quantities and notations that we have studied in section 2:

p0 =2− γf ′

0 − 2√1− γf ′

0

γ2, τ0 =

f ′0 − γp02p0d4

,

α0 = t(2, f ′0 + γp0), α∗

0 =1

4(1− p0d4)t(2,−d4(f ′

0 + γp0)).

F ′′(0)α20 = t(f ′′

0 22, 0) = t(4f ′′

0 , 0), b0 =f ′′0

1− γf ′0

t(γ, 1),

b2 =f ′′0

9(1− γf ′0)t(1− 2f ′

0/p0, 1).

In addition,(5.5)

M1 = − 1

2(1− p0d4)

{19

9· γ(f

′′0 )

2

1− γf ′0

+ f ′′′0 − 2

9· (f ′′

0 )2f ′

0

p0(1− γf ′0)

}and M2 =

1

1− p0d4.

Then simple calculation yields v1 =4f ′′

0

1−γf ′0

t(f ′0/p0, 1) and

(5.6) ω1 =1

1− p0d4

{− 1

18· γ(f

′′0 )

2

1− γf ′0

+ f ′′′0 +

19

9· (f ′′

0 )2f ′

0

p0(1 − γf ′0)

}.

We next calculate the value of γ1 defined in section 2. From Lemma 2.2, we knowthat γ1 = 〈 De1,α

∗0 〉. Let e be a vector satisfying (−τ0D + B)e = −Dα0. Since

ker(−τ0D + B) = span{α0}, it follows that e1 = e + aα0 with a being a constant.Consequently, γ1 = 〈 De1,α

∗0 〉 = 〈 De,α∗

0 〉 + a 〈 Dα0,α∗0 〉 = 〈 De,α∗

0 〉. Solving(−τ0D +B)e = −Dα0 yields e = t(1, 12 (f

′0 + γp0)− 2p0d4) and

(5.7) γ1 = 〈 De,α∗0 〉 = p0d

24(f

′0 + γp0)

1− p0d4> 0.

In particular if f ′′0 = 0, some simpler expressions can be derived as follows:

M1 = − f ′′′0

2(1− p0d4), ω1 =

f ′′′0

(1− p0d4).

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Thus, we see f ′′′0 < 0 is a condition for the stability of planar solutions as stated in

Theorem 3.1 and Corollary 3.3.A typical example satisfying f ′′

0 = 0 and f ′′′0 < 0 is f(u) = u(1 − u2); in this

case M1 > 0, M2 > 0, and ω0 < 0 hold, so through a pitchfork bifurcation we obtain

two stable planar stationary solutions, say W±(y) = ±√

M2ηM1

cosC1yα0 + O(|η|) for

η > 0, provided that (d1, d′2) is sufficiently close to (d3, 1). Also, the constant γ′1 is

close to γ1.For the FitzHugh–Nagumo model (5.1), the approximation function constructed

in Theorem 4.1 is Σ(√ηR0( 4

√ηx))(y) =

√ηR0( 4

√ηx) cosC1yα0 + O(|η|). Figure 5.3

gives a numerical result for the standing wave W (x, y) = (u(x, y), v(x, y)) of (5.1)

joining with W±(y). Figure 5.4 shows that the graph of u-component of W (x, 0)quite coincides with that of the approximate function 2

√ηR0( 4

√ηx). In addition to

numerical treatment, the standing wave W (x, y) will be further studied by means ofvariational method.

Fig. 5.3. Wavefront (u-component) of (5.1). In case d1 = d3 = 0.308, d′2 = 1.0, d4 = 0.333,f(u) = 1.6u(1 − u2), η = 0.01, γ = 0.35, and l = 2.184.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Fig. 5.4. Graphs for u-component of W (x, 0) (solid line) and 2√ηR0( 4

√ηx) (broken line).D

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5.2. Existence of standing wavefronts by variational approach. For theexistence of standing waves of (5.1), our attention turns to the heteroclinic solutionsof the elliptic system

−d1uxx − d3uyy − f(u)− ηu+ v = 0,(5.8)

−d2vxx − d4vyy − u+ γv = 0,(5.9)

limx→−∞(u(x, y), v(x, y)) =W−(y),(5.10)

limx→∞(u(x, y), v(x, y)) =W+(y).(5.11)

Here f(u) := f ′(0)(u − u3) with f ′(0) = 2√

d3d4

− d3γd4

, d2 := d′2d4, W− = −W+, and

W+ is a Turing pattern of

−d3uyy − f(u)− ηu+ v = 0,(5.12)

−d4vyy − u+ γv = 0,(5.13)

uy(0) = uy(l) = vy(0) = vy(l) = 0.(5.14)

In what follows, assume that the following conditions are satisfied:

γ <

√d4d3,(A1)

π2

l2=

√1

d3d4− γ

d4.(A2)

It is not difficult to verify that (A2) is equivalent to (5.4); that is, we look for patternsas well as wavefronts near a neighborhood of Turing instability. Based on a variationalstructure possessed in (5.8)–(5.9), we give an additional viewpoint for the existenceof standing wavefronts joining with Turing patterns.

Theorem 5.1. Assume that d1 > d3 and γ ≥√

d2d3. If η is sufficiently small,

there exists a standing wave solution W (x, y) of (5.1) with asymptotic properties that

W (x, y) →W±(y) as x→ ±∞.A Turing patternW+(y) can be obtained from a variational argument: Let E0 :=

H1(0, l). For a given u ∈ E0, we denote by H0u the unique solution of

−d4vyy + γv = u,(5.15)

vy(0) = vy(l) = 0.

Substituting v = H0u in (5.15) and integrating by parts, we yield∫ l

0

uH0udy =

∫ l

0

∣∣∣∣∂H0u

∂y

∣∣∣∣2 + γ(H0u)2dy.

Define, for u ∈ E0,

J0(u) :=

∫ l

0

1

2

(d3

∣∣∣∣∂u∂y∣∣∣∣2 + uH0u

)+ F (u)dy,

where F (ξ) := − ∫ [f(ξ) + ηξ]dξ. Note that there exist c1 > 0 and c2 > 0 suchthat F (ξ) ≥ 1

2c1ξ2 − c2. It is easily seen that J0 is bounded from below and

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infξ∈R lF (ξ) < infu∈E0J0(u) <∞. Clearly (0, 0) is a trivial solution of (5.12)–(5.14). If(A1) and (A2) are satisfied, then simple calculation shows that infu∈E0 J0(u) < J(0).By adding a constant to F (u) if necessary, we may assume that

infu∈E0

J0(u) = 0.

Let u∗ be a global minimizer of J0 over E0 and v∗ = H0u∗. Observe that F is aneven function and J0(−u∗) = J0(u∗). By [7] we know that in a neighborhood ofTuring instability u∗ and −u∗ are the only minimizers of J0. It will be seen that thereis a solution (u, v) of (5.8)–(5.9) with the properties that uy(x, 0) = vy(x, 0) = 0,uy(x, l) = vy(x, l) = 0, and (u(x, y), v(x, y)) → (u∗, v∗) as x→ ∞. Indeed, if

W+(y) := (u∗(y), v∗(y))

and

(5.16) W (x, y) :=

{(u(x, y), v(x, y)) for x ≥ 0,

−W (−x, y) for x < 0,

then, as in Theorem 5.1, W is a standing wave solution joining with Turing patternsW+ and W−.

To obtain W (x, y) for x > 0, we turn to a variational problem on Ω := (0,∞)×(0, l). Let v be a C∞-function satisfying the following properties:

(i) ∂v∂y (x, 0) =

∂v∂y (x, l) = 0 and v(x, y) = 0 if x ∈ [0, 14 ].

(ii) v(x, y) = v∗(y) if x ≥ 12 .

Let E := {ψ; ψ ∈ H1(Ω) and ψ(0, y) = 0}. Denote by L∗ the differential operator

γ − d2∂2

∂x2 − d4∂2

∂y2 . For a given ψ ∈ E, we denote v by Hψ, the unique solution of

L∗v = ψ,∂v

∂y(x, 0) =

∂v

∂y(x, �) = 0, v ∈ E.

Define u := L∗v and

J(ψ) :=

∫Ω

1

2

[d1

∣∣∣∣∂(u+ ψ)

∂x

∣∣∣∣2 + d3

∣∣∣∣∂(u+ ψ)

∂y

∣∣∣∣2 + (u+ ψ)(v +Hψ)

]+F (u+ψ)dydx

for ψ ∈ E. If ψ is a minimizer of J over E, then (u + ψ, v + Hψ) is a solution of(5.8)–(5.9) and (u+ ψ, v +Hψ) → (u∗, v∗) as x→ ∞.

By straightforward calculation∫Ω

ψHψdydx =

∫Ω

d2

∣∣∣∣∂Hψ

∂x

∣∣∣∣2 + d4

∣∣∣∣∂Hψ

∂y

∣∣∣∣2 + γ(Hψ)2dydx

and

J(ψ) =

∫Ω

1

2

[d1

∣∣∣∣∂(u+ ψ)

∂x

∣∣∣∣2 + d3

∣∣∣∣∂(u+ ψ)

∂y

∣∣∣∣2 + d2

∣∣∣∣∂(v +Hψ)

∂x

∣∣∣∣2+d4

∣∣∣∣∂(v +Hψ)

∂y

∣∣∣∣2 + γ(v +Hψ)2

]+ F (u+ ψ)dydx.

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To simplify the notation, ‖ · ‖ will denote the norm for H1(Ω). Likewise, denoted by‖ · ‖2 is the norm for L2(0, l). By the simple eigenvalue bifurcation theorem [7], weknow that u∗ is a nondegenerate critical point of J0 if η is small. Since u∗ is a globalminimizer of J0, we have the following lemma.

Lemma 5.2. There exist ρ ∈ (0, 12‖u∗‖2) and ν0 > 0 such that

J0(u) ≥ ν0

if ‖u− u∗‖2 ≥ ρ and ‖u+ u∗‖2 ≥ ρ.For ψ ∈ E and α ∈ [0, σ), define

Jα,σ(ψ) =

∫ σ

α

∫ l

0

1

2

[d1

∣∣∣∣∂(u+ ψ)

∂x

∣∣∣∣2 + d3

∣∣∣∣∂(u+ ψ)

∂y

∣∣∣∣2 + d2

∣∣∣∣∂(v +Hψ)

∂x

∣∣∣∣2+ d4

∣∣∣∣∂(v +Hψ)

∂y

∣∣∣∣2 + γ(v +Hψ)2

]+ F (u+ ψ)dydx

and

Jα,∞(ψ) = limα→∞ Jσ,α(ψ).

Below is a key estimate for showing that J is bounded from below.Proposition 5.3. If ψ ∈ E, then

(5.17) J(ψ) ≥∫ ∞

0

[J0(u+ ψ) +

1

2(d1 − d3)

∫ l

0

(∂

∂x(u+ ψ)

)2

dy

]dx.

Proof. Let

φm,j = sin2mπx

ksin

2jπy

l, m, j ∈ N ,

φ−m,j = cos2mπx

ksin

2jπy

l, m ∈ Z \N , j ∈ N ,

φm,−j = sin2mπx

kcos

2jπy

l, m ∈ N , j ∈ Z \N ,

and

φ−m,−j = cos2mπx

kcos

2jπy

l, m, j ∈ Z \N .

Set u = u + ψ, φm,j = (∫ k0

∫ l0 |φm,j |2dydx)−

12 φm,j and am,j =

∫ k0

∫ l0 uφm,jdydx for

m, j ∈ Z. By straightforward calculation∫ k

0

∫ l

0

u2xdydx =

∞∑m=−∞

(2mπ

k

)2 ∞∑j=−∞

a2m,j ,

∫ k

0

∫ l

0

u2ydydx =

∞∑j=−∞

(2jπ

l

)2 ∞∑m=−∞

a2m,j ,

∫ k

0

∫ l

0

uH0udydx =

∞∑j=−∞

(d4

(2jπ

l

)2

+ γ

)−1 ∞∑m=−∞

a2m,j ,

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and ∫ k

0

∫ l

0

uHudydx ≥∞∑

m,j=−∞

(γ + d2

(2mπ

k

)2

+ d4

(2jπ

l

)2)−1

a2m,j .

Moreover,

∫ k

0

J0(u)dx =1

2

∞∑j=−∞

⎡⎣d3(2jπ

l

)2

+

(d4

(2jπ

l

)2

+ γ

)−1⎤⎦ ∞∑m=−∞

a2m,j

+

∫ k

0

∫ l

0

F (u)dydx

and∫ k

0

∫ l

0

[1

2(d1u

2x + d3u

2y) +

1

2uHu+ F (u)

]dydx

≥∫ k

0

∫ l

0

F (u)dydx+1

2d3

∞∑m,j=−∞

[(2mπ

k

)2

+

(2jπ

l

)2]a2m,j

+1

2

∞∑m,j=1

(γ + d2

(2mπ

k

)2

+ d4

(2jπ

l

)2)−1

a2m,j +

∫ ∞

0

∫ l

0

1

2(d1 − d3)u

2xdydx.

By virtue of γ ≥√

d2d3

and (A1), it is easy to verify that d3((2mπk )2 + (2jπl )2) +

(γ + d2(2mπk )2 + d4(

2jπl )2)−1 ≥ d3(

2jπl )2 + (d4(

2jπl )2 + γ)−1 for all m, j. Therefore,

J0,k(ψ) =

∫ k

0

∫ l

0

[1

2(d1u

2x + d3u

2y) +

1

2uHu+ F (u)

]dydx

≥∫ k

0

[J0(u) +

1

2(d1 − d3)

∫ l

0

u2xdy

]dx.

Letting k → ∞ yields (5.17).As an immediate consequence of Proposition 5.3, we see that c := infψ∈E J(ψ) >

0. Below we give lower bound estimates of J(ψ) for wavefront type oscillation.Lemma 5.4. Let u = u+ψ. Assume that ‖u(β, ·)+u∗‖2 = ρ, ‖u(x, ·)+u∗‖2 > ρ,

and ‖u(x, ·)− u∗‖2 > ρ for x ∈ (β, α).

(i) If ‖u(α, ·)−u∗‖2=ρ, then∫ βα [J0(u)+

12 (d1−d3)

∫ l0 u

2xdy]dx≥2ρ

√2ν0(d1 − d3).

(ii) If ‖u(α, ·)‖2 = 0, then∫ βα[J0(u) +

12 (d1 − d3)

∫ l0u2xdy]dx ≥ ρ

√2ν0(d1 − d3).

Lemma 5.4 directly follows from Lemma 5.2 and the Schwartz inequality.Remark 7. If the hypotheses of Lemma 5.4 are satisfied, roughly speaking there

is a cross from −u∗ to u∗ in [β, α]. Such a cross is referred to as a layer in phasetransition models, or a wavefront in case u is a standing wave. The same estimateshold if the roles of u∗ and −u∗ are interchanged.

Set um = u+ψm and vm = v+Hψm, where {ψm} is a minimizing sequence of J .It is clear from Lemma 5.4 and (5.17) that the number of wavefront type oscillations inum are uniformly bounded for m ∈ N . Let ‖·‖H1(α,β) denote the norm of H1((α, β)×(0, l)). By Proposition 5.3 we know for any fixed k ∈ (1,∞) there is a c = c(k) > 0

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Fig. 5.5. The u-component of a Turing pattern W∞(y) of (5.18) on the interval [−l, l].

Fig. 5.6. Wavefront (u-component) joining with ±W∞(y).

such that

supm≥1

(‖um‖H1(0,k) + ‖vm‖H1(0,k)) < c.

Furthermore, a comparison argument based on energy estimates shows that for allum the wavefront type oscillation must take place in a compact subset of Ω. Usingthe fact that u∗ is a nondegenerate minimizer of J0, we obtain a ψ ∈ E such thatlimm→∞ ‖ψm − ψ‖ = 0 and ‖Hψm −Hψ‖ → 0 as m → ∞, and ψ is a minimizer ofJ over E.

Let u = u + ψ and v = v + Hψ. Then the function W defined by (5.16) is astanding wavefront heteroclinic to Turing patterns.

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5.3. A model with two inhibitors. We now turn to a reaction-diffusion model[3] with one activator and two inhibitors being in action:

(5.18)

⎧⎨⎩ ut = d1uxx + d3uyy + f(u) + ηu − v − w,vt = d2vxx + d4vyy + u− v,wt = wxx + d5wyy + u− γw.

In case d1 = 10, d2 = d4 = 1, d3 = 0.1511, d5 = 2, f(u) = 0.845u(1− u2), η = 0.005,γ = 0.5, and l = 2; a Turing pattern W∞(y) has been worked out numerically inFigure 5.5. Also, a wavefront joining with W∞(y) and −W∞(y) is demonstrated inFigure 5.6.

6. Discussions. We study connecting orbits for reaction-diffusion systems in aneighborhood of Turing instability. In particular, a standing wave joining two Turingpatterns in a FitzHugh–Nagumo type model has been established by means of numer-ical analysis and variational method. For more general systems like (3.1), we obtainedsome approximate functions for such standing waves.

A related important subject is to find traveling front solution; i.e., the existence ofsolutions of (1.5). An additional unknown to be treated here is the traveling velocityc. In this section, based on formal arguments, we discuss how to find more informationabout the velocity of traveling wave.

Suppose in (1.5) there is a scalar function V such that F (U) = −Q∇V (U), whereQ is a symmetric and invertible matrix satisfying DQ = QD. If U(z, y) is a solutionof (1.5), then

−c∫Ω

〈 Uz, Q−1Uz 〉 dzdy

=

∫Ω

〈 DΔU,Q−1Uz 〉 dzdy −∫Ω

V (U)zdzdy

=1

2

∫Ω

∂

∂z〈 DUz, Q−1Uz 〉 dzdy

+

∫ ∞

−∞

{[〈 DUy, Q−1Uz 〉]l0 −

∫ l

0

〈 DUy, Q−1Uzy 〉 dy}dz −

∫ l

0

[V (U)]∞−∞dy

=1

2

(∫ l

0

[〈 DUz, Q−1Uz 〉]∞−∞dy −∫ l

0

∫ ∞

−∞

∂

∂z〈 DUy, Q−1Uy 〉 dzdy

)−∫ l

0

[V (U)]∞−∞dy

= −1

2

∫ l

0

{〈 DW+y , Q

−1W+y 〉 − 〈 DW−

y , Q−1W−

y 〉} dy − ∫ l

0

{V (W+)− V (W−)}dy

= −1

2

∫ l

0

{〈 DW+y , Q

−1W+y 〉 − 〈 DW−

y , Q−1W−

y 〉} dy − ∫ l

0

{V (W+)− V (W−)}dy.

Letting E(U) :=∫ l0

{12 〈 DUy, Q−1Uy 〉+ V (U)

}dy, we have

(6.1) c

∫Ω

〈 Uz, Q−1Uz 〉 dzdy = E(W+)− E(W−).

If the matrix Q is positive definite, the coefficient of c in (6.1) is positive; in otherwords, (1.1) is a gradient system and the sign of c is determined by the differencebetween E(W+) and E(W−).

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In case Q has eigenvalues with opposite sign, (1.1) is referred as a skew gradientsystem [35]. Since the integral in the left-hand side of (6.1) does not have a fixed sign,the structure of traveling waves in a skew gradient system seems to be more delicate.

Lastly, we remark that the methods used in section 4 can be applied to thetraveling wavefront for Turing patterns as well. Indeed, with slight modification inthe derivation of (4.1), we obtain the following reduced equation:

(6.2) RT = −γ1Rζζζζ +R(M2 −M1R2) + δ, T > 0, ζ ∈ R.

Here the nonlinearity h(R) := R(M2 −M1R2) + δ is asymmetric due to δ �= 0. In

studying traveling waves on the moving coordinates ξ := ζ − cT , R(T, ζ) is taken of

the form R(ζ − cT ) with c being the traveling velocity, where R(ξ) satisfies

(6.3) −cR′ = −γ1R′′′′ + h(R), ξ ∈ R.

Multiplying both sides of (6.3) by R′ and integrating over R, we get

−c∫ ∞

−∞(R′)2dξ = −γ1

∫ ∞

−∞R′′′′R′dξ +

∫ ∞

−∞h(R)R′dξ

= −γ1[R′′′R′ − 1

2(R′′)2

]∞−∞

+

∫ R+

R−h(R)dR

=

∫ R+

R−h(R)dR

with R± := R±(δ) being the only two stable equilibria of (6.2). The calculation ofM1, M2, and δ can be carried out in the same manner as in section 4, thus the sign oftraveling velocity c is determined. The profile of a traveling wavefront in the movingframe looks like, for instance, the one we have seen in Figure 5.2.

REFERENCES

[1] R. Asai, E. Taguchi, Y. Kume, M. Saito, and S. Kondo, Zebrafish Leopard gene as a com-ponent of putative reaction-diffusion system, Mech. Dev., 89 (1999), pp. 87–92.

[2] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Inst. H. Poincare Anal. NonLineaire, 9 (1992), pp. 497–572.

[3] M. Bode, A. W. Liehr, C. P. Schenk, and H.-G. Purwins, Interaction of dissipative soli-tons: Particle-like behaviour of localized structures in a three-component reaction-diffusionsystem, Phys. D, 161 (2002), pp. 45–66.

[4] J. Carr, Applications of Centre Manifold Theory, Appl. Math. Sci. 35, Springer-Verlag,New York, 1981.

[5] R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations withNeumann boundary conditions, J. Differential Equations, 27 (1978), pp. 266–273.

[6] C.-N. Chen and X. Hu, Stability criteria for reaction-diffusion systems with skew-gradientstructure, Comm. Partial Differential Equations, 33 (2008), pp. 189–208.

[7] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal.,8 (1971), pp. 321–340.

[8] E. N. Dancer and S. Yan, A minimization problem associated with elliptic systems ofFitzHugh-Nagumo type, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), pp. 237–253.

[9] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Bio-phys. J., 1 (1961), pp. 445–466.

[10] D. G. De Figueiredo and E. Mitidieri, A maximum principle for an elliptic system andapplications to semilinear problems, SIAM J. Math. Anal., 17 (1986), pp. 836–849.

[11] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972),pp. 30–39.

Dow

nloa

ded

12/3

1/12

to 1

28.1

48.2

52.3

5. R

edis

trib

utio

n su

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cens

e or

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yrig

ht; s

ee h

ttp://

ww

w.s

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.org

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[12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,2nd ed., Springer-Verlag, Berlin, 1983.

[13] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840,Springer-Verlag, Berlin, New York, 1981.

[14] S. Jimbo and Y. Morita, Stability of nonconstant steady-state solutions to a Ginzburg-Landauequation in higher space dimensions, Nonlinear Anal., 22 (1984), pp. 753–770.

[15] G. A. Klaassen and E. Mitidieri, Standing wave solutions for system derived from theFitzHugh–Nagumo equations for nerve conduction, SIAM J. Math. Anal., 17 (1986), pp.74–83.

[16] G. A. Klaasen and W. C. Troy, Stationary wave solutions of a system of reaction-duffusionequations derived from the FitzHugh–Nagumo equations, SIAM J. Appl. Math., 44 (1984),pp. 96–110.

[17] S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish po-macanthus, Nature, 376-31 (1995), pp. 765–768.

[18] O. Lopes, Radial and nonradial minimizers for some radially symmetric functionals, Electron.J. Differential Equations, 1996, 14 pp.

[19] H. Matano, Asymptotic behavior and stability of solutions of semilinear elliptic equations,Publ. Res. Inst. Math. Sci., 15 (1979), pp. 401–454.

[20] P. K. Maini, K. J. Painter, and H. N. P. Chau, Spatial pattern formation in chemical andbiological systems, J. Chem. Soc. Faraday Trans., 93 (1997), pp. 3601–3610.

[21] A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. DifferentialEquations, 110 (1994), pp. 322–355.

[22] M. Mimura, K. Sakamoto, and S.-I. Ei, Singular perturbation problems to a combustionequation in very long cylindrical domains, in Nonlinear Evolutionary Partial DifferentialEquations, AMS Stud. Adv. Math. 3, AMS, Providence, RI, 1997, pp. 75–84.

[23] J. S. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulatingnerve axon, Proc. Inst. Radio Engineers, 50 (1962), pp. 2061–2070.

[24] Y. Nishiura, Far-from-Equilibrium Dynamics, in Transl. Math. Monogr. 209, AMS, Provi-dence, RI, 2002.

[25] Y. Nishiura and M. Mimura, Layer oscillations in reaction-diffusion systems, SIAM J. Appl.Math., 49 (1989), pp. 481–514.

[26] Y. Oshita, On stable nonconstant stationary solutions and mesoscopic patterns for FitzHugh-Nagumo equations in higher dimensions, J. Differential Equations, 188 (2003), pp. 110–134.

[27] C. Reinecke and G. Sweers, A positive solution on Rn to a system of elliptic equations ofFitzHugh-Nagumo type, J. Differential Equations, 153 (1999), pp. 292–312.

[28] X. Ren and J. Wei, Nucleation in the FitzHugh-Nagumo system: Interface-spike solutions, J.Differential Equations, 209 (2005), pp. 266–301.

[29] F. Rothe and P. de Mottoni, A simple system of reaction-diffusion equations describingmorphogenesis: Asymptotic behavior, Ann. Mat. Pura Appl. (4), 122 (1979), pp. 141–157.

[30] H. Shoji, Y. Iwasa, A. Mochizuki, and S. Kondo, Directionality of stripes formed byanisotropic reaction-diffusion models, J. Theoret. Biol., 214 (2002), pp. 549–561.

[31] H. Shoji, A. Mochizuki, Y. Iwasa, M. Hirata, T. Watanabe, S. Hioki, and S. Kondo,Origin of directionality in the fish stripe pattern, Dev. Dyn., 226 (2003), pp. 627–633.

[32] J. Smoller, Shock Waves and Reaction Diffusion Equations, 2nd ed., Springer-Verlag, NewYork, 1994.

[33] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London B, 237(1952), pp. 37–72.

[34] J. M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains,Comm. Partial Differential Equations, 18 (1993), pp. 505–531.

[35] E. Yanagida, Mini-maximizers for reaction-diffusion systems with skew-gradient structure, J.Differential Equations, 179 (2002), pp. 311–335.

[36] E. Yanagida, Standing pulse solutions in reaction-diffusion systems with skew-gradient struc-ture, J. Dynam. Differential Equations, 4 (2002), pp. 189–205.

[37] J. Wei and M. Winter, Clustered spots in the FitzHugh-Nagumo system, J. Differential Equa-tions, 213 (2005), pp. 121–145.

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turing patterns and wavefronts for reaction-diffusion systems in an infinite channel

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