数理解析研究所講究録
第 1924 巻 2014 年 1-10

1An $(N-2)$ -dimensional surface with positive principal
curvatures gives an $N$-dimensional traveling front inbistable reaction-diﬀusion equations
Masaharu Taniguchi*
Department of Mathematics, Faculty of Science, Okayama University

3-1-1, Tsushimanaka, Kita-ku, Okayama City, 700-8530, JAPANAbstract
This paper is a preliminary report of the forthcoming paper [21]. This paper
studies traveling fronts to the Allen-Cahn equation infor $N\geq 3$ . We consider$(N-2)$ -dimensional smooth surfaces as boundaries of strictly convex compact sets in, and define an equivalence relation between them. We prove that there exists
a traveling front associated with a given surface and that it is asymptotically stablefor given initial perturbation. The associated traveling fronts coincide up to phasetransition if and only if the given surfaces satisfy the equivalence relation.
$\mathbb{R}^{N}$$\mathbb{R}^{N-1}$

AMS Mathematical Classifications: $35C07,$ $35B20,$ $35K57$
Key words: traveling front, Allen-Cahn equation, non-symmetricAs a preliminary report of the forthcoming paper [21] we briefly state the results. We
study the following reaction-diﬀusion equation$\frac{\partial u}{\partial t}=\triangle u+f(u) x\in \mathbb{R}^{N}, t>0$,(1)

$u(x, 0)=u_{0} x\in \mathbb{R}^{N}.$Herewith
andfor $1\leq j\leq N$ . Now $N\geq 3$ is agiven integer, andis a given bounded and uniformly continuous function fromtoThe assumption on is as follows.
$\triangle=\sum_{j=1}^{N}D_{jj}$$D_{jj}=(\partial/\partial x_{j})^{2}$$D_{j}=\partial/\partial x_{j}$$\mathbb{R}^{N}$$u_{0}$$f$(A1) $f\in C^{1}[-1, 1]$ satisfies $f(1)=0,$ $f(-1)=0,$ $f’(1)<0,$ $f’(-1)<0$ and$\int_{-1}^{1}f(s)ds>0.$(A2) There exists $a_{*}\in(-1,1)$ such that$f(s)<0$$f(s)>0$

for all $s\in(-1, -a_{*})$ ,for all $s\in(-a_{*}, 1)$ .

$\mathbb{R}.$2Figure 1: The graph of $f.$

See Figure 1. Equation (1) is called the Nagumo equation [15] or the unbalanced AllenCahn equation [1]. For this equation, multi-dimensional traveling fronts have been studied
by many mathematicians. Two-dimensional -form fronts are studied by Ninomiya andmyself [16, 17], Hamel, Monneau and Roquejoﬀre [8, 9] and Haragus and Scheel [10] and so
are studied by [8, 9]. Raveling fronts
on. Cylindrically symmetric traveling fronts inof pyramidal shapes and convex polyhedral shapes are studied by [18, 19, 13, 20]. See [14]for a related work. haveling fronts associated with strictly convex compact domain inin [20]. Thewith a smooth boundary are studied for the Allen-Cahn equation inwitha smoothcompactsetin
strictlyaconvex
purpose of this paper is to show thatby using a clear andboundary gives a traveling front in the Allen-Cahn equation inconcise argument. Since the Allen-Cahn equation is one of the simplest reaction-diﬀusion
equations, the argument in this paper might be useful for studies on other reaction-diﬀusion
equations or reaction-diﬀusion systems that admit comparison principles.
The profile equation of a one-dimensional traveling front with speed is given by$V$$\mathbb{R}^{N}$$\mathbb{R}^{3}$$\mathbb{R}^{2}$$\mathbb{R}^{N-1}$$\mathbb{R}^{N}$$k$$-\Phi"(y)-k\Phi’(y)-f(\Phi(y))=0 -\infty<y<\infty,$(2)$\Phi(-\infty)=1, \Phi(\infty)=-1.$

It is known that (2) has a solution under (A1) and (A2), and it is unique up to translation.
One can refer to [2, 3, 11, 12, 6, 4] for instance. See Figure 2. Now (A1) gives $k>$ O.
Especially one has $k=\sqrt{2}a_{*}$ and $\Phi(x)=-\tanh(x/\sqrt{2})$ when $0<a_{*}<1$ and $f(u)=$$-(u+1)(u+a_{*})(u-1)$ .
The Allen-Cahn equation by a moving coordinate system with speed toward thedirection is given by$\Phi$$c$$(D_{t}-\Delta-cD_{N})w-f(w)=0 x\in \mathbb{R}^{N}, t>0,$$w(x, 0)=u_{0}(x) x\in \mathbb{R}^{N}.$$x_{N^{-}}$(3)

Here we assume $c>k$ . We denote the solution of (3) by $w(x, t;u_{0})$ . The profile equation
is given byof a traveling front in$\mathbb{R}^{N}$$(-\triangle-cD_{N})v-f(v)=0 x\in \mathbb{R}^{N}$*taniguchi-m@okayama-u. ac.jp

.

(4)

3

Figure 2: A one-dimensional traveling front$\Phi.$

Here we put $x’=(x_{1}, \ldots, x_{N-1})\in \mathbb{R}^{N-1}$ and $x=(x’, x_{N})$ .
with
for $|s|>1$ . Setting
We extend as a function of class$f$$f’(\mathcal{S})<0$$C^{1}(\mathbb{R})$$\beta=\frac{1}{2}\min\{-f’(-1), -f’(1)\}>0,$

we choose

with$\delta_{*}\in(0,1/4)$

if $|s+1|\leq 2\delta_{*}$ or $|s-1|\leq 2\delta_{*}.$$-f’(s)>\beta$

In this paper we assume $c>k$ . Let$M = \max_{|s|\leq 1+\delta_{*}}|f’(s)|>0,$$\sqrt{c^{2}-k^{2}}$$m_{*} =$and define$\theta_{*}\in(0, \pi/2)$$\overline{k}$ ’by$\tan\theta_{*}=m_{*}.$

Let $n\geq 2$ be a given integer andfor $i\neq j$ . Thensatisfies$1et\{a_{j}\}_{j=1}^{n}$be a set of unit vectors in$\mathbb{R}^{N-1}$

with$a_{i}\neq a_{j}$$a_{j}=(a_{j}^{1}, \ldots, a_{j}^{N-1})$$|a_{j}|^{2}= \sum_{i=1}^{N-1}(a_{j}^{i})^{2}=1$

for all $1\leq j\leq n.$

Here we put $x’=(x_{1}, \ldots, x_{N-1})\in \mathbb{R}^{N-1}$ and $x=(x’, x_{N})=(x_{1}, \ldots, x_{N})\in \mathbb{R}^{N}$ with$|x’|=\sqrt{\sum_{i--1}^{N-1}x_{i}^{2}}$and$|x|=\sqrt{\sum_{i=1}^{N}x_{i}^{2}}$, respectively. For$x’\in \mathbb{R}^{N-1}$$h_{j}(x’) = m_{*}(a_{j}, x$$h(x’)$$=$nmaxl j$\leq$$\leq$$h_{j}(x’)=m_{*} \max_{1\leq j\leq n}(a_{j}, x$

we set(5)(6)

4Here $(a_{j}, x’)$ denotes the inner product of vectors
a pyramid. Setting$a_{j}$

and $x’$ . In this paper we call $\{(x’, x_{N})\in$$\mathbb{R}^{N}|x_{N}\geq h(x’)\}$$\Omega_{j}=\{x’\in \mathbb{R}^{N-1}|h(x’)=h_{j}(x’)\}$

for $j=1$ , . . . , , we have$n$$\mathbb{R}^{N-1}=\bigcup_{j=1}^{n}\Omega_{j}.$We denote the boundary ofby$\Omega_{j}$$\partial\Omega_{j}$. Now we put$S_{j}=\{x\in \mathbb{R}^{N}|x_{N}=h_{j}(x’)$

for each , and call$j$$\bigcup_{j}^{n}S_{j}\subset \mathbb{R}^{N}$$n$$\bigcup_{j=1}^{n}\Gamma_{j}$$x’\in\Omega_{j}\}$the lateral faces of a pyramid. We put$\Gamma_{j}=\{x\in \mathbb{R}^{N}|x_{N}=h_{j}(x’)$

for $j=1$ , . . . , . Then

for

for

$x’\in\partial\Omega_{j}\}$represents the set of all edges of a pyramid. For $\gamma>0$ let$D(\gamma)=\{x|$dist$(x, \bigcup_{j=1}^{n}\Gamma_{j})>\gamma\}.$Now we define $v(x)$ by$\underline{v}(x)=\Phi(\frac{k}{c}(x_{N}-h(x’)))=\max_{1\leq j\leq n}\Phi(\frac{k}{c}(x_{N}-h_{j}(x’)))$.

Figure 3: The graph of a level set of a pyramidal traveling front ([18, 19])
Pyramidal traveling fronts are stated as follows. See Figure 3. For the proof see [16] for$n=2$ and see [13] for $n\geq 3.$

5Theorem 1 ([16], [13]) Let$h$be given in (6). Let$V(x)= \lim_{tarrow\infty}w(x, t;\underline{v})$

Then

$V$

$V$

be defined byfor all$x\in \mathbb{R}^{N}.$satisfies(7)$(-\triangle-cD_{N})V-f(V)=0 x\in \mathbb{R}^{N}$ .

with$\lim_{\gammaarrow\infty}\sup_{x\in D(\gamma)}|V(x)-\underline{v}(x)|=0,$$-1<\underline{v}(x)<V(x)<1 forallx\in \mathbb{R}^{N}.$Figure 4: The graph of a level set of $U.$

Cylindrically symmetric traveling front $U(r, z)$ satisfies$(-D_{rr}- \frac{N-2}{r}D_{r}-D_{zz}-cD_{z})U-f(U(r, z))=0$ ,$U_{r}(0, z)=0$

for $r>0,$for$z\in \mathbb{R}$,(8)$z\in \mathbb{R},$$U(0,0)=0.$

HereandThe following is the main assertion in this paper.
$D_{r}U=\partial U/\partial r,$$D_{rr}U=\partial^{2}U/\partial r^{2},$$D_{z}U=\partial U/\partial z$$D_{zz}U=\partial^{2}U/\partial z^{2}$. See Figure 4.

Theorem 2 ([21]) Let $g\in C^{2}(S^{N-2})$ satisfy $g(\xi)>0$ for all $\xi\in S^{N-2}$ . Assume thatis a convex compact set inand all principal$D_{g}=\{r\xi|0\leq r\leq 9(\xi), \xi\in S^{N-2}\}$$\mathbb{R}^{N-1}$

6

Figure 5: The graph of a level set of$\tilde{U}.$curuatures of $\partial D_{g}=\{g(\xi)\xi|\xi\in S^{N-2}\}$ are positive at every point oftoexists a unique solution$\partial D_{g}$. Then there$\tilde{U}$$(- \sum_{i=1}^{N}\frac{\partial^{2}}{\partial x_{i}^{2}}-c\frac{\partial}{\partial x_{N}})\tilde{U}-f(\tilde{U})=0 in\mathbb{R}^{N}$$\lim_{sarrow\infty}\sup_{|x|\geq s}|\tilde{U}(x)-\min_{\xi\in S^{N\underline{2}}}U(|x’-g(\xi)\xi|, x_{N})|=0$

Let satisfy the assumption stated above and let
respectively. One has$g_{j}$$\tilde{U}_{j}$

(9)

,(10)

.

be the associated solution for $j=1$ , 2,(11)$\tilde{U}_{2}(x_{1}, \ldots, x_{N-1}, x_{N})=\tilde{U}_{1}(x_{1}, \ldots, x_{N-1}, x_{N}-\zeta)$

for some$\zeta\in \mathbb{R}$if and only if$g_{1}\sim g_{2}.$be as inLet be the set of all that satisfies the assumption of Theorem 2. Let. We define an equivalence relation in . Roughly speaking, we defineTheorem 2 forwith a constant width and the expanded one
if and only if one can expand
with a constant width and the expanded one equals
equalsor one can expand
See Figure 6.
Let $g\in C^{2}(S^{N-2})$ satisfy $g(\xi)>0$ for all $\xi\in S^{N-2}$ . We set$\mathcal{G}$$D_{9}$$g$$\mathcal{G}$$g\in \mathcal{G}$$D_{91}$$g_{1}\sim g_{2}$$D_{g_{2}}$$D_{g_{2}}$$C_{g} = \{g(\xi)\xi|\xi\in S^{N-2}\},$$D_{g} = \{r\xi|0\leq r\leq g(\xi), \xi\in S^{N-2}\},$$D_{g_{1}}.$

7. For some neighborhood of $g(\xi)\xi\in C_{g}$ with $\xi\in S^{N-2}$ weand haveas $(y, \psi(y))$ with $\psi(y^{0})=0$ and $\nabla\psi(y^{0})=0$ , where $y=(y_{1}, \ldots,y_{N-2})$ . Here wewrite$g(\xi)\xi=(y^{0},\psi(y^{0}))$ withput
Let $v(y)$ be the unit normal vector oftoat $(y, \psi(y))$ pointing fromWe have$v(y)= \frac{1}{1+|\nabla\psi(y)|^{2}}(^{-\nabla\psi(y)}1)$ ,$C_{g}=\partial D_{g}\subset \mathbb{R}^{N-1}$$C_{g}$$y^{0}\in \mathbb{R}^{N-2}.$$C_{g}$$D_{9}$$\mathbb{R}^{N-1}\backslash D_{g}.$where$\nabla\psi(y)=t(D_{1}\psi(y), \ldots, D_{N-2}\psi(y))$

The eigenvalues$\kappa_{1}(y^{0})$

,...,

.

of the Hessian matrix$\kappa_{N-2}(y^{0})$$-D^{2}\psi(y^{0})=-(D_{ij}\psi(y^{0}))_{1\leq i,j\leq N-2}$

are the principal curvatures of. We take the basis ofatas the eigenvectors of the Hessian matrix. Using this principal coordinate system, we have$\mathbb{R}^{N-1}$$(y^{0}, \psi(y^{0}))$$C_{9}$$-D^{2}\psi(y^{0})=$diag ($\kappa_{1}(y^{0}), \ldots, \kappa_{N-2}(y^{0}))$and$D_{j}v_{i}(y^{0})=\kappa_{i}(y^{0})\delta_{ij} 1\leq i, j\leq N-2.$

We define{$\mathcal{G}$by$g\in C^{2}(S^{N-2})|g\geq 0$

For any$g\in \mathcal{G}$, all principal curvature ofand $a\geq 0$ we define$g_{1}=\tau_{a}g$$C_{9}$

are positive at every point of$C_{g}$}.

by$C_{g_{1}}=\{x’\in C_{g}\cup(\mathbb{R}^{N-1}\backslash D_{9})|$dist $(x’, C_{9})=a\}.$See Figure 6.
Then we have the following lemma.
Lemma 1 For any $a\geq 0,$$\tau_{a}$is a mapping in$\mathcal{G}$. Moreover one has(12)$\tau_{b}(\tau_{a}g)=\tau_{b+a}g$

for any$a\geq 0,$ $b\geq 0$

and$g\in \mathcal{G}.$Now we define an equivalence relation. We define
if andfor
only if one has either
gives
for some $a\geq 0$ . We will show thatora traveling front of (1).
Theorem 2 says that each element of a quotient set
gives an $N$-dimensional
traveling front
in the Allen-Cahn equation. Figure 5 shows the graph of a level set$g_{1}\sim g_{2}$$g_{1}=\tau_{a}g_{2}$$g_{1},$$g_{2}\in \mathcal{G}$$\mathcal{G}/\sim$$\tilde{U}$$\{x\in \mathbb{R}^{N}|\tilde{U}(x)=-a_{*}\}.$We choose $\eta>0$ large enough such that we have$\eta>_{1}\max_{\leq j\leq N-2\xi\in}m_{S^{N-2}}\frac{1}{\kappa_{j}(\xi)}$$g_{1}\sim g_{2}$$\mathcal{G}/\sim$$g_{2}=\tau_{a}g_{1}$8Figure 6: The graphs of$C_{g}$

and$C_{g_{1}}.$is included in the closure of a circumscribed ball ofandat$\xi\inS^{N-2}$every. Let
be the unit normal vector ofatfor $\xi\in S^{N-2}.$
Now we define a weak subsolutionas$D_{9}$$C_{9}$$\nu(\xi)$with radiuspointing from$g(\xi)\xi$$g(\xi)\xi$$C_{g}$$\eta$$D_{g}$

forto$\mathbb{R}^{N-1}\backslash D_{g}$$\underline{v}(x)$$\underline{v}(x’, x_{N})=\xi S^{N\underline{2}}\max_{\in}U(|x’-g(\xi)\xi+\eta\nu(\xi)|, x_{N}+m_{*}\eta)$

The stability of$\tilde{U}$for all$(x’, x_{N})\in \mathbb{R}^{N}$.

(13)

is as follows.

Corollary 3 (Stability [21]) Let and be as in (13) and Theorem 2,satisfy
a bounded and uniformly continuous function$\tilde{U}$$\underline{v}$pectively. Let$re\mathcal{S}$$u_{0}$$\lim_{Rarrow\infty}\sup_{|x|\geq R}|u_{0}(x)-\tilde{U}(x)|=0,$$\underline{v}(x)\leq u_{0}(x)\leq 1$for all$x\in \mathbb{R}^{N}.$Then one has$\lim_{tarrow\infty}\sup_{x\in R^{N}}|w(x, t;u_{0})-\tilde{U}(x)|=0.$

This work is supported by JSPS Grant-in-Aid for Scientific Research (C), Grant Number

26400169.

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[21] M. Taniguchi, An (N $-1)$ -dimensional convex compact set gives an $N$ -dimensional
traveling front in the Allen-Cahn equation, submitted.