数理解析研究所講究録 1425 巻 2005 年 122-129

122

Dynamics of Turing patterns in cylindrical domains
on 2D

九州大学数理学研究院 栄 伸一郎 (Shin-Ichiro Ei)

Faculty of Mathematics, Kyushu University
Abstract. Reaction-diffusion systems in an infinitely long strip-like domain with finite width
inare treated. We construct the solution connecting different types of stationary solutionsat infinity by considering the neighborhood of Turing instability. We also derive 4th orderequations of buckling type which shows the dynamics of the connecting solutions.
$2\mathrm{D}$1Introduction

In 1995, Konclo and Asai[1] show ed some chemical waves are observed in the skinof real fishes. Their simulations by using reaction-diffusion systems reappear very well
the growth of patterns on the skin. One of the typical patterns is the rearrangement of
the stripe patterns. For example, when the width of skin are different in locations, thenumber of stripes are different depending on the width. Then patterns with different
numbers of stripes are connected and some defects appear. According to the growthof skin, the location of the defects change. After their work, there have been manysimulations related to such phenomena, but no theoretical results yet. To give a theoretical framew ork to it, we first consider a problem on a fixed domain and construct asolution connecting two different stable stripe patterns as follows:
Let $\Omega=(-\infty, \infty)\mathrm{x}(0, l)\in R^{2}$ and consider a reaction-diffusion system(1.1)$u_{t}=D\Delta u$ $+F(u)$, $t>0$ ,$x$$=(x,y)\in\Omega$with the homogeneous Neumann boundary conditions, where $u=$ $(u_{1}, \cdots,u_{N})\in$$R^{N},F$ is a smoothfunction onanddenotes a diagonal matrix with positiveelements , that is,.
Let us also consider ID problem of (1.1)$R^{N}$$d_{j}$$D$$D=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{d}1, \cdots,d_{N})$(1.2)$u_{t}=Du_{yy}+F(u)$ , $t>0$ , $y\in(0, l)$with the boundary conditions

stationary solutions, say$\frac{\partial u}{\partial y}=0$$U^{\pm}(y)$at $y=0$ , . We assume (1.2) have two different$l$with $U^{+}(y)\neq U^{-}(y)$ . Under these assumptions, we123look for the solutions $u(t$ , $ of (1.1) satisfying$)$(1.3)$u(t, \pm\infty, y)=U^{\pm}(y)$.

There have been few works related to the problem (1.1) with (1.3) (e.g. [2], [3],have been considered. If we[4] . Specially, traveling front solutions connecting, then satisfies$u(t,x,y)=U(z,y)$for$z=x-ci$
asolution
intheform
take$U^{\pm}(y)$$)$

$U$

(1.4)

$0=D(U_{zz}+U_{yy})+cU_{z}+F(U)$ ,$U(\pm\infty, y)=U^{\pm}(y)$.

When the problem (1.1) is a gradient system or skew gradient system, we can know
in a formal way how is determined, which will be mentioned in the following sections. It gives the movement of the connecting solution. In fact, [4] showed rigorously
the existence and its stability of connecting traveling wavefront solution for a sealer
equation.
On the other hand, if there are no gradient structure in any sense for the system,
there have been no theoretical results even in formal arguments.
In this paper, we consider the problem (1.1), (1.3) for general function without
any gradient structure, but in the neighb orhood of Turing instab ility. By it, it is showthat a pitchfork type bifurcation occurs in the system (1.2) on ID and we can get two. We can construct approximate soludifferent stable stationary solutions, say$xarrow\pm\infty$ . We can show the dynamics is essentiallyattions of (1.1) connectingofdynamics
governed by the$c$$F$$\mathrm{n}$$U^{\pm}(y)$$U^{\pm}(y)$(1.5)$R_{T}=-\gamma_{1}R_{\zeta\zeta\zeta\zeta}+R(M_{2}-M_{1}R^{2})$, $T>0$ ,$($$\in R$,

where all coefficients are positive constants.
In the following sections, we give preliminaries and results.

2

traveling solutions

In this section, we consider the problem (1.4) and show how the velocity is determined, All are by formal discussions
$c$

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Gradient systems

2.1

Suppose that $F(u)=-\nabla W(u)$ holds for a function $W(u)\in R$ and that (1.4) has asolution $U(z, y)$ . Then according to [4] we have$-c \int_{\Omega}$$\langle\langle U_{z}, U_{z}\rangle$dzdy$=$$\int_{\Omega}$$\langle D\Delta U, U_{z}\rangle dzdy-\oint_{\Omega}W(U)_{z}dzdy$$=$$\int_{\Omega}$$\langle D\triangle U, U_{z}\rangle$$=$$\frac{1}{2}\oint_{\Omega}\frac{\partial}{\partial z}$$\langle$

DU,,$dzdy- \int_{\Omega}W(U)_{z}dzdy$$U_{z}\rangle$$dzdy+I_{-\infty}^{\infty}\{[\langle$

DU,,$U_{z}\rangle]_{0}^{l}-l^{l}$$\langle$

DU,,$U_{zy}\rangle$$dy\}dz$$- \int_{0}^{f}[W(U)]_{-\infty}^{\infty}d\mathrm{y}$$=$$\frac{1}{2}$($\int_{0}^{l}[\langle DU_{z}, U_{z}\rangle]_{-\infty}^{\infty}dy-\int_{0}^{l}\oint_{-\infty}^{\infty}\frac{\partial}{\partial z}$$\langle DU_{y}, U_{y}\rangle dzdy$)$- \int_{0}^{l}[W(U)]_{-\infty}^{\infty}dy$$=$$- \frac{1}{2}f_{0}^{l}\{\langle DU_{y}^{+}, U_{y}^{+}\rangle-\langle DU_{y}^{-}, U_{y}^{-}\rangle\}dy-l^{l}\{W(U^{+})-W(U^{-})\}dy$$=$$- \frac{1}{2}\int_{0}^{l}\{\langle DU_{y}^{+}, U_{y}^{+}\rangle-\langle DU_{y}^{-}, U_{y}^{-}\rangle\}dy-\int_{0}^{l}\{W(U^{+})-W(U^{-})\}dy$$=$$E(U^{-})-E(U^{+})$ ,where$E(U):= \int_{0}^{l}\{\frac{1}{2}$(2.1)$\langle DU_{y}, U_{y}\rangle+W(U)\}dy$$c$’$(U_{z}, U_{z})$. Thus$dzdy=E(U^{+})-E(U^{-})$

holds. This means the sign of is determined the difference of energies $E(U^{\pm})$ ofon the interval $(0, l)$ .
$c$

2.2

$U^{\pm}(y)$Skew-gradient systems

Suppose that $F(u)=-Q\nabla W(u)$ holds for a function $W(u)\in R$ and a matrix
which is symmetric and invertible satisfying $DQ=QD$ (e.g. [5]). Assume (1.4) has asolution $U(z, y)$ . Then similarly to the previous subsection, we have$Q$(2.2)$-c \int_{\Omega}$$\langle U_{z}, Q^{-1}U_{\mathit{2}}\rangle$

dzdy

$=$

$\int_{\Omega}$$=$$E(U^{+})-E(U^{-})$ ,$\langle D\triangle U, Q^{-1}U_{z}\rangle dzdy-\int_{\Omega}$$\langle Q\nabla W(U), Q^{-1}U_{z}\rangle$dzdy

125
where
$E(U):= \int_{0}^{l}\{\frac{1}{2}$$\langle DU_{y}, Q^{-1}U_{y}\rangle+W(U)\}dy$.

But the coeffi cient of in the left hand side of (2.2) does not have fixed sign due to .
Thus, we need more informations on the solution $U(z,y)$ in order to know the sign of.
$Q$

$c$

$c$

3

Preliminaries for Turing Instability

3.1

Bifurcation diagram in ID problems

Let 0 be a linearly stable equilibrium of , that is, $F(0)=0$ is satisfied and all
eigenvalues of the linearized matrix $B:=F’(0)$ are negative real parts. Consider the 1
dimensional problem (1.2) and be a linearized operator with respect to 0. Expanding$C_{n}ya_{n}(a_{n}\in R^{N})$ and substituting it to thesolutions in Fourier series
eigenvalue problem $LU=\lambda U$ , we have
$F$

$L$

$U= \sum_{n=0}^{\infty}\cos$$\{-C_{n}^{2}D+B\}a_{n}=\lambda a_{n}$whereby$C_{n}:= \frac{n\pi}{\ell}$,. Therefore, we first consider the matrix –(-\mbox{\boldmath $\tau$}) $:=-\tau D+B$ parametrized. Let Xj(r) $(j=1, \cdots, N)$ be eigenvalues of —(\mbox{\boldmath $\tau$}). We assume there exist,, $N$ andfor $j=2$ ,positive constants 70, such that
. Moreover, weand Ai (r) $=0$ if and only ifAi(r)for any$\tau\geq 0$$Re(\lambda \mathrm{i}(\tau))<-\gamma \mathfrak{o}$$\cdots$$\tau_{0}$$\leq 0$$\tau=\tau \mathrm{o}(\mathrm{F}\mathrm{i}\mathrm{g}.3.1)$$\tau\geq 0$

Figure 3.1: Eigenvalues$\lambda_{1}(\tau)$

and others$\lambda_{1}(\tau)\in R$

126
assume.
is a simple eigenvalue of —(\mbox{\boldmath $\tau$}) with the associated eigenvector$L\phi_{1}=0$
If we take the width of such that, thenand takeholds, where $\phi_{n}(y):=\cos C_{n}ya_{n}$ and. Thus, 1n the system (1.2), only 1
mod solution close to
can be unstable.
theUnderabove assumptions, we consider the system$\alpha(\tau)\in R^{N}$$\lambda_{0}(\tau)$$( \frac{\pi}{l})^{2}=\tau_{0}$$\Omega$$l$$a_{1}=\alpha_{0}$$\alpha_{0}=\alpha(\tau_{0})$$\phi_{1}(y)$(3.1)$u_{t}=D\triangle u+F(u)+\eta G(u)$ ,for smallof (3.1)$\eta$$t>0$ , $x$ $=(x, y)\in$ St

and a function $G(u)$ with$\mathrm{h}G(0)=0$ . Then we can show for the ID problem(3.1)$u_{t}=u_{yy}+F(u)+\eta G(u)$ ,$t>0,0<y<l$has a pitch-fork type bifurcation diagram in super critical as follows.

Proposition 3.1 For sufficiently , there exist constants $M_{1;}M_{2}$ and a function
with $||\sigma||=O(|\eta|+r^{2})$ such that$\eta$$\sigma=$$\sigma(r;\eta)(y)$$u(t, y)=r(t)\varphi_{1}(y)+\sigma(r(t);\eta)(y)$with$\dot{r}=-fll_{1}r^{3}+M_{2}r\eta+O(|\eta|^{2}+r^{4})$.

This proposition is easily proved by a standard center manifold theory.
Hereafter, we assume bothandare positive, which means a super criticaliy pitch-fork type bifurcation diagram occurs. In fact, there exist stable stationary$M_{1}$

solutions$U^{\pm}(y)$corresponding to$M_{2}$$r_{\pm}:=\pm\sqrt{\frac{M_{2}\eta}{M_{1}}}$for small $\eta>0$ .

$U^{\pm}(y)$$U^{\pm}(y)=r_{\pm}\phi_{1}(y)+\sigma(r_{\pm \mathrm{j}\eta)(y)=\pm\sqrt{\frac{M_{2}\eta}{M_{1}}}}\cos C_{1}y\alpha_{0}+O(|\eta|)$

3.2

are given by.

Stability of planar solutions in 2D

In this subsection, we give the stability conditions ofas the solutions of (3.1) in
211 Let $u^{\pm}(x, y):=U^{\pm}(y)$ and Jet
be the eigenvectors satisfying$=1$ . Define,and. Then we have$U^{\pm}(y)$$t—(\tau)\alpha^{*}(\tau)=$$\alpha^{*}(\tau)$$\lambda_{1}(\tau)\alpha^{*}(\tau)$$\langle$$\alpha(\tau)$

Theorem 3.1 Let$\omega_{1}$$\omega_{1}:=\frac{1}{8}$where$v_{1}$

and$b_{2}$$\alpha^{*}(\tau)$$)$$\alpha_{0}^{*}$

be the constant given by$\langle F’(0)\alpha_{0}.

(v_{1}-b_{2}), \alpha_{0}^{*}\rangle+\frac{1}{32}$$\omega_{1}<0\{>0$$\langle F’(0)\alpha_{0}^{3}, \alpha_{0}^{*}\rangle$,

are unique vectors determined by$(-4 \tau_{0}D+B)b_{2}+\frac{1}{4}F’(0)\alpha_{0}^{2}=0$

If$:=\alpha^{*}(\tau_{0})$) then$u^{\pm}$, $(-2\tau_{0}D+B)v_{1}+F’(0)\alpha_{0}^{2}=0$.

are stable (unstable) as the stationary solutions in$2\mathrm{D}$127
Corollary 3.1 Assume$u=(\begin{array}{l}uv\end{array})$$\in R^{2}$

aanndd $F(u)+\eta G(u)=(f(u)-v+\eta g(u,v)\mathit{5}(u-\gamma v))$

with $f(0)=g(0,0)=0$ and $f’(\mathrm{O})=0$ , where and are positive constants. Thenare stable (or unstable ) if $f’(0)<0$ $(or>0)$ .
$\delta$$\gamma$$u^{\pm}(x, y)$

Typical examples of are cubic-like functions such as $f(u)=u(1-u^{2})$ . Thenarethe conditions in Corollary 3.1 are easily satisfied and we find planar solutions
stable.
$f$$u^{\pm}$4Solutions of (3.1) connecting

Define $\Sigma(r)(y):=r\phi_{1}(y)+\sigma(r;\eta)(y)$ . Thensuch thatof (3.2), that is, there exists$u^{\pm}$$\Sigma(r)(y)$is invariant for the dynamics$H(r,\cdot\uparrow?)$$H\Sigma_{r}=\mathcal{L}(\Sigma))$where $\mathcal{L}(u):=u_{yy}+F(u)+?7G(u)$ . Proposition3.1 suggests tl at $H(r;\eta)=-M_{1}r^{3}+$$+O(|\eta|^{2}+r^{4})$ holds. Especially,.
are given bymentioned in Section 3.1, we mayof the matrix
For the eigenvalue
assume$U^{\pm}(y)$$II_{\mathit{2}}r\eta$(4.1)$\Sigma(r_{\pm}; \eta)$$\cup--(\tau)$$\lambda_{1}(\tau)$$\lambda_{1}(\tau_{0}+\epsilon)=-\gamma_{1}\epsilon^{2}+O(\epsilon^{3})$for positive and small (Fig.3.1).
Using the above, we define approximate functions by$\epsilon$$\gamma_{1}$$u^{*}(t, x, y)$ $:=\Sigma(\sqrt{\eta}R(T, \zeta))(y)$and define constants$R\pm:=\pm\sqrt{\frac{M_{2}}{M_{1}}}$, $T:=\eta t$ ,$\zeta:=\sqrt[4]{\eta}x$. ThenTheorem 4.1$u_{t}^{*}-\{D\triangle u^{*}+F(u’)+\eta G(u^{*})\}=O(|\eta|^{3/2})$

holds and $R(T, |;)$ satisfies(4.2)$R_{T}=-\gamma_{1}R_{\zeta\zeta\zeta\zeta}+R(M_{2}-M_{1}R^{2})+O(\sqrt{\eta})$

unifo rmly for $0\leq T\leq T^{*}$ and $(x, y)\in\Omega$ , where$T^{*}$

is an arbirrarily given positiveconstant.
Suppose there exists the solution, say $R_{0}(T, ()$ , of (L5), the equation (4.2) with.
satisfying $R_{0}(T, \pm\infty)=R\pm for0\leq T\leq T^{*}$ . Let, then the solution of (3.1)
If the initial date $u(0,x, y)$ is sufficiently close to$\eta\prec$$u_{0}^{*}(t, x_{7}y):=\Sigma(\sqrt{\eta}R_{0}(T, \zeta))(y)$$0_{f}$$u_{0}^{*}(0, x, y)$satisfies$u(t, x,y)-u_{0}^{*}(t,x, y)=o(1)$uniformly for $0\leq T\leq T^{*}$ and $(x, y)\in\Omega$ .

128Remark 4.1$u_{0}^{*}(t, \pm\infty, y)=$holds. Ttvus,

I $(\sqrt{\eta}R_{0}(T, \pm\infty))(y)=\Sigma(\sqrt{\eta}R_{\pm})(y)=$ I $(r_{\pm})(y)=U^{\pm}(y)$$u_{0}^{*}(t, x, y)$is the approximate function which connects$U^{\pm}(y)$at $xarrow\pm\infty$ .

Remark 4.2 For the symmetry of the equations (3.1) with respect to -axis,
may be taken as a stationary
are symmetric each other for $y=l/2$ . Hence$say\=R_{0}(\zeta)$.
solution of (1.5),$y$$U^{\pm}(y)$$R_{0}(T, \langle)$are explicitly given though we don’t
All coefficients mentioned hitherto such as
touch on it here because of the restriction of total pages. By using such informations,
we can compare thhe approximate functions constructed here quantitatively with the(insolutions of (3.1). Fig4.1 denotes the stationary solution of (3.1) connecting$M_{j}$$U^{\pm}(y)$Figure 4.1: Solution of (3.1) connecting$U^{\pm}(y)$.

numerical sense ), say $u_{0}(x, y)$ . Fig4.2 show the graph of values of $u_{0}(x, y)$ at $y=0$ .
$\mathrm{s}$$x$Figure 4.2: Graph of the edge $u_{0}(x, 0)$ .

It is quite coincident with the graph of$\sqrt{\eta}R_{0}(\sqrt[4]{\eta}x)$.

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