数理解析研究所講究録
第 1901 巻 2014 年 69-78

69

On a population model with a free boundaryand related elliptic problems
早稲田大学大学院基幹理工学研究科 兼子裕大 (Yuki Kaneko)
Department of Pure and Applied Mathematics, Waseda University

早稲田大学理工学術院 山田義雄 Yoshio Yamada)
Department of Pure and Applied Mathematics, Waseda University$($

1

Introduction

The spreading of invasive or new species is one of the most important topics
in mathematical ecology. Since Skellam’s work [12], a lot of researchers havestudied the population dynamics of the species (see e.g. Shigesada-Kawasaki[11] and Cantrell-Cosner [1]). Recently Du and Lin [4] proposed a new mathematical model to understand the spreading of the species:$\{\begin{array}{ll}u_{t}-du_{xx}=u(a-bu) , t>0, 0<x<h(t) ,u_{x}(t, 0)=0, u(t, h(t))=0, t>0,h’(t)=-\mu u_{x}(t, h(t)) , t>0,h(O)=h_{0}, u(O, x)=u_{0}(x) , 0\leq x\leq h_{0},\end{array}$

(1.1)

where
and are given positive numbers andis a nonnegative
function. In (1.1), $u=u(t, x)$ represents a population density of the species in
one dimensional habitat. A free boundary $x=h(t)$ is a spreading front of thespecies, while $x=0$ is the fixed boundary. The dynamics of the free boundary
is determined by Stefan-like condition $h’(t)=-\mu u_{x}(t, h(t))$ . This conditionmeans that the spreading speed is proportional to the population pressure at
the free boundary (the spreading front).
It is characteristic of this model that the asymptotic behaviors of solutionsfor (1.1) are divided into two cases:$\mu,$$h_{0},$$d,$(i) Spreading:$(0, \infty)$

$a$

$b$

$\lim_{tarrow\infty}h(t)=\infty$$u_{0}$

and $\lim_{tarrow\infty}u(t, x)=a/b$ locally uniformly in;(ii) Vanishing:$\lim_{tarrow\infty}h(t)\leq(\pi/2)\sqrt{d/a}$and$\lim_{tarrow\infty}\Vert u(t, \cdot)\Vert_{C(0,h(t))}=0.$

Here the spreading means that the species succeed to spread to a whole region, while the vanishing means that the species cannot survive in the region.
Such a model has been developed by many researchers. See e.g. Du-Guo[2], Du-Lou [5], Kaneko-Oeda-Yamada [8] and Kaneko-Yamada [9].
$(0, \infty)$$*p_{artially}$ supported by Grant-in-Aid for Scientific Research (C-24540220), Japan Society
for the Promotion of Science70
In this article, we will set more realistic environments and consider a freeboundary problem describing the population dynamics of biological species$:=\{x\in \mathbb{R}^{N}|R_{1}<$which desires a new environment in a limited area
$R_{2}=\infty$$r<R_{2}\}(R_{1}, R_{2}>0, r=|x|, N\in \mathbb{N})$ . Here we allow. For simplicity,
we assume that the distribution and the habitat of the species are radiallysymmetric. The problem (FBP) is given by (1.2) and (1.3):$(R_{1}, R_{2})$u_{r}(t, R_{1})=0) , t>0,u(t, h(t))=0, t>0,h(t)\leq R_{2}, t>0,h(O)=h_{0}, u(O, r)=u_{0}(r) , R_{1}\leq r\leq h_{0}\end{array}$(1.2)$h’(t)=\{\begin{array}{ll}-\mu u_{r}(t, h(t)) for t> Osuch that h(t)<R_{2},0 for t>0 such that h(t)\geq R_{2},\end{array}$(1.3)$\{\begin{array}{ll}u_{t}-d\triangle u=uf(u) , t>0, R_{1}<r<h(t) ,u(t, R_{1})=0 (resp.

and

where

$\mu,$$d$and$R_{1}$

are positive constants,$R_{2}$

is a positive parameter and$\triangle u :=u_{rr}(t, r)+\frac{(N-1)}{r}u_{r}(t, r)$Moreover initial datainand$(u_{0}, h_{0})$.

satisfies $h_{0}\in(R_{1}, R_{2})$ , $u_{0}\in C^{2}(R_{1}, h_{0})$ , $u_{0}>0$$(R_{1}, h_{0})$$u_{0}(R_{1})=u_{0}(h_{0})=0$$($

resp. $u_{0}’(R_{1})=u_{0}(h_{0})=0)$ .

We assume that the nonlinear function satisfies, $f(u)>0$ for $0\leq u<1,$ $f(u)<0$ for $u>1,$
and $f’(u)<0$ for $u\geq$ O.
$f\in C^{1}(\mathbb{R})$(1.4)

A typical example of this nonlinearity is a logistic term, $uf(u)=u(1-u)$ .

Figure 1. the habitat of species

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In (FBP), we denote by $u(t, r)$ the population density of the species. The
area is not initially occupied by the species, and the habitat of the species is
described as $(R_{1}, h(t))$ , where $r=h(t)$ is the free boundary representing the
spreading front of the species. The condition (1.3) on $h(t)$ is same as thatin (1.1) for $N=1$ and $t>0$ such that $h(t)<R_{2}$ . However, when the freeboundary reaches $r=R_{2}$ at some time $t=\tau*$ , it must stop at the point andwe will consider a fixed boundary problem infor $t\geq\tau*$ . We also
note that the region $[0, R_{1}]\cup[R_{2}, \infty]$ is a hostile environment, and the species
cannot inhabit the region.
When $R_{2}=\infty$ , we can replace the Stefan-type condition of (1.3) by$[R_{1}, R_{2}]$$h’(t)=-\mu u_{r}(t, h(t)) , t>0$

.

(1.5)

Problem (P) given by (1.2) and (1.5) was studied by Kaneko [7].
It has been proved that the spreading behaviors (stationary states) of solutions for (FBP) and (P) are closely related to the following elliptic problems,
respectively:(SP1)$\{\begin{array}{l}d\triangle v+vf(v)=0, R_{1}<r<R_{2},v(R_{1})=v(R_{2})=0 (resp.

v_{r}(R_{1})=v(R_{2})=0)\end{array}$

and

(SP2)

$\{\begin{array}{l}d\triangle v+vf(v)=0, R_{1}<r<\infty,v(R_{1})=0 (resp.

v_{r}(R_{1})=0) .\end{array}$

We will show such relations and present some results on (SP2) obtained in [7]in this paper.
The purposes of this article are as follows:(i) Show the asymptotic behaviors of solutions for (FBP);(ii) Make clear the diﬀerences on spreading and vanishing between (FBP)
and (P);(iii) Give some suﬃcient conditions for spreading and vanishing.
(iv) Show the existence and uniqueness of solutions for (SP2).

2

Main Results

2.1

Spreading and vanishing in a limited area

In this section, we discuss the existence and uniqueness of solutions for(FBP) and the asymptotic behaviors of solutions as $tarrow\infty$ . We first obtainthe following results.

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Theorem 2.1. Let $fsati_{\mathcal{S}}fy(1.4)$ . The free boundary problem (FBP) has aunique solution $(u, h)$ satisfying

for $R_{1}<r<h(t)$ ,such that $h(t)<R_{2},$for
for $t>0,$$0<u(t, r)\leq C_{1}$$t\geq 0,$$0<h’(t)\leq\mu C_{2}$$t\geq 0$$h_{0}<h(t)\leq R_{2}$where$C_{1}$

andare positive constants depending only on, respectively. Moreover the limit of$C_{2}$$h(t)a\mathcal{S}tarrow\infty exi_{\mathcal{S}}ts$$\Vert u_{0}\Vert_{C^{1}(R_{1},h_{0})}$$belong_{\mathcal{S}}$toandand it$\Vert u_{0}\Vert_{C(R_{1},h_{0})}$$(h_{0}, R_{2}].$The proof of this theorem is almost similar to that for (P), and we omit thedetails here (see e.g. [2] and [7]).
We next prepare some positive number
to show the asymptotic behaviors of solutions. Let
be the least eigenvalue of$R^{*}$$\lambda_{1}(d, R_{1}, l)$$\{\begin{array}{l}-d\triangle\phi=\lambda\phi, R_{1}<r<l,\phi(R_{1})=\phi(l)=0.\end{array}$Here is a given positive number. It is well known thatand decreasing with respect to , and moreover it satisfies$l$$\lambda_{1}(d, R_{1}, l)$is continuous$l$$\lim_{larrow R_{1}+0}\lambda_{1}(d, R_{1}, l)=+\infty$

Thus, for givensuch that$d,$$R_{1}$

and$\lim_{larrow+\infty}\lambda_{1}(d, R_{1}, l)=0.$

and , there exists a positive number $R^{*}=R^{*}(d, R_{1}, f(O))$$f$$f(O)=\lambda_{1}(d, R_{1}, R^{*})$

and$f(O)>\lambda_{1}(d, R_{1}, l)$for $l>R^{*}$(2.1)

atisfy ( $1.4)$ and let $(u, h)$ be any solution of (FBP). Then
Theorem 2.2. Let
there exists $R^{*}=R^{*}(d, R_{1}, f(O))>0$ determined by (2.1) with the following
properties.
$f_{\mathcal{S}}$(I) Suppose$R_{2}\leq R^{*}$. ThenVanishing:$\lim_{tarrow\infty}h(t)\leq R_{2}$and$\lim_{tarrow\infty}\Vert u(t, \cdot)\Vert_{C(R_{1},h(t))}=0$

occurs for any initial data.
(II) Suppose $R_{2}>R^{*}$ . Then either (A) or (B) holds true:(A) Spreading : $h(t)=R_{2}$ for all $t\geq T$ with some $T\in(O, \infty)$ and, where$\lim_{tarrow\infty}u(t, r)=v(r)$ uniformly in$[R_{1}, R_{2}]$is a unique positive solution of (SP1);(B) Vanishing :and$\lim_{tarrow\infty}h(t)\leq R^{*}$$\lim_{tarrow\infty}\Vert u(t, \cdot)\Vert_{C(R_{1)}h(t))}=0.$

$v$

73
We need the following lemma.

Lemma 2.1. Let $(u, h)$ be any solutions of (FBP). If$\lim_{tarrow\infty}h(t)<R_{2}$, then$\lim_{tarrow\infty}\Vert u(t, \cdot)\Vert_{C(R_{1},h(t))}=0.$

We can easily prove this lemma by using [7, Theorem 2].
Proof of Theorem 2.2. Suppose that

$R_{2}\leq R^{*}$. Let$(\overline{u}, \overline{h})$be a solution of$\{\begin{array}{ll}\overline{u}_{t}-d\triangle\overline{u}=\overline{u}f(\overline{u}) , t>0, R_{1}<r<R_{2},\overline{u}(t, R_{1})=0, \overline{u}(t, R_{2})=0, t>0,\overline{u}(0, r)=u_{0}(r) , R_{1}\leq r\leq R_{2}.\end{array}$(2.2)

Then the standard comparison principle shows$u(t, r)\leq\overline{u}(t, r)$

Since$\Vert\overline{u}(t, \cdot)\Vert_{C(R_{1},R_{2})}$converges tofor $t>0,$ $R_{1}\leq r\leq h(t)$ .

as

$0$

$tarrow\infty$(cf. Henry [6]), we have$\lim_{tarrow\infty}\Vert u(t, \cdot)\Vert_{C(R_{1},h(t))}=0.$

We next assume $R_{2}>R^{*}$ . Since $h(t)$ is strictly increasing for $t>0$ as
long as $h(t)<R_{2}$ , we find thator $h(T)=R_{2}$ for some$h_{\infty}$$:= \lim_{tarrow\infty}h(t)<R_{2}$$T\in(0, \infty].$When$h_{\infty}<R_{2}$, it holds from Lemma 2.1 that$\lim_{tarrow\infty}\Vert u(t, \cdot)\Vert_{C(R_{1},h(t))}=0$

.

(2.3)

To complete the proof of part (B), we must show
. Otherwise there$T_{1}>0$
exists somesuch that $l:=h(T_{1})\in(R^{*}, R_{2})$ . Consider a solutionof$h_{\infty}\leq R^{*}$$\underline{u}(t, r)$$\{\begin{array}{ll}\underline{u}_{t}-d\triangle\underline{u}=\underline{u}f(\underline{u}) , t>0,R_{1}<r<l,\underline{u}(t, R_{1})=0, \underline{u}(t, l)=0, t>0,\underline{u}(T_{1}, r)=u(T_{1}, r) , R_{1}\leq r\leq l.\end{array}$

Then the comparison principle shows$u(t, r)\geq\underline{u}(t, r)$

Since$\underline{u}(t, r)$for$t\geq T_{1},$$R_{1}\leq r\leq l.$converges to the unique positive solution $q(r)$ of$\{\begin{array}{l}d\triangle q+qf(q)=0, R_{1}<r<l,q(R_{1})=q(l)=0\end{array}$as$tarrow\infty$, we have$\lim\inf u(t, r)tarrow\infty\geq q(r)>0$

for $R_{1}<r<l.$74This is a contradiction to (2.3), and henceif
We next consider the case that $h(T)=R_{2}$ for some $T\in(0, \infty$ ]. To provepart (A), we will show $T<\infty$ . Indeed, by the assumption, there exists some$T_{2}>0$ such that $h(T_{2})>R^{*}$ . Let $(v(t, r), s(t))$ be a solution of (P) with initial
data $(u(T_{2}, r), h(T_{2}))$ . Then we can easily show by a comparison principle (see[7, Lemma 3]) that$h_{\infty}\leq R^{*}$$s(t)\leq h(t+T_{2})$$h_{\infty}<R_{2}.$for $t\geq 0$ and $v(t, r)\leq u(t+T_{2}, r)$ for $t\geq 0,$ $R_{1}\leq r\leq s(t)$ .

By [7, Theorem 5], we find that $s(T_{3})=R_{2}$ for some $T_{3}<\infty$ . Hence it holds
for $T:=T_{2}+T_{3}$ that$h(t)=R_{2}$for $t\geq T$ and $u(T, r)>0$ for $R_{1}<r<R_{2}.$Thus we consider a fixed boundary problem with initial data $u(T, r)$ , andobtain the uniform convergence of to the positive solution of (SP1) as$tarrow\infty.$$u$

口, vanishing occurs for any initial data. We
By Theorem 2.2, when
can also give suﬃcient conditions for spreading and vanishing when $R_{2}>R^{*}.$$R_{2}\leq R^{*}$

Proposition 2.1. Suppose $R_{2}>R^{*}$
Then the following results hold true:(i) Suppose$h_{0}\geq R^{*}$

Let $(u, h)$ be any solution of (FBP).

Then spreading occurs.

(ii) Suppose $h_{0}<R^{*}$ There exists a positive function in, then vanishing occurs andif $u_{0}(r)\leq w(r)$ in
as
for
$w$

$[R_{1}, h_{0}]$$O(e^{-\beta t})$$\mathcal{S}ome\beta>0$$[R_{1}, h_{0}]$such that,$\Vert u(t, \cdot)\Vert_{C(R_{1},h(t))}=$$tarrow\infty.$

Proof. We first prove part (i). Since $h(t)$ is strictly increasing andby the assumption, we see $h(t)>R^{*}$ for all $t>0$ . By Theorem 2.2, we have$h_{0}\geq R^{*}$$h(t)=R_{2}$for $t\geq T$ and $T<\infty,$$\lim_{tarrow\infty}u(t, r)=v(r)$uniformly in$[R_{1}, R_{2}].$

It remains to prove part (ii). Define $(v(t, r), s(t))$ by$s(t)=\mathcal{S}_{0}(1+\delta(1-e^{-\alpha t}))$

) and
where
eigenvalue for the problem:$s_{0}\in[h_{0}, R^{*}$$\varphi(y;\gamma)$

and$v(t, r)= \epsilon_{0}e^{-\beta t}\varphi(\frac{s_{0}}{s(t)}r;\gamma)$

is an eigenfunction corresponding to the least$\{\begin{array}{l}-d\triangle_{y}\varphi=\lambda_{1}\varphi, \varphi>0, \gamma<y<s_{0},\varphi(\gamma)=\varphi(s_{0})=0 (resp.

with suﬃciently close tosuitable constants$\gamma$$\alpha,$$\beta,$$\delta,$$R$,\varphi_{y}(\gamma)=\varphi(s_{0})=0)\end{array}$and $0< \delta<\min\{R_{1}/\gamma-1, R_{2}/s_{0}-1\}$ . Choosing
and small initial data such that $u_{0}(r)\leq w(r)$ $:=$$\epsilon_{0}$75, we can regard $(v, s)$ as an upper solution of (FBP) (see the proof of [7,Theorem 5 and we have$\epsilon_{0}\varphi(r)$$h(t)\leq s(t)$Hence we concludeas. 口

and $u(t, r)\leq v(t, r)$ for $t>0,$ $R_{1}\leq r\leq h(t)$ .
$\lim_{tarrow\infty}h(t)\leq \mathcal{S}_{0}(1+\delta)<R_{2}$and$\Vert u(t, \cdot)\Vert_{C(R_{1},h(t))}=O(e^{-\beta t})$$tarrow\infty$

We can show a threshold on initial data which separates spreading andvanishing. Let
and $\phi(R_{1})=\phi(h_{0})=0$
satisfy $\phi>0$ in
resp. $\phi_{r}(R_{1})=\phi(h_{0})=0)$ . Then we have the following result.
$\phi\in C^{2}(R_{1}, h_{0})$$(R_{1}, h_{0})$$($Corollary 2.1. Suppose $h_{0}<R^{*}<R_{2}$ . Consider the solution of (FBP) with] suchinitial data
. Then, there exists a numberthat spreading occurs if, while vanishing occurs if$\sigma^{*}=\sigma^{*}(u_{0}, h_{0})\in(0, \infty$$(u_{0}, h_{0})$$u_{0}\leq\sigma^{*}\phi.$$u_{0}>\sigma^{*}\phi$

The proof is almost similar to that in [7, Corollary 1].

An elliptic problem in an exterior domain

2.2

In this section, we will discuss elliptic problem (SP2). We remark that
(SP2) is concerned with the stationary state of solutions for (P) (which is(FBP) with $R_{2}=\infty$ ). In other words, when spreading occurs, the solutions
converge to some solution of (SP2) as $tarrow\infty$ . Moreover the stationary stateis uniquely determined because of the unique existence of solutions for (SP2).
The proofs of results in this section are shown in [7].
Our purpose of this section is to prove the following theorem.

Theorem 2.3. Let. Then there exists a unique positive solution$v_{r}(r)>0$(SP2).
The solution satisfies
offor all $r\geq R_{1}$ and $\lim_{rarrow\infty}v(r)=1$
with $v_{r}(r)=o(1/r^{N-1})$ as $rarrow\infty.$$fsati_{\mathcal{S}}fy(1.4)$$v$

We need the following proposition.
Proposition 2.2. Suppose that satisfies (1.4). Let $v\in C^{2}(R_{1}, \infty)$ be any
positive solution of (SP2). Then $v_{r}(r)>0$ for all $r\geq R$ and $\lim_{rarrow\infty}v(r)=1$$f$with $v_{r}(r)=o(1/r^{N-1})$ as $rarrow\infty.$

Proof of Theorem 2.3. We first prove the existence of solutions for the
problem by the standard monotone method. Let$w(r)=\{\begin{array}{ll}\phi(r) , r\in[R_{1}, l],0, r\in(l, \infty) ,\end{array}$where is a positive number satisfying $l>R^{*}$ and$l$$\phi$$\{\begin{array}{l}-d\triangle\phi=\lambda_{1}\phi, R_{1}<r<l,\phi(R_{1})=\phi(l)=0.\end{array}$is a positive solution of

76
Then we find that, for any small
is a lower solution of (SP2) in the
distribution sense. On the other hand,is an upper solution of (SP2).
Hence, by the standard monotone method (see Sattinger [10] and Smoller [13]),
there exists a solution such that $\delta w(r)\leq v(r)\leq 1$ for $r\in[R, \infty$ ). Moreoversatisfies (SP2) in the classical sense.
We next prove the uniqueness of solutions for (SP2). Since is aIly suﬃciently small positive number, the uniqueness of solutions for (SP2) satisfying$\delta w(r)\leq v(r)\leq 1$ for $r\in[R, \infty$ ) enables us to get the conclusion. Suppose(resp.
) is a minimal (resp. maximal) positive solution of (SP2),
that(resp. 1). Then
which is generated by$\delta>0,$$\delta w$$v\equiv 1$

$v$

$v$

$\delta$$v$$w^{*}$$w_{*}$$\delta w(r)$$($$d(r^{N-1}w_{*,r}(r))_{r}+r^{N-1}w_{*}(r)f(w_{*}(r))=0,$$R_{1}<r<\infty,$$w_{*}(R_{1})=0$

resp. $d(r^{N-1}w_{r}^{*}(r))_{r}+r^{N-1}w^{*}(r)f(w^{*}(r))=0,$$R_{1}<r<\infty,$$w^{*}(R_{1})=0)$

with$w_{*}(r)\leq w^{*}(r)$Multiplying the equation by
the equations, we obtain$w^{*}$

for $R_{1}<r<\infty.$(resp.

$w_{*}$) and subtracting the both sides of$r^{N-1}w^{*}(r)w_{*}(r)\{f(w^{*}(r))-f(w_{*}(r))\}$$=d\{(r^{N-1}w_{*,r}(r))_{r}w^{*}(r)-(r^{N-1}w_{r}^{*}(r))_{r}w_{*}(r)\}.$

Moreover integrating the equation in$[R_{1}, \rho]$

for $\rho>R_{1}$ leads to$\frac{1}{d}\int_{R_{1}}^{\rho}r^{N-1}w^{*}(r)w_{*}(r)\{f(w^{*}(r))-f(w_{*}(r))\}dr$$= \int_{R_{1}}^{\rho}(r^{N-1}w_{*,r}(r))_{r}w^{*}(r)-(r^{N-1}w_{r}^{*}(r))_{r}w_{*}(r)$

$dr$

.

Integrating by parts the right-hand side of the above identity implies$\frac{1}{d}\int_{R_{1}}^{\rho}r^{N-1}w^{*}(r)w_{*}(r)\{f(w^{*}(r))-f(w_{*}(r))\}dr$$=\rho^{N-1}w_{*,r}(\rho)w^{*}(\rho)-\rho^{N-1}w_{r}^{*}(\rho)w_{*}(\rho)$.

By Proposition 2.2, it holds that$\lim_{\rhoarrow\infty}\rho^{N-1}w_{*,r}(\rho)=\lim_{\rhoarrow\infty}\rho^{N-1}w_{r}^{*}(\rho)=0,$$\lim_{\rhoarrow\infty}w^{*}(\rho)=\lim_{\rhoarrow\infty}w_{*}(\rho)=1.$

Taking$\rhoarrow\infty$, we have$\int_{R_{1}}^{\infty}r^{N-1}w^{*}(r)w_{*}(r)\{f(w^{*}(r))-f(w_{*}(r))\}dr=0.$

77
It follows from $f’(u)<0$ for $u\geq 0$ and $w^{*}\geq w_{*}>0$ in, and we complete the proof.
in$[R_{1}, \infty$) that$w^{*}\equiv w_{*}$$\square$$[R_{1}, \infty)$

We will show the existence and uniqueness of positive solutions for (SP2)under the Neumann boundary condition at $r=R_{1}.$Theorem 2.4. Suppose thatsatisfies (1.4). Then there exists a uniquepositive solutionfor (SP2) under the Neumann boundary condition at$f$$v\equiv 1$$r=R_{1}.$We can prove this theorem by the following proposition.
Proposition 2.3. Suppose that $fsati\mathcal{S}fies(1.4)$ . Let $v\in C^{2}(R_{1}, \infty)$ be any
positive solution of (SP2) under the Neumann boundary condition at $r=R_{1}.$Then$v\equiv 1.$

Concluding Remarks

3

In this section, we will give some remarks.
(i) We can extend the results on spreading and vanishing to the case ofgeneral nonlinearity. When we consider a bistable term like $uf(u)=$$u(u-c)(1-u)(0<c<1/2)$ , we also get spreading and vanishing
behaviors, but it is diﬀerent from the logistic case.

(ii) If the area and the distribution of the species are not radially symmetric,
then the problem becomes more complicated. The case $R_{2}=\infty$ wasdiscussed by Du-Guo [3].

(iii) For general nonlinearities, we may define spreading and vanishing ofsolutions for (FBP) as follows.
Definition 3.1. Let $(u, h)$ be any solution of (FBP).

(I)

Spreading of species is the case when$h(t)=R_{2}$(II)

for $t\geq T$ with some $T \in(0, \infty], \lim_{tarrow\infty}u(t, x)>0$ for $R_{1}<r<R_{2}$ ;Vanishing of species is the case when$\lim_{tarrow\infty}h(t)\leq R_{2}$and$\lim_{tarrow\infty}\Vert u(t, \cdot)\Vert_{C(R_{1},h(t))}=0.$78

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