数理解析研究所講究録1102 巻 1999 年 91-111

91

Chauchy problem of nonl.inear wave equations
with small and smooth initial data
久保英夫 (Hideo Kubo)

静大工In this note, we consider Chauchy problem for
(0.1)
(0.2)

$\coprod_{\dot{\iota}}u^{i}\equiv\partial_{t}^{2}u^{i}-di\Delta u^{i}=F^{i}(u, \partial u, \partial 2u)$$u^{i}(x, 0)=\epsilon f^{:}(x)$,$\partial_{t}u^{i}(X, \mathrm{o})=\epsilon g^{i}(x)$inin$\mathrm{R}^{n}\cross(0, \infty)$$\mathrm{R}^{n}$,,

is a real valued unknown function,. Besides,
and$u=(u^{1},\cdots, u^{m})$and. We also denoted
andstand for the first and second derivatives with respect to,$(j=1, \cdots , n)$ . Roughly speaking, we would like to compare the behavior of thesolution $u(x, t)$ to the problem (0.1) and (0.2) with that of $u_{0}(x, t)=(.u_{0}^{1}(x, t),$$u^{m}(0tX,))$ ,which is the solution to the homogeneous wave equation
where $i=1,$$\cdots,$$m,$ $u^{i}(x,t)$$\partial u,$$\epsilon>0$$\mathrm{C}:>0$$F^{i}\in C^{\infty}(\mathrm{R}^{(}n+1)m\cross \mathrm{R}^{(n+1)^{2}m})$$f^{i},$$g^{i}\in C_{0}^{\infty}(\mathrm{R}^{n})$$\partial^{2}u$$\partial_{t}=\partial/\partial t(\equiv\partial_{0})$$\partial_{j}=\partial/\partial x_{j}$$\cdots,$(0.3)$\coprod_{i}u^{i}(X,t)=0$

in$\mathrm{R}^{n}\cross(0, \infty)$,

satisfying (0.2), provided the parameter is sufficiently small. If $n=2$ or $n=3$ , the-norm ofcan be controled as follows:$\epsilon$$L^{\infty}$$u_{0}^{i}(X, t)$(0.4)$|u_{0}(x,t)|\leq C\epsilon(1+r+t)^{-\frac{n-1}{2}(1}+|c_{i}t-r|)^{-\frac{n-1}{2}}$provided$f^{i},$$g^{i}\in C_{0}^{\infty}(\mathrm{R}^{n})$,. In the following, we shall only consider the case where $n=2$

or $n=3$ . To my knowledge, the imprtance of the factor $(1+|\mathrm{q}t-r|)$ is firstly pointed
out by Professor F. John in [17]. And the factor also plays an essential role in our analisis.
We divide our argument into two parts. First one is concerned with the quasilinear case.
This part is a collabolation work with Professor A. Hosiga in [15]. While the second part,which is ajoint work with Professor M. Ohta in [30], is concerned with the semilinear case.
The author wishes to be thankful to Professor R. Agemi and Professor K. Kubota for theirvaluable comment.

Key words. A unique global smooth solution, Critical exponent, Smffi Data Global

Existence, Small Data Blow-up, Null condition, Different propagation speeds

1

Quasilinear CaseIn this section we study the quasilinear case. Namely, we assume(1.1)$F^{i}=F^{\dot{*}}( \partial u,\partial^{2}u)=\sum\sum_{\delta\iota=10}^{n}m\gamma,=H^{\gamma\delta}d(\partial u)\partial\partial\gamma\delta u^{\mathrm{t}}+.K_{*}.(\partial u)$

,

92

where

$H_{il}^{\gamma\delta},$

(1.2)

Here

satisfy$K_{i}\in C^{\infty}(\mathrm{R}^{(}n+1)m)$$H_{il}^{\gamma\delta}(\partial u)=O(|\partial u|^{p-1})$$p$

is an integer with $p>1$ .

(1.3)

,

order to$\mathrm{h}$near$K_{i}(\partial u)=O(|\partial u|^{p})$$\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}_{\mathrm{V}}\mathrm{e}$$\partial u=0$.

an energy aet’ate, we need to assume$\mathrm{i}\mathrm{m}$.

$H_{1l}^{\gamma\delta}.(\partial u)=H_{\mathrm{t}}\gamma\delta(i\partial u)=H_{1}\gamma(\delta\partial lu)$

Since the existence and the uniqueness of the local smooth solution of (0.1) and (0.2)
are already known (see e.g. S. Klainerman [23]), we aim at the global solvbility of the
problem. For this purpose, we shffi establish a uniform a priori estimate of a suitable
for some large integer $N$ . More precisely, we wish to prove-norm
weighted
for $0\leq t<T$ ,, then we must haveforthat if
is a constantprovided is sufficiently small. Here is a given positive number and. Once we could obtainand the functions
depending only on the initial data
the above proposition, we get a global solution due to the blow-up criterion. (See e.g.
[34] . Althogh our basic strategy to get a uniform a priori estimate is based on the method
developed by Professor S. Klainerman in [26], we need some modification to handle the casewhere the system (0.1) has different propagation speeds.
$[\partial u(t)]_{N}$$L^{\infty}$$0\leq \mathrm{t}<T$$[\partial u(t)]_{N}\leq 3M_{N}\epsilon$$[\partial u(t)]_{N}\leq 2M_{N}\epsilon$$T$$\epsilon$$f^{i},$$M_{N}$$H_{\dot{*}l}^{\gamma\delta},$$g^{i}$$K_{1}.$$)$$\mathrm{N}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{S}}$. We introduce the following vector fields:(1.4)$\Gamma=(\mathrm{r}_{0}, \cdots, \mathrm{r}_{N\mathrm{o}})=(\partial, \Omega,s)$

where $N_{0}=n+ \frac{n(n-1)}{2}+1$ and$\partial=(\partial_{0}, \cdots, \partial_{n})$$\partial_{0}=\partial_{t}$,,$\Omega=(\Omega_{i,\mathrm{j}})$,,$S=t\partial_{t}+r\partial$$\partial_{j}=\partial/\partial_{x_{\dot{g}}}(1\leq j\leq n)$,$\Omega_{i\dot{o}j}=X:^{\partial\partial(1i}-X_{ji}\leq<j\leq n)$

Remark. In our analysis, and will play a crucial role. Indded,
efeectively an.d will be used in (1.32), (1.37) and (1.38).
$u^{m}(x,t))$ , we set
For a vector valued function $u(x,t)=(u^{1}(X,t),$$\Omega$$S$$S\mathrm{w}\mathrm{i}\mathbb{I}$.

be used in (1.39)$\Omega$$\cdots,$$||u(t)||_{k}=$$|u(x,t)|_{k}=$$|a| \leq\sum_{k}\sum_{l=1}^{m}||\mathrm{r}_{u}^{al}(\cdot.’t)||_{L^{2}}$$\sum_{|a|\leq k}\sum_{=\iota 1}^{m}|\Gamma^{a}u(\iota x,t)|$,,is a nonnegative integer, $a=(a_{0}, \cdots, a_{N})0$ is a multi-index,$|a|=a_{0}+\cdots+a_{N_{0}}$ . Moreover, we shffi use the following weighted

where

$k$

$\Gamma^{a}=\Gamma_{0N0}^{a_{1}}\ldots\Gamma a_{N_{0}}$$L^{\infty}$

(1.5)

where

$[u(t)]_{k}= \sum\sum_{1\mathrm{I}a\mathfrak{l}\leq kl=}^{m}|.|wl(|\cdot|,t)\mathrm{r}_{u}al(\cdot,t)||_{\iota}\infty$$w_{l}$,

is a weight function associated with the l-th component of$w_{l}(r,t)=(1+r)^{\frac{n-1}{\mathit{2}}-\gamma}(1+t+r)^{\gamma}(1+|c_{l} \mathrm{t}-r|)\frac{n-1}{2}$-norm:$u$defined by

for $r\geq 0,t\geq 0$ ,

and

93
where $0<\gamma<(n-1)/2$ . Moreover, we ako use$||u||k,T|= \sup_{0<t<T}|u(t)||k$ ’$[u1_{k,T^{-\sup}}-[u(t)]_{k}0<t<T^{\cdot}$

Energy Estimates I. In order to assure the hyperbolicity of the system (0.1), we

will consider the following assumption on the solution(1.6)$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0,T))$:

for $0\leq t<T’$’$\sup_{x\in \mathrm{R}^{n}}|\partial u(x,t)|_{k}\leq\delta$

where a nonnegative integer and $\delta(0<\delta<1)$ is a real number. Note that the following
commutator relations hold:
$k$

(1.7)

$[\Gamma_{\sigma}, \square _{\dot{l}}]=-2\delta 4\sigma\coprod_{1}$for$\sigma=0,$ $\cdots,N_{0},$$i=1,$

Here $[ , ]$ denotes the usual commutator of linear operators andTherefore, if satisfies, then we have$u^{i}$$\cdots$, $m$ .

$\delta_{\alpha\beta}$

is Kronecker’s delta.

$\square _{i}u^{:}=F^{:}$(1.8)$\square _{i}\Gamma^{a}u^{i}=\sum_{b\leq a}cb\Gamma^{bi}F$

for some suitable constant . Then following a standard argument, we obtain the energy
estimate as follows. (For the details, see e.g. the proof of Proposition 5.1 in [15]).
$C_{b}$

Proposition 1. 1Let
be a solution of (0.1) and (0.2). Supposethat $(1.1)-(1.3)$ hold. Then there is a sufficiently small positive number such that if (1.6)
with $k=[(N+1)/2]$ holds, then we have for $0\leq t<T$$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0, T))$$\delta$(1.9).

$|| \partial u(t)||_{N}\leq C_{N}||\partial u(\mathrm{o})||N\exp[C_{N}\int_{0}^{t}\{\sup_{x\in \mathrm{R}^{n}}|\partial u(x, S)|[\frac{N+1}{2}]\}p-1ds]$

Following a formal argument for the moment, we will introduce a
the nonlinerlity. Having (0.4) in mind, we suppose that$‘(\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{C}\mathrm{a}}1$

exponent” of$F^{i}(\partial u, \partial^{2}u)$(1.10)$|\partial u(X,t)|_{[\frac{N+1}{2}]}\leq C(1+t+r)^{-\frac{n-1}{2}}$for$(x, t)\in \mathrm{R}^{n}\cross(0, \infty)$

holds. Then we have.

$\int_{0}^{t}\{\sup_{x\in \mathrm{R}n}|\partial u(X, S)|_{[\frac{N+1}{2}]}\}^{p-}1d_{S}\leq C\int^{t}\mathrm{o}(1+s)^{-\frac{\mathfrak{n}-1}{2}}(p-1)dS$

Therefore, if $(n-1)(p-1)/2>1$ , i.e., $p>1+2/(n-1)$ , we get a bound of(1.9). Such a bound indicates the existence of a global solution to the problem (0.1) and(0.2). Hence it is interesting to consider whether or not a global solution to the problem
exists when $p=p_{c}$ . Here is a critical exponent defined by$||\partial u(t)||N\mathrm{h}_{\mathrm{o}\mathrm{m}}$$p_{c}$(1.11)

2$(n=3)$ ,3$(n=2)$$p_{c}:=1+ \frac{2}{n-1}=\{$.

94
Known results. The folowing results assure that the number deffied by (1.11)
has real meaning: Let $m=1$ or the system (0.1) has common propagation speeds; i.e.,$c_{1}=\cdots=c_{m}$ . Suppose that $(1.1)-(1.3)$ hold.
$p_{\mathrm{C}}$,If $p>p_{c}$ , then [Small Data Global Existence] holds. Namely, for any
Moreover,.
sufficiently
small
globalsolution forthe problem (0.1) and (0.2) has a smooth
if $p=p_{c}$ , then [Small Data “Almost” Global Existence] holds. Namely, for any, the problem (0.1) and (0.2) has a smooth local solution whose life span is estimatedfrom below for sufficiently small . (See F. John and S. Klainerman [21],by. Klainerman [25] and M. Kodyov [27], for instance).
On the other hand, if $1<p\leq p_{c}$ , then [Small Data Blou-up] holds. Namely, [Small
Data Global Existence] dose NOT holds. (See R. Agemi [1], S. Alinhac [4], L. H\"ormander.
[12], A. Hoshiga [14] and F.
, the following interesting results are known. If
Furthermore, in the critical casethe nonlinearity has a special form, a global solution of (0.1) and (0.2) exists, instead of analmost global solution. (See D. Christodoulou [6], P. Godin [11], A. Hoshiga [13], F. Jolm[19], S. Katayama [22] and S. Klainerman [26], for instance). We shall call the restriction
on the nonlinearlities null condition, according to S. Klainerman [24]. We shall give its
definition for the 2-dimensional case in (1.43) below.
We now turn our attention to the case where $p=p_{c},$ $m\geq 2$ and the propagationeach other. This case was studied by M. Kovalyov in [30] andspeeds are differentin [3]. They found some nonlinearities, which do not satisfy
R. Agemi and K.
null condition (1.43), and proved the existence of lthe global solution to (0.1) and (0.2). In
the following, we shall investigate to find a wider class of nonlinearity which garantees theglobal solvability of the problem (0.1) and (0.2). (See also (1.42) below.)$f^{i},$$g^{1}\in C^{\infty}(\mathrm{R}^{n})$$\epsilon$$f^{i},$$g^{i}\in$$C^{\infty}(\mathrm{R}^{n})$$\exp(C/\epsilon^{p_{\mathrm{C}}-1})$$\epsilon$$.\mathrm{S}$$\mathrm{J}\mathrm{o}\mathrm{h}^{(}1[18])$$p=p_{\mathrm{C}}$$\mathrm{h}\mathrm{o}\mathrm{m}$$\mathrm{Y}\mathrm{o}\mathrm{k}_{0}\mathrm{y}\Re \mathrm{I}\mathrm{l}\mathrm{a}$Basic estimate. Since one can treat the case where the system (0.1) has common

propgation speeds less hard, we shall consider the case where the speeds are disitinct.
Namely, we assume.
(1.12)$c_{1}>c_{2}>\cdots>\mathrm{q}_{n}$Under this situation, we prepear basic estimates of the solution to the inhomogeneous wave
equations as in Proposition 1. 2 below, so that we will be able to get a variant of (1.10).
In what follows, we restict ourselves to the case where $n=2$ . (For the -dimensional case,see K. Yokoyama [41] .
for each $i(i=1, \cdots, m)$ as follows:
We strat with splitting the region$)$$(0, \infty)\cross(0, \infty)$$\Lambda_{i}=${ $(r,t)\in(0,$$\infty)\cross(0,$ $\infty)$:$\frac{1}{3}(2+\frac{C_{\dot{*}}}{c_{t-1}})r\leq c_{\dot{*}}t\leq\frac{1}{3}(2+\frac{\mathrm{q}}{C_{*+1}}.)r$

and$r\geq 1$}, where we have set
and $\omega_{+1}=u/4$ . Because ofand(1.12), this definition is meaningffi. In particular, we have$\mathrm{q}\mathrm{l}=4c_{1}$$\Lambda_{i}^{c}=(0, \infty)\cross(0, \infty)\backslash \Lambda_{i}$(1.13)$\Lambda_{\dot{\epsilon}}\cap\Lambda_{l}=\emptyset$

Notice that $1+r$ is equivalent to $1+t+r$ for

if$i\neq l$$(r, t)\in\Lambda_{i}$.

, while so is $1+|\alpha t-r|$ for$(r,t)\in\Lambda_{i}^{c}$.

95
Next we introduce the following weight functions:$r$$z^{(i)}(\lambda, S)=\{$,$(1+\lambda+S)(1+|\lambda-C_{\mathrm{j}}S|)$if$(\lambda, s)\in\Lambda_{\mathrm{j}},$$(1+\lambda+S)1+\mu(1+|\lambda-c_{t}s|)^{1-\mu}$if$(\lambda, s)\in\Lambda_{i}$$(1+\lambda)1-2\gamma(1+S+\lambda)^{1+}2\gamma$

if$(\lambda, s)\in(\mathrm{o}, \infty)\cross(0, \infty)\backslash \cup m\Lambda j=1j$$j\neq i$,and$(1+\lambda+s)(1+|\lambda-Cj\mathit{8}|)$if$(1+\lambda)1-2\gamma(1+S+\lambda)^{1+}2\gamma$

if$(\lambda, s)\in\Lambda_{j}$,$z(\lambda, S)=\{$

where $0<\mu<1$ and$\gamma$$(\lambda, s)\in(\mathrm{o}, \infty)\cross(\mathrm{o}, \infty)\backslash \cup m\Lambda j=1j$,

is the number in the definition of $w_{l}(r,t)$ . Then we have

Proposition 1. 2 Let $u(x,t)$ be a solution tois a multi-index urith $|b|=1$ .
(i) Letand $t<T$ . Then we have$\square _{:}u=\theta F$with the zero initial data, where$b$$(|x|,t)\in\Lambda^{c}.\cdot$(1.14)

where$w_{i}(r,t)|u(X,t)|\leq CM_{1}(F)(r,t)$$w_{i}(r, t)$,

is definded in (1.5) and we have set for a nonnegative integer$k$.

$M_{k}(F)(r,t)= \sum 1^{a}\mathrm{I}\leq k<\sup_{0s<t\in}\sup_{y\mathrm{R}}||2y|^{\frac{1}{2}}Z(|y|, S)\mathrm{r}^{a}F^{i}(y, S)|$(ii) Let $(x,t)\in \mathrm{R}^{2}\cross[0, T)$ . Then we have

(1.15)$w_{i}(r,t)|u(x,t)|\leq CM_{1}^{()}|(F)(r,t)$where we have set for a positive integer$M_{k}^{(i)}(F)(r,t)$

$=$

,

$k$

$|a| \leq\sum_{k}\sup_{<0S<ty\in}\sup_{\mathrm{R}2}||y|^{\frac{1}{2}}z^{(}i)(|y|, s)\mathrm{r}^{a}F^{i}(y, S)|$

Outline of the proof: Without loss of generality, we may assume, we have.
$\mathrm{q}=1$. Then if$F\in C^{\infty}(\mathrm{R}^{2}\cross[0,T))$(1.16)$u(x,t)= \frac{1}{2\pi}\int_{\mathrm{I}^{x}-y1}\leq t\frac{\partial^{b}F(y,s)}{\sqrt{t^{2}-|x-y|2}}dy$

Switching to polar coordinates as $x=(r\cos\theta,r\sin\theta)$ andas in Section 2 in [27], we have
(1.17)$u(x,t)$$=$$\frac{1}{2\pi}\iint_{D},$.

$y=(\lambda \mathrm{c}\mathrm{o}\mathrm{e}(\theta+\psi), \lambda\sin(\theta+\psi))$$\lambda d\lambda dS\int_{-\varphi}\varphi 1\theta F(\lambda\xi, S)Kd\psi$$+ \frac{1}{2\pi}H(t-r)\iint_{D^{j}},$$\lambda d\lambda d_{S}\int_{-\pi}^{\pi_{\partial^{b}F}}(\lambda\xi, s)K1d\psi$,

96
where $H$ is the Heviside function and we have set$\xi=$$(\cos(\theta+^{\psi}),\sin(\theta+\psi))$,$=K_{1}(\lambda, s,\psi;r,t)=\{(t-S)^{2}-r-\lambda 22+\mathrm{r}\lambda\cos\psi\}^{-\frac{1}{2}}2$$K_{1}$

for$\varphi=\varphi(\lambda, s;r,t)=\arccos[\frac{r^{2}+\lambda^{2}-(\mathrm{t}-s)^{2}}{2r\lambda}]$,.

$(\lambda, s)\in D’$

Moreover, the domains $D’$ and $D”$ are defined as follows:$D’$$=\{(\lambda, S)\in(0, \infty)\cross(0, \infty) :$D”$$=\{(\lambda, S)\in(0, \infty)\cross(\mathrm{o}, \infty)$

where(1.18)$\lambda_{-=}|t-.s-r|$,0<s<t, \lambda_{-}<\lambda<\lambda_{+}\}$: $0<s<t-r,$,$0<\lambda<\lambda_{-\}}$,$\lambda+=t-s+r$

The key point to get such estimates as in Proposition 1. 2 is to integrate by parts withrespect to and . Following [27]and [ , we shffi sketch this procaes briefly. To begin$D”$:
into subregions as
with, we split the regions of integration $D$ and$\lambda$

$3|$

$s$

$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$D’=blue\cup white$ ,(1.19)$blue=$

black

{

$=$

$(s,$ $\lambda)\in D’${$D”=black\cup red$:$(s,$ $\lambda)\in D’$$\lambda_{-}<\lambda\leq\lambda_{-}+\delta_{1}$:$\lambda_{-}-\delta_{2}\leq\lambda<\lambda_{-}$

oror$\lambda+-\delta_{1}\leq\lambda<\lambda_{+}$$0<\lambda\leq\delta_{2}$},},where we have set $\delta_{1}=\min\{r, 1/2\}$ and $\delta_{2}=\min\{(t-r)/2,1/2\}$ . Notice thta white is
empty, if $0<r\leq 1/2$ and that red is empty, if $0<t-r\leq 1$ .
in (1.17). Then, according to the above decompositions, weLet
have$\partial^{b}=\partial_{\alpha}(\alpha=0,1,2)$(1.20)$2\pi u(x,\mathrm{t})$$= \int I_{u_{ue}}\lambda d\lambda dS\int-\varphi s(\partial_{\alpha}F)(\lambda\xi,)\varphi K1d\psi$$+H(r- \frac{1}{2})j0\sum_{=}^{1}\int\int white\lambda d\lambda d_{S}\int_{0}^{1}(\partial_{\alpha}F)(\lambda_{-}--j’ s)K_{2}d\mathcal{T}$$+H(t-r) \int\int_{uack}\lambda d\lambda ds\int_{-\pi}^{\pi}.(\partial_{\alpha}F)(\lambda\xi, s)K_{1}d\psi$$+H(t-r-1) \int\int_{r\epsilon d}\lambda d\lambda d_{S}\int_{-\pi}^{\pi}(\partial_{\alpha}F)(\lambda\xi, S)K1d\psi$,

where we have changed the variable as $\psi=\Psi$ in the second term and set$\Psi$$—j$$K_{2}$$=\Psi(\lambda,s,\tau;r,t)=\arccos[1-(1-\infty \mathrm{s}\varphi)\tau]$,$=—j(\lambda_{S,\mathcal{T}},;r,t)=(\cos(\theta+(-1)j\Psi),\sin(\theta+(-1)^{j}\Psi))$,$=K_{2}( \lambda, S,\mathcal{T};r,t)=\{2r.\lambda \mathcal{T}(1-\mathcal{T})(2-(1-\mathrm{c}\mathrm{o}\mathrm{e}\varphi)\mathcal{T})\}-\frac{1}{2}$.

Carrying out the integration by parts in the second and fourth terms, we get the following:

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Lemma 1. 1

Let $u(x, t)$ be thetowith the$|u(t,X)|$, thenis dominated by the followings:$solut\tilde{i}_{on}$initial data.

$\square _{i}u=\partial_{\alpha}F$$.ze\mathfrak{w}$$F\in C^{\infty}(\mathrm{R}^{2}\cross[0,T))$If,$I_{1}(F)(X, t)$$= \int\int_{u_{ue}}\lambda d\lambda dS\int_{-\varphi}\varphi|(\partial_{\alpha}F)(\lambda\xi, S)|K_{1}d\psi$$I_{2}(F)(X;t)$$= \int_{\partial(e}whit)d\lambda\sigma\int_{0}^{1}|F(\lambda---j, S)|K_{2}d\tau$$I_{3}(F)(X,t)$$= \int\int_{whit}\mathrm{G}d\lambda ds\int^{1}0--_{j},$$s|+|(\Omega F)(\lambda_{-}--_{j}, S)\{|F(\lambda^{-})|\}K_{2}d\mathcal{T}$$I_{4}(F)(_{X}, t)$$= \int\int_{white}\lambda d\lambda dS\int_{0}^{1}|F(\lambda^{-}--j, s)|\{|\partial_{s}K_{2}|+|\partial_{\lambda}K_{2}|\}d_{\mathcal{T}}$$I_{5}(F)(x, t)$$= \int\int_{whte}|.\lambda d\lambda ds\int_{0}1|(\Omega F)(\lambda_{-}--_{j}, s)|K_{2}\{|\partial_{s}\Psi|+|\partial_{\lambda}\Psi|\}d\tau$$J_{1}(F)(_{X}, t)$$= \int\int_{\mathrm{H}ack}\lambda d\lambda ds\int_{-\pi}\pi||(\partial_{\alpha}F)(\lambda\xi, S)K_{1}d\psi$$J_{2}(F)(_{X},t)$$= \int_{\partial\langle_{t}e}d)\lambda d\sigma\int_{-\pi}\pi|F(\lambda\xi, s)|K_{1}d\psi$$J_{3}(F)(x,t)$$= \int\int_{\mathrm{r}ed}d\lambda dS\int_{-\pi}\pi|\{|F(\lambda\xi, s)|+|(\Omega F)(\lambda\xi, s)\}K_{1}d\psi$$J_{4}.(F)(x,t)$$= \int\int_{ted}\lambda d\lambda ds\int_{-}^{\pi}\pi\psi|F(\lambda\xi, S)|\{|\partial\epsilon K1|+|\partial_{\lambda}K1|\}d$

,
,

,
,

,

,
,

.

Proof: It is easy to see that the first and second terms in (1.20) are dominated by $I_{1}(F)$and $J_{1}(F)$ , respectively. Since,$( \nabla F)(\lambda\xi, \mathit{8})=\xi\partial\lambda(F(\lambda\xi)s))-\frac{\xi^{\perp}}{\lambda}(\Omega F)(\lambda\xi, S)$$\xi^{\perp}=(\sin(\theta+\psi), -\cos(\theta+\psi))$,

we find that the fourth term in (1.20) is dominated by $J_{j}(F)(j=2,3,4)$ , by integration
by parts.
To deal with the second tem in (1.20), we use the following identities:$(\partial_{S}F)(\lambda---_{j}, S)=\partial_{s}(F(\lambda-_{\mathrm{j}}--, s))-(-1)j\partial s\Psi(\Omega F)(\lambda_{-}^{-_{j}}-, s)$,$( \nabla F)(\lambda^{-}--j’ S)=---j(\partial\lambda(F(\lambda_{-}^{-}-j’ S))-(-1)j\partial\lambda\Psi(\Omega F)(\lambda_{-j}--,s))-\frac{---j\perp}{\lambda}(\Omega F)(\lambda^{-}--j, S)$,. Again by integration by parts, we findthat the second term is dominated by $I_{j}(.F)(j=2, \cdots , 5)$ . The proof is complete.
For the further d\‘etails of the proof of Proposition 1. 2, see the proof of Propositionin [15].
Here we prepear the following proposition for the latter use. For the proof, see e.g. theproof of Proposition 4.4 in [15].
$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}^{-\perp}-_{j}-=(\sin(\theta+(-1)^{j}\Psi), -\cos(\theta+(-1)^{j}\Psi))$$4.3\square$

Proposition 1. 3 Let $u(x, t)$ be the solution to$0\leq\mu<1/2$ . Then we have forwith $t<T$$\square _{:}u=F$

urith the zero initial data. Let$(|x|,t)\in\Lambda_{i}$(1.21)$(1+t+r)^{\mu}|u(x,t)|\leq CM_{0}(F)(r,t)$where $M_{0}(F)(r, t)$ is defined in (1.14).

,

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Application of the Basic estimates. We now tem our attention to the origin4

problem (0.1) and (0.2). Since $p_{c}=3$ for $n=2$ and the higher order terms inare harmless as long as we consider the small solution, we will assume that is cubic with, namely,andrespect to$P^{\dot{\mathrm{B}}}(\partial u, \partial^{2}u)$$F^{i}$$\partial u$(1.22)$\partial^{2}u$$F^{\dot{*}}(\partial u, \partial^{2}u)$$=j,k,l= \sum_{1\alpha,\rho,0}^{m}\cdot\cdot\sum_{\gamma,\delta=}^{2}A_{ij}^{\alpha}\rho_{l}\gamma\delta\partial_{a}ku\partial jk\partial\partial u\iota\rho u\gamma\delta$$+ \sum_{j,k,l=1\alpha,\beta,\gamma 0}^{m}\sum_{=}B\alpha\beta\gamma\partial_{\alpha}ku2.\partial j\partial u|\mathrm{j}l\rho u^{k.l}\gamma$

’

where

$A_{1jkl}^{\alpha\beta\gamma}.\delta$

and$B_{ijkl}^{\alpha\rho_{\gamma}}$are constants and(1.23)$A_{ijk\iota}^{\alpha\beta\gamma\delta}$

satisfy$A_{1jkl}^{\alpha\beta\gamma\delta}.=A^{\alpha}l\mathrm{j}ki\rho\gamma\delta A_{ijk\iota}^{\alpha}\beta\delta\gamma=$.

As we have mentioned before, in order to get a global solution, we need to establish
for some large integer $N$ . By (1.7) and (0.1),
a uniform a priori estimate ofsatisfies$1^{\partial u}(t)]_{N}$$\Gamma^{a}\partial^{b}u^{i}(x, t)$(1.24)$\square _{:}\Gamma^{a}\theta u^{1}(X,t)=\tilde{F}^{i}(\partial u,\partial^{2}u)$in$\mathrm{R}^{2}\cross(0,T)$

and
where we have set
are determined byMoreover, the initial vfiues of$a=0$ andinstance,(0.1).
whenFor
by using the equation$\tilde{F}^{i}(\partial u, \partial^{2}u)=\sum_{d\leq a}C_{a},b\theta\Gamma dF^{i}(\partial u, \partial 2u)$$\Gamma^{a}\theta u^{i}(x,t)$$a,$$\epsilon,$,

and are multi-indices.
and $g^{j}(j=1, \cdots, m)$, we have$d$$b$$f^{j}$$\partial^{b}=\partial_{t}$$(\partial_{t}u^{i})(X, 0)=\epsilon g^{\dot{*}}(_{X})$,$(\partial_{t}^{2}u^{i})(X,0)=\epsilon C^{2}i\Delta fi(_{X)}+Fi(\partial u, \partial 2u)(x,0)$We can solve the second equation with respect tosmall.
as follows:
Based on thi.s, we decompose$(\partial_{t}^{2}u^{i})(x, 0)$, if$\delta$.

in (1.6) is sufficiently$\Gamma^{a}\partial^{b}u(x,t)$(1.25) $\Gamma^{a}\partial^{b}u(x,t)=u_{0}(x,t)+u_{1}(x,t)$

is a solution to
where
and
solution to$u_{1}^{i}$$\coprod_{i}u_{0}^{\dot{t}}=0$$f^{j}(x),$$g^{j}(x)\in C_{0}^{\infty}(\mathrm{R}^{2})$when $|a|+|b|\leq N$ ,(1.26)

with$u_{0}=(u_{0}^{1}, \cdots , u_{0}^{m}),$$\square _{i}u\dot{\mathrm{i}}=\tilde{F}^{1}(\partial u, \partial^{2}u)$with the zero initial data, while$u_{0}^{i}(x,\mathrm{o})=(\Gamma^{a}\theta u)(x, \mathrm{o}),$, the initial values of$u_{0}^{i}$$u_{1}=(u_{1}^{1}, \cdots, u_{1}^{m})$$\partial_{t\%^{i}}(x, \mathrm{o})=(\partial_{t}\Gamma^{a}\partial bu)(x,0)$

are also belongs to$C_{0}^{\infty}(\mathrm{R}^{2})$,$u_{0}^{*}$

is a. Since. Therefore,

we have$|u_{\mathrm{o}(x,t}^{i})|\leq M_{N}\epsilon(1+t+r)^{-\frac{1}{2}}(1+|\mathrm{q}t-r|)^{-\frac{1}{2}}$for$(x,\mathrm{t})\in \mathrm{R}^{2}\cross[0, \infty)$,, and their finite times derivatives. (See Lemma
depends on -norms of
where. Glassey [10], and also Lemma 4 in [27] and [32] .
1 inestimates.
Then we have the following$M_{N}$$L^{1}$$f^{j}$$g^{j}$$)$$\mathrm{R}.\mathrm{T}$$L^{\infty}- L^{\infty}$

be a solution of (0.1) and (0.2). Suppose
Proposition 1. 4 Let
be the constant in (1.26).
that (1.22) and (1.23) hold. Let(i) There is a sufficiently small positive number such that if (1.6) with $k=[(N+2)/2]$
urith $t<T$ and $|a|\leq N$
holds, then we have for$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0, T))$

$M_{N}$$\delta$$(|x|,t)\in\Lambda_{i}^{c}$(1.27)$w_{*}(|x|,t)| \Gamma^{a}\partial u^{i}(x,t)|\leq M_{N}\epsilon+c_{N}[\partial u]^{2}[\frac{N+2}{2}],t||\partial u||_{N+}4,\iota$.

99(ii) Let $0\leq\mu<1/2$ . There is a sufficiently small positive number such that if (1.6) urith$k=[(N+1)/2]$ holds, then we have forwith $t<Tand|a|\leq N$,$\delta$$(|x|,t)\in.\Lambda_{i}$(1.28)$(1+t+r)^{\mu}|\mathrm{r}_{u^{i}}^{a}(X,t)|$$\leq$.

$M_{N}\epsilon+C_{N}[\partial u]_{[\frac{N+1}{2}1t}^{2},||\partial u||N+3,t$Proof: First we shall show (1.27). Using the decomposition (1.25) with $|b|=1$ and theestimates (1.26) and (1.14), we have
(1.29) $w_{i}(r,t)|\mathrm{r}^{a}\partial u^{*}(X,t)|\leq M_{N}\epsilon+C_{N}M!a1+1(\dot{p})(r,t)$ for$(|x|,t)\in\Lambda_{i}^{c}$with $t<T$.

Therefore, it suffices to show
(1.30)$M_{N+1}( \dot{P})(r,t)\leq C_{N}[\partial u]_{[}^{2}\frac{N+2}{2}],t||\partial u||N-\mu,t$.

It follows that for $|a|\leq N+1$.

(1.31)$| \Gamma^{a}\dot{P}(y, S)|\leq C\sum_{j,k,l=1}^{m}\frac{1}{(w_{j}w_{k})(|y|,s)}[\partial u(S)]\iota 2\sum_{1}\frac{N+2}{2}1|b\mathrm{I}\leq|a1+|\Gamma^{b}\partial ul(y, S)|$

Employing the following imbedding theorem(1.32)

for$|x|^{\frac{1}{2}}|f(x)| \leq\sum_{|a|\leq 2}||\Gamma^{a}f||\iota 2$$x\in \mathrm{R}^{2}$(for the proof, see e.g. Lemma 6 in [27]), we get (1.30).
Moreover, we can prove (1.28) in a similar fashion, if we use (1.21) instead of
The proof is complete.

Now we are in a position to derive aestimate for any
By (1.25) with $|b|=1,$ $(1.26)$ and (1.15), it suffices to control
with $s<t$ , we have from (1.31) and (1.32)$L^{\infty}- L^{\infty}$$(1.14)\square$$(x, t)\in \mathrm{R}^{2}\cross[0, T)$$M_{N1}^{(i)}+(F^{i})(r,t)$.
.

. When$(|y|, S)\in\Lambda_{i}^{c}$.

(1.33)$|y|^{\frac{1}{2}}z^{11)}(|y|, \mathit{8})|\Gamma aFi(y, S)|\leq C_{N}[\partial u]_{[\frac{N+\mathit{2}}{2}1t}^{2},||\partial u||N+4,t$In order to deal with the case where
as follows:(1.34)$N^{i}(\partial u, \partial^{2}u)$(1.35)$R^{i}(\partial u,\partial^{2}u)$$(|y|, S)\in\Lambda_{i}$with $s<t$ , we divide$F^{:}$

into$N^{\dot{*}}$$=, \sum_{a,\rho_{\gamma,\delta}=0}^{2}A_{ii}^{\alpha}.\beta.\cdot\gamma\delta\partial\alpha u*|\partial\rho u\partial_{\gamma}i\partial_{\delta}u^{i}+\sum_{\alpha,\beta,\gamma=0}^{2}B_{i}^{a}\rho_{i}\gamma\partial_{\alpha}iiu^{i}\partial\rho u\partial*u^{1}\gamma$$=\dot{P}(\partial u, \partial^{2}u)-N^{i}(\partial u, \partial^{2}u)$and$R^{*}$’.

Firstly, we shall show for $|a|\leq N+1$
(1.36),$|y|^{\frac{1}{2}}z(i)(|y|, S)|\mathrm{r}^{a}\dot{H}(y, S)|\leq C_{N}\delta^{2}M_{N}\epsilon+C_{N}[\partial u]_{\iota_{\frac{N+4}{2}],t}}^{2}||\partial u||N+6,t$

provided (1.6) with $k=[(N+4)/2]$ . Since there is at least one index among $j,$which does not coincide with in each term of , we have$i$$|\Gamma^{a}R^{i}(y, \mathit{8})|$

$k$

and

$R^{1}$$\leq$$C \sum_{(j,k)\neq.(1*)}.,\cdot\sum^{m}l=1\frac{1}{(w_{\mathrm{j}}w_{k})(|y|,s)}1\partial u(S)1[2\sum_{1}a_{2}\perp+\underline{1}]||b1\leq a|+1\mathrm{r}^{b}\partial ul(y, s)|$$+C \sum_{)(j,k),\iota\overline{\neq}}.\cdot.,.\cdot\frac{1}{(w_{i}w_{l})(|y|,s)}.\frac{1^{\partial u}(S)]_{\iota\frac{|a|+1}{2}:}2}{.\cdot w_{i}(|y|,S)}-(\cdot b|\leq 1\sum_{|a|+1}w\iota(|y|, S)|\mathrm{r}b\partial u^{l}(y, s)|$

.

$l$

100
By (1.32), (1.6) with $k=[(N+2)/2]$ and $w_{i}(|y|, s)\geq|y|^{1/2}$ , we get$|y..|^{\frac{1}{2}}z^{(i}.()|y|,$$s)|\mathrm{r}aR.i(y,s:.).|$

Moreover, since$\Lambda_{i}\subset\Lambda_{l}^{c}$for.

$|a|\leq N+1$$\leq C[\partial u(S)]^{2}[\frac{N+\mathit{2}}{2}1||\partial u(S)||N+4+c\delta 2\sum_{+|b1\leq N2}w\iota(|y|,S)|\Gamma^{b}\partial u^{l}(y, s)|$.

by (1.13), we get from (1.27),$w\iota(|y|, s)|\mathrm{r}b\partial u\iota(y, s)|\leq MN\epsilon+oN[\partial u]_{[\frac{N+4}{2}],t}^{2}|[\partial u||_{N}+6,t$for$(|y|, S)\in\Lambda_{i}$with $s<t$ and $|b|\leq N+2$ , provided

we get (1.36).
In order to treat$N^{:}$, we need a concept of

Null condition. The idea to deal with
(1.37)$\partial_{1}u^{*}=$(1.38)$\ u^{i}=$(1.39)$\mathrm{R}^{-}u^{:}=$$‘(\mathrm{N}\mathrm{u}\mathrm{l}1$$N^{i}$$(1.6)_{\mathrm{W}}.\mathrm{i}\mathrm{t}\mathrm{h}k=[(N+4)/2]$. Hence,condition” below.

is to decompose$\partial_{a}u^{i}$

as follows:,$\frac{x_{1}}{r}\partial,u^{i}-\frac{x_{2}}{r^{2}}\Omega u^{i}$$\frac{x_{2}}{r}\partial_{f}u^{i}+\frac{x_{1}}{r^{2}}\Omega u^{:}$$-c_{t} \partial_{t}u^{i}+\frac{c_{i}t-r}{t}\partial_{f}u^{i}+\frac{1}{t}Su^{:}$,

where $r=|x|,$and.
First we handle the last trems in the right hand saide of (1.37), (1.38) and (1.39). If we
set for a nonnegative integer$\Omega=X_{1\mathrm{a}\partial_{1},S}-x2=t\partial_{t}+r\partial_{t}$$\mathrm{R}=\partial_{t}$$k$(1.40)$\langle u(t)\rangle_{k}=\sum_{1l=1}^{m}\sum_{x ,a1\leq k\in}2\sup_{\{\mathrm{R}:(x,t)\in\Lambda:\}}|\Gamma^{a}u^{l}(x,\mathrm{t})|$,we see that those terms are dominated by$C(1+t+r)^{-1}\langle u(t)\rangle_{1}$Since$\langle u(t)\rangle_{1}$$t+r)^{-1}$.

for$(|x|, \mathrm{t})\in\Lambda_{i}$.

can be estimated by using (1.28), we get an additional decaying factor $(1+$Nest we consider the second trem in the right hand saide of (1.39). Since$\frac{|c_{i}t-r|}{t}\leq\frac{C(w_{i}(r,t))2}{(1+t+r)^{2}}$for$(|x|,t)\in\Lambda_{i}$we get a good decaying factor $(1+t+r)^{-1}$ instead ofbom the above consideration, we can approximate(1.41)$\partial_{\alpha}u^{i}\sim\omega_{\alpha}\partial_{r}u^{i}$where we have set
get$\omega_{0}=-\mathfrak{g},$$\omega_{1}.=x_{1}/r$

and,$(1+|\mathrm{q}\mathrm{t}-r|)^{-1}$$\partial_{\alpha}u^{i}$

as follows:

ffom this term.

,$\omega_{2}=x_{2}/r\sim$. Substituting (1.41) into (1.34), we.

$N^{i}( \partial u, \partial^{2}u)\sim\sum_{\delta\alpha,\rho,\gamma,=0}^{2}A\{iii\omega\alpha\omega\beta\omega_{\gamma}\omega\delta(a\beta\gamma\delta\partial_{\mathrm{r}}u^{i})2\partial,2+u^{i},\rho_{\gamma},=0\sum_{\alpha}^{2}B_{iii\alpha}^{\alpha}.\beta\gamma\omega\omega\beta\omega_{\gamma}|(\partial \mathrm{f}u^{i})3$

101
In order to kill these terms, we naturally arrive at the following condition.
Definition. We fix $i(i=1, \cdots, m)$ . When (1.12) holds, we sa.y that
atisfies .he “Nullcondition”, if$\mathrm{t}$$P.\mathrm{s}$(1.42)$\sum_{\alpha,\beta,\gamma,\delta=0}^{2}A_{i}\alpha_{i}\rho\gamma\delta x_{\alpha}iiX_{\beta\gamma\delta}XX=0$

and$\sum_{\alpha,\beta,\gamma=0}^{2}B.\alpha\rho\gamma Xii\alpha X\rho X_{\gamma}=0*i$hold on the hyper-surface $(X_{0})^{2}-di\{(X_{1})^{2}+(X_{2})^{2}\}=0$ .
In addition, when
, we say. that $P$ satisfies the
any $j,$ $l(j, k,.l=1, \cdots,m)$$c_{1}=\cdots=\mathrm{q}_{n}$($‘ \mathrm{N}\mathrm{u}\mathrm{l}1$

condition”, if for$k,$(1.43)$\sum_{a,\beta,\gamma,\delta=0}^{2}4_{j\beta}^{\alpha\rho\gamma}k\mathrm{t}\mathrm{x}\alpha \mathrm{x}\mathrm{x}_{\gamma}X_{\delta}=0\delta$

and$\sum_{a,\beta,\gamma=0}^{2}B_{\dot{\}}\alpha\rho\gamma xxkl\rho Xj\alpha\gamma=0$

hold on the hyper-surface $(X_{0})^{2}-c^{2}\{i(X_{1})2+(X_{2})^{2}\}=0$.
By assuming thatsatisfies the null condition (1.42), we obtain$\dot{P}$$| \Gamma^{a}N^{i}(\partial u^{i}, \partial 2u)i(x,t)|\leq\frac{C}{(1+t+r)^{2}}[\partial u(t)]_{[\frac{|a|+1}{2}}2\sum_{1}1|b1\leq|a|+|\partial \mathrm{r}^{b}u(:x,t)|$$+ \frac{C}{(1+t+r)w_{i}(r,t)}[\partial u(t)]1\frac{|a|+1}{2}](u(t)\rangle_{\mathrm{I}a|+2}\sum_{||b\leq|a\mathrm{I}+1}|\partial\Gamma^{b}u(_{X}:,)t|$f\‘Or$(|x|, t)\in\Lambda_{:}$

havewith $t<T$ . (See also the proof of Proposition 3.1 [15]). Using (1.32), we$|x|^{\frac{1}{2}}z((i)|_{X|,t})|\mathrm{r}a_{N}:(\partial u^{;2}, \partial u^{i})(_{X},t)|\leq C[\partial u(t)]_{[\frac{1a|+1}{2}]}2||\partial u(t)||_{|\mathrm{I}+3}a$,$+C[\partial u(t)]\iota_{\frac{\mathrm{I}a|+1}{2}]}\langle u(t)\rangle|a\mathrm{I}+2||\partial u(t)||_{|a|+3}$

hence, by the aid of (1.28), we get(1.44)$|y|^{\frac{1}{2}(i)}Z(|y|, s)|\mathrm{r}a_{\dot{W}(}y,$$S)| \leq C_{N}\delta M_{N^{\xi||}}\partial u||_{N+}4,t+C_{N[1^{2}}\partial u\iota\frac{N+4}{2}],t||\partial u||2N+a,t$

’

for
with
and $|a|\leq N+1$ , provided (1.6) with $k=[(N+4)/2]$ . As aconclusion, it follows from (1.33), (1.36) and (1.44) that$(|y|, S)\in\Lambda_{\dot{\mathrm{t}}}$$s<\mathrm{t}$Proposition 1. 5Letbe a solution of (0.1) and (0.2). Supposethat (1.22), (1.23) and (1.42) hold. Let
be the constant in (1.26). Then there is asufficiently small number $\delta=\delta(n)>0$ such that if (1.6) with $k=[(N+4)/2]$ holds, thenwe have for $(x, t)\in \mathrm{R}^{2}\cross[0, T)$ and $|a|\leq N$$u^{*}\in C^{\infty}(\mathrm{R}^{2}\cross[0, T))$

$M_{N}$(1.45)$w_{i}(|X|,t)| \Gamma^{a}\partial u^{:}(x,t)|\leq MN(\frac{3}{2}+\frac{1}{2}||\partial u||_{N+}4,t)\mathit{6}+cN[\partial u]_{[}^{2}\frac{N+4}{2}1,t||\partial u||^{2}N+6,t$.

Note that we have from Propositions 1.1 with $p=3$(1.46),$||\partial u||_{N,t}\leq C_{N}\epsilon(1+t)^{C_{N}[}\partial u121*1t\mathrm{l}$for $0\leq t<T$ and the solution
to (0.1) and (0.2), provided (1.6)
with $k=[(N+1)/2]$ holds. Combining this estimate with (1.45) and (1.28), we obtain thefollowing.
$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0,T))$

102
be a solution of (0.1) and (0.2). Suppose that
Corollary 1. 1 Let(1.22), (1.23) and (1.42) hold. Then there is a sufficiently small positive number such thatif (1.6) with $k=[(N+7)/2]holdS_{f}$ then we have for $0\leq t<T$$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0,T))$$\delta$(1.47)$[\partial u(t)]N\leq C_{N}\epsilon(1+t)\mathrm{l}\Delta \mathrm{F}c_{N[]}\partial u21$

and(1.48),‘.

$\langle u(t)\rangle_{N+1}\leq C_{N}\epsilon(1+t)^{c_{N}}1\partial u]2\mathfrak{l}*N1^{\iota},$

We thus obtain a upper bound ofin , we .eed to work a little bit.
$t$$[\partial u(t)]_{N}.,$$\mathrm{n}$.

H.O.wever, since the estimate is not uniform

Energy Estimates II. In oder to derive a uniform a priori estimate ofwithrespect to , we need the following another enrgy estimate. For the proof, see the proof of$[\partial u(t)]_{N}$$t$

Proposition 5.2 in [15].

Letbe a solution of (0.1) and (0.2). Suppose
Proposition 1. 6that (1.22), (1.23) and (1.42) hold. Then we have for $0\leq t<T$$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0,T))$(1.49)$||\partial u(\mathrm{t})||_{N}^{2}$$C_{N\{N}^{2}||\partial u(0)||^{2}+$$\leq$$+ \int_{0}^{t}(1+s)-\frac{5}{4}([\partial u(s)]^{2}N+1+(u(_{S)}\rangle^{2}N+2)||\partial u(s)||_{N1\}}^{2}+dS$

Now$\mathrm{i}\dot{\mathrm{t}}$follows$\mathrm{b}\mathrm{o}\mathrm{m}(1.49)$.

, Corollary 1. 1 and (1.46) that

Corollary 1. 2 Let
be a solution of (0.1) and (0.2) and $0<\epsilon\leq 1$ .
Suppose that (1.22), (1.23) and (1.42) hold. Then there is a sufficiently small positive
number such that if (1.6) with $k=[(N+14)/2]$ holds, then we have for $0\leq t<T$$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0, T))$$\delta$.

(1.50)$|| \partial u(t)||_{N}^{2}+6\leq C_{N}^{2}\epsilon^{2}\{1+\int_{0}^{t}(1+s)^{-\frac{5}{4}+4C1\mathrm{J}_{[\neq],\epsilon}^{2}}ds\}N\partial \mathrm{u}N1$

Main result. Now we are in a position get a uniform a priori estimate of
We fix an integer satisqing $N\geq 13$ , which guarantees $[(N+14)/2]\leq N$ . We take$[\partial u(t)]_{N}$$N$be(1.51)$0<\epsilon_{0}\leq 1$, .

,$3C_{N}\epsilon_{0}\leq 1$$3M_{N}\epsilon_{0}\leq\delta$

and$12C_{N}M_{N} \epsilon_{0}\leq\frac{1}{8}$$\epsilon_{0}$

.

to

,

where
is the constant in (1.26),
is the constant in (1.50) and is the smallest number
taken in Proposition 1.5 and Corollary 1.2. We will fix an inin the following.
Suppose that the problem (0.1) and (0.2) has a solution
satisfying. Then we have
$C_{N}$$M_{N}$$\delta$$\epsilon$$[0,\epsilon_{0})$$u^{i}\in C^{\infty}(\mathrm{R}^{2}\cross[0, T))$$[\partial u]_{N},\tau\leq 3M_{N}\epsilon$$[ \partial u1[\frac{N+14}{2}],\tau\leq 1\partial u]_{N,\tau\leq}\delta(<1)$.

Therefore, by (1.50) and (1.51), we have for $0\leq t<T$$||\partial u(t)||N+6$$\leq$$\leq$$C_{N} \epsilon(1+\int_{0}^{t}(1+s)^{-\frac{9}{8}}ds)\frac{1}{2}$

$1$

.

103
Substituting this into (1.45), we have.

$[\partial u]_{N,\tau}\leq 2MN\xi+3cNM_{N^{\mathcal{E}}}1^{\partial u}]_{1\frac{N+4}{2}],\tau}$Hence, by (1.51), we have
(1.52)$[\partial u]_{N,T_{\epsilon}}\leq 2M_{N^{\xi}}$

We thus obtain a uniform a priori estimate of.

$[\partial u]_{N}.\tau$. Now we state our main theorem.

Theorem 1. 1 ([ Hoshiga-K]) Let $n=2$ andif $i\neq j$ . Suppose that (1.1), (1.2)
with $p=3,$ $(1.3)$ and (1.42) hold. Then there nists a positive constantsuch that theinitial value problem (0.1) and (0.2) has a unique-solution in.
for$c_{i}\neq c_{\mathrm{j}}$$\epsilon_{0}$$C^{\infty}$$\mathrm{R}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}^{\mathrm{I}}$2$\mathrm{R}^{2}\cross[0, \infty)$$0<\epsilon\leq\epsilon_{0}$. Recently, K. Yokoyama extended this result for 3-dimension4 case in [41].

Semilinear CaseIn this section we study the semilinear case, namely, we assume $F^{i}=\dot{P}(u)$ . We consider
the following simple system of semilinear wave equations as a model case of the problem(0.1) and (0.2):
(2.1)
(2.2)
(2.3)
(2.4)

$\partial_{t}^{2}u-\Delta u=|v|^{p}$$\partial_{t}^{2}v-\Delta v=|u|^{q}$,$v(X,0)=\epsilon f^{2}(x)$ ,$u(x,\mathrm{O})=\epsilon f^{1}(x)$inin$\mathrm{R}^{n}\cross(0, \infty)$$\mathrm{R}^{n}\cross(0, \infty)$$\partial_{t}u(x, 0)=\epsilon g^{1}(x)$

,
,
in
in

$\partial_{t}v(x, 0)=\epsilon g^{2}(x)$$\mathrm{R}^{n}$$\mathrm{R}^{n}$,,where $1<p\leq q,$ $n=2,3,$
andis a small parameter. It is regarded
as a natural extension of the following Cauchy problem:$f^{\mathrm{j}},$

(2.5)
(2.6)

$\partial_{t}u-\Delta u=|u|^{p}$$u(x, 0)=\epsilon f(x)$where $f,$$g\in C_{0}^{\infty}(\mathrm{R}^{2})$$\epsilon>0$$g^{j}\in C^{\infty}(\mathrm{R}^{2})$,in$\mathrm{R}^{n}\cross(0, \infty)$$\partial_{t}u(x, 0)=\epsilon g(X)$,in$\mathrm{R}^{n}$,.

Known results. The problem $(2.5)-(2.6)$ has been extensively studied by many au-

thors (see, e.g., [1], [2], [5], [9], [10], [17], [31], [33], $[35]-[40],$ $[42],$ $[43]$ ). Set
(2.7)

Let

$\alpha_{0}=w^{*}-1$,$p^{*}= \frac{n-1}{2}p-\frac{n+1}{2}$.

be the positive root of the quadratic equation, namely, $p_{0}(3)=1+\sqrt{2}$ and. Note that when $p>1,$ $p>p_{0}(n)$ and $1<p<p\mathrm{o}(n)$ are equivfient to,andresepectively. Then we have$p_{0}(n)$$\alpha_{0}=0$$p\mathrm{o}(2)=(3+\sqrt{17})/2$$\alpha_{0}>0$$\alpha_{0}<0$, then [Small Data Global Enistence] holds. While, if
Theorem 2. 1 Ifthen [Small Data Blow-up] holds.
$\alpha_{0}>0$$\alpha_{0}<0$,

For the proof, see F. John [17], [20] for $n=3$ and. Glassey [9], [10] for $n=2$ . Here
we give an observation why
implies [Small Data Global Existence] based on thefollowing basic estimate.
$\mathrm{R}.\mathrm{T}$$\alpha_{0}>0$104

Proposition 2. 1
Then we have for

Let $u(x, t)$ be a solution to$\partial_{t}^{2}u-\Delta u=F$

urith the zero initial data.

$(x,t)\in \mathrm{R}^{n}\cross(0, \infty)$(2.8)$|u(x,t)|(1+r+t)^{\frac{n-1}{2}/\Phi_{\mathfrak{n}}(t}r,;\nu)$$\leq$$C \sup_{0<s<ty\in}\sup\{\mathrm{R}^{n}|y|^{\frac{n-1}{2}(+|y|)}1+S1+\nu(1+|s-|y||)^{1}+\mu|F(y, \mathit{8})|\}$

for any $\mu>0$ and $\nu>0$ . Here $r=|x|$ and we have set
(2.9)

,

$\Phi_{3}(r,t;\nu)=(1+|t-r|)^{-\nu}$and(2.10)$(1+|\mathrm{t}-r|)^{-\nu}$if
if$0<t\leq r$

$r<t$

.

$|||u|||=(x,t) \in \mathrm{R}^{n}\mathrm{x}\sup_{(0,\infty)}\{|u(x,t)|(1+t+r)^{\frac{n-1}{2}}/\Phi n(r,t;p^{*})\}$.

$\Phi_{2}(r,t;\nu)=\{$$(1+t-r)^{-\frac{1}{2}}(1+t-r)^{1^{\frac{1}{2}}\nu}-1+$,

In addition, $[a]_{+}= \max\{a, 0\}$ and $A^{[01+}=1+\log A$ .
We introduce the norm of $u(x,t)$ as follows:(2.11)
Then, if$\alpha_{0}>0$(2.12), we get$\sup_{0<s<ly\mathrm{R}^{n}}\sup_{\in}\{|y|^{\frac{n-1}{2}(1}+S+|y|)^{1+\mathrm{P}^{\mathrm{s}}}(1+|s-|y||)^{1+\mu}|u(y, \mathit{8})|^{p}\sim\}\leq C|||u|||^{p}$.

Indeed, this estimate immidiately follows, if we choose such that $0<\mu<\alpha_{0}$ . Having theso that a a priori estimate
estimate (2.8) with $\nu=p^{*}$ in mind, we need to assume
holds.
$\mu$$\alpha_{0}>0$

Next we consider the critical case$\alpha_{0}=0$, namely, $p=n(n)$ .

, as for the life
, then [Small Data Blow-up] holds.
Theorem 2. 2 If
spanof classical solutions of $(2.5)-(2.6)$ , there nist positive constants and $B$ suchthat$\exp(A\epsilon^{-p(p1}-))\leq T(\epsilon)\leq\exp(B\epsilon^{-p(-})p1)$.
(2.13)$Mo7\mathrm{e}over$$\alpha_{0}=0$$A$$T(\epsilon)$

For the proof, see J. Schaeffer [35], Y. Zhou [42] and [43], and also H. Takmura [36].

The$rq\mathrm{s}\mathrm{y}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{m}$. We tum our attention to the Cauchy problem$\Gamma=\alpha+p\beta$,$\alpha=pq^{*}-1$$p^{*}= \frac{n-1}{2}p-\frac{n+1}{2}$,,$\beta=qp^{*}-1$$(2.1)-(2.4)$

. Set

,

$q^{*}= \frac{n-1}{2}q-\frac{n+1}{2}$.

Then we have
Theorem 2. 3
If $\Gamma>0$ and $0<p^{*}\leq q^{*}$ , then [Small Data Global Existence] holds.
While, if $\Gamma<0$ , then then [Small Data Blow-up] holds.

andFor the proof, see D. Del Santo, V. Georgiev and E. Mitidieri [7] for the case.
K. Deng [8] for the caseand ,In the following, we consider the critical case $\Gamma(p,q,n)=0$ . Here, given
as the supremum of all such that a -solution of $(2.1)-(2.4)$
we define the life span
and $0\leq t<T$ . Then our main result of this section is as follows.
exists for all$\Gamma\neq 0$$\Gamma<0$$p,$ $q,$$T^{*}(\epsilon)$$x\in \mathrm{R}^{2}$$T$$C^{2}$$f_{\mathrm{j}}$$g_{j}$

105
Theorem 2. 4 ([ $K$-Ohta]) Let $1<p\leq q,$ $n=2,3,$ $\Gamma(p, q,n)=0,$. Assume that$g_{j}\in C^{\infty}(\mathrm{R}^{2})$and$0<\epsilon\leq 1$(2.14)$f_{j}(X)=0$,$g_{j}(x)\geq 0$$(x\in \mathrm{R}^{2},j=1,2)$,$g_{2}(0)>0$.

Then the classical solution of $(2.1)-(2.4)$ blows up in a finite
and there nists a positive
constant $C$ , independent of such that the lifeof the classical solution of $(2.1)-$(2.4) satisfies(2.15)$(0<\epsilon\leq 1)$
if $p<q$ ,$time_{J}$$\mathit{8}panT^{*}(\epsilon)$$\epsilon_{J}$$T^{*}(\epsilon)\leq\exp(,C\epsilon-p\omega-1))$(2.16)

if $p=q$ .

$(0<\epsilon\leq 1)$$T^{*}(\epsilon)\leq\exp(c_{\epsilon^{-p}}\mathrm{t}p-1))$

Remark. For 3-dimensional case, Y. Kurokawa and H. Tdfflmura in [32] obtained thesame estimates as in (2.15) and (2.16) and the lower bounds of the life span, independently.

Key lemma. The key point of the proof of Theorem 2. 4 is to reduce the blowup
problem for $(2.1)-(2.4)$ to that for a system of integral equations $(2.17)-(2.18)$ below.
(2.17)$\varphi(z)\geq 1+\gamma\lambda^{a}\int^{z}0(1-e^{-\lambda}(z-\zeta))e^{\alpha\lambda}|\phi(\zeta)|^{p}\zeta d\zeta$(2.18)

where$\phi(z)\geq\gamma\lambda^{b}\int_{0}^{l}(1-e^{-\lambda}(z-\zeta))e\rho_{\lambda}\zeta|\varphi(\zeta)|^{q}d\zeta$

$a,$

$b,$

$p,$ $q,$$\alpha,$$\beta,$$\gamma$and(2.19)$\lambda$

be constants$1<p\leq q$,$(z\geq 0)$,$(z\geq 0)$,$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\Psi\dot{\mathrm{m}}\mathrm{g}$$\beta\geq 0$,$\gamma>0$,$\lambda\geq 1$.

Then we have

Lemma 2. 1 Let(2.18). Assume that either$(\varphi(Z), \phi(Z))$(2.20)

be a solution of the system of integml inequalities (2.17)-$a+p(b-1)\geq 0$ ,$\alpha+p\beta\geq 0$

,

$or$

(2.21)$a+pb\geq 0$

Then the life span of
only onand .
$p_{f}q_{f}\beta$$(\varphi(z), \phi(Z))$,$\alpha\geq 0$.

is bounded fiom above by a positive constant depending$\gamma$Lemma 2. 1 is a key lemma to prove Theorem 2. 4. The terms
andwith
and $\beta>0$ in $(2.17)-(2.18)$ , which never appear in the problem for the single equation $(2.5)-$(2.6), make the problem difficult. And our method is also applicable to the problem forthe single equation and the assumptions (2.20) and (2.21) lead the different estimates ofthe life span (2.15) and (2.16) in the theorem. For the further details of the proof of thetheorem, see [30].
$e^{\alpha\lambda\zeta}$$e^{\beta\lambda\zeta}$$\alpha<0$

106
Proof of the key lemma. Here we give the proof of Lemma 2. 1. In what follows,

we always assume that
Lemma 2. 2$(\varphi(z), \phi(z))$

is a solution of $(2.17)-(2.18)$ under the condition (2.19).

Assume (2.19). Let $A>0,0<h\leq 1$ and $Z\geq 0$ . Suppose that

(2.22)$(z\geq Z)$$\varphi(z)\geq A$

Then there nists a positive constant$C_{4}$

depending only on(2.23)$\phi(z)\geq C_{4}A^{q}h^{2\triangleright 1\beta\lambda z}\lambda e$(2.24)$\phi(z)\geq C_{4}A^{a}h^{2}\lambda b$

Proof: First we show (2.23). Since.that for $z\geq Z+h$(2.25)$\phi(z)$.

$\lambda\geq 1$

and$\beta$such that$\gamma$$(z\geq Z+h)$ ,$(z\geq Z+h)$.

and $0<h\leq 1$ , it follows from (2.18) and (2.22)$\geq\gamma\lambda^{b}\int_{z-}^{z_{h/\lambda}}(1-e^{-\lambda \mathrm{t}z}-\zeta))eA\rho_{\lambda}\zeta qd\zeta$$\geq\gamma\lambda^{b}e^{\beta\lambda}-h\lambda)A(z/q\int_{z}^{z}-h/\lambda(1-e-\lambda(z-\zeta))d\zeta$.

$\geq\gamma e^{-\beta}A^{q}\lambda^{b}e\rho_{\lambda}z\int_{z-h/\lambda}^{z}(1-e^{-\lambda(z-\zeta)}.)d\zeta$

Thus (2.23) follows from (2.25) and the following fact:$(0<h\leq 1)$$\int_{z-h/}^{z}\lambda(1-e^{-\lambda 1-\zeta)})zd\zeta=\frac{e^{-h}-1+h}{\lambda}\geq\frac{h^{2}}{e\lambda}$

Next we show (2.24). Again(2.26)$\mathrm{h}\mathrm{o}\mathrm{m}(2.18)$$\phi(z)$.

and (2.22) we have for $z\geq Z$$\geq\gamma\lambda^{b}\int_{Z}^{z}(1-e^{-\lambda}(z-\zeta))e\rho\lambda\zeta A^{q}d\zeta$.

$\geq\gamma\lambda^{b}A^{q}\int_{Z}z-(1e-\lambda(z-\zeta))d\zeta$

Thus (2.24) follows from (2.26) and the fact that for $z\geq Z+h$ we have$(2.27) \int_{z}^{z}(1-e^{-\lambda(z-\zeta)})d\zeta\geq\int_{z-h}^{z}(1-e^{-(-\zeta)})zd\zeta=e^{-h}-1+h\geq\frac{h^{2}}{e}$This completes the proof.
Lemma 2. 3

(2.20)

and$(z\geq Z)$$\phi(z)\geq B\lambda^{b1\beta\lambda z}-e$$or$(2.21)(2.29)

Then there(2.30)(2.31)$w\dot{\kappa}ts$.
$\square$

Let $B>0_{f}0<h\leq 1$ and $Z\geq 0$ . Suppose that either(2.28)$(0<h\leq 1)$

anda positive constant$C_{6}$$\phi(z)\geq B\lambda^{b}$$(z\geq Z)$

depending only on$\varphi(z)\geq C_{5}B^{p}(z-z-1)$$\varphi(z)\geq C_{5}B^{p}h^{2}$$\gamma$,.

such that$(z\geq Z+1)$ ,$(z\geq Z+h)$.

107
Proof: Under the assumption (2.28) or (2.29), we show
(2.32)$\varphi(z)\geq\gamma B^{\mathrm{P}\int_{z}^{z}}(1-e^{-\lambda(z}-\zeta))d\zeta$$(z\geq Z)$.

First we assume (2.28). It follows from (2.17) and (2.28) that for $z\geq Z$(2.33)$\varphi(z)$$\geq\gamma\lambda^{a}\int_{Z}^{z}(1-e^{-\lambda}-\zeta))(zeB\alpha\lambda\zeta p\lambda p(b-1)p\rho\lambda\zeta ed\zeta$.

$= \gamma B^{\mathrm{P}}\lambda^{a}+p(b-1)\int_{Z}^{z}(1-e-\lambda(z-\zeta))e^{(})\lambda\zeta\not\subset\alpha+p\beta$

By the assumption that $a+p(b-1)\geq\prime 0$ andin (2.20), we have (2.32).
Next we assume (2.29). It follows from (2.17) and (2.29) that for $z\geq Z$$\alpha+p\beta\geq 0$(2.34)$\varphi(z)$$\geq\gamma\lambda^{a}\int_{Z}^{z}(1-e^{-\lambda(}z-\zeta))eB\alpha\lambda\zeta p\lambda^{pb}d\zeta$.

$= \gamma B^{p}\lambda^{a+\phi}\int_{z}^{z}(1-e^{-}-\zeta))\lambda(zeda\lambda\zeta\zeta$

By the assumption that $a+pb\geq 0$ andin (2.21), we have (2.32).
Thus, (2.30) follows
and the fact that for $z\geq Z+1$ we have$\alpha\geq 0$$\mathrm{h}\mathrm{o}\mathrm{m}(2.32)$$I_{Z}^{z}(1-e^{-\lambda(z-\zeta}))d \zeta=z-z-\frac{1}{\lambda}+\frac{1}{\lambda}e^{-}-Z)\geq(zz-\lambda Z-1$.

While (2.31) follows ffom (2.32) and (2.27).

$\square$From Lemmas 2. 2 and 2. 3, we have the following lemma.

Lemma 2. 4 Assume (2.19) and either (2.20) or (2.21). Let $A>0,0<h\leq 1$ and$Z\geq 0$ . Suppose that
(2.35)$(z\geq Z)$ .
$\varphi(z)\geq A$

Then there exists a positive constant(2.36)(2.37)

Lemma 2. 5

depending only on$\varphi(z)\geq C_{6}A^{pq}h^{2p+}2$$p,$$\beta$and$\gamma$such that$(z\geq Z+2)$ ,$\varphi(z)\geq C_{6}A^{p}q(Z-z-2)$$(z\geq Z+2h)$.

For any $L>0$ there exists a constant $Z_{0}=Z_{0}(L)>0$ such that

(2.38)$\varphi(z)\geq L$

Proof: From (2.17) we havetake $A=1$ and $Z=0$ in L$(z\geq Z_{0})$.

for all $z\geq 0$ . Thus it follows
2. 4) that$\varphi(z)\geq 1$

emm-a(2.39)$C_{6}$$\varphi(z)\geq C_{6}(z-2)$

Thus Lemma 2. 5 follows from (2.39).

$(z\geq 2)$$\mathrm{h}\mathrm{o}\mathrm{m}$Lemma 2. 4 (we.
$\square$108
Lemma 2. 6
such that

$-$

Let

$j$

be a non-negative integer. Suppose that there nist constants$Z_{j}$(2.40)$(z\geq z_{j})$$\varphi(z)\geq A_{j}$$A_{j}$

and.

Then there nists a constant $M>1$ such that
(2.41)$(z\geq Z\mathrm{j}+1)$$\varphi(z)\geq A_{\mathrm{j}}+1$

where(2.42)$A_{j+1}= \frac{A_{\mathrm{j}}^{pq}}{M(j+1)^{44}p+}$,,$Z_{j+1}=Z_{j}+ \frac{2}{(j+1)^{2}}$.

Proof: Rom (2.40) and Lemma 2. 4 (we take $A=A_{j},$ $h=1/(j+1)^{2}$ and $Z=Z_{j}$ inLemma 2. 4), we have
(2.43)$(z \geq Z_{j}+\frac{2}{(j+1)^{2}})$$\varphi(_{Z})\geq\frac{C_{6}A_{j}^{pq}}{(j+1)^{4+4}\mathrm{P}}$.

Thus we obtain (2.41) and (2.42).

Lemma 2. 7
then we haveLet$\square$be the sequence defined by (2.42). If. Here $\nu=1/(pq-1)$ and $m= \sum_{k=2}^{\infty}(pq)-k\log k$ .

)$A_{0}>L_{0}:=M^{\nu(4_{\mathrm{P}}}e$$+4m_{J}$$\{A_{j}\}_{j=}^{\infty}0$$\lim_{jarrow\infty}A_{j}=\infty$

Proof: Rom (2.42) we have$\log A_{j}$$=$$(_{\mathrm{P}q})^{j}\log A0^{-\frac{(pq)^{j}-1}{pq-1}\log M}$$-(4p+4)\{\log j+(pq)\log(j-1)+(pq)^{2}\log(j-2)+\cdots+(pq)j-2\log 2\}$$\geq$$(pq)^{j}( \log A_{0}-\frac{\log M}{pq-1}-(4p+4)k\sum_{=2}\infty(pq)-k\log k)$$=$$(pq)^{j}( \log\frac{A_{0}}{M^{\nu}}-(4p+4)m)$.

Since $pq>1$ , this completes the proof.

$\square$

We are now in a position to give the proof of Proposition 2. 1.
Proof of Lemma 2. 1: Put $A_{0}=L_{0}+1$ . If we take $Z_{0}=Z_{0}(A_{0})$ in Lemma 2. 5, we. Moreover, it follows from (2.42) that $Z_{j}=Z_{0}+\Sigma_{k=1}^{j}2/k^{2}$ forhave
for$j\geq 1$ . Thus if we put, we have $Z^{*}<\infty$ . Rom Lemma
2. 6 for any $j\geq 1$ we havefor all $z\geq Z^{*}$ . Therefore, from Lemma 2. 7 we seethat the life span ofis less than or equal to . Since the positive constantdepends only on
and , this completes the proof.
$z\geq Z_{0}$$\varphi(z)\geq A_{0}$$Z^{*}:= \sup_{\mathrm{j}\geq 1}Z_{j}=Z_{0}+\sum_{k=1}^{\infty}2/k^{2}$$\varphi(z)\geq A_{j}$$(\varphi(z), \phi(z))$$Z^{*}$$z_{\square }^{*}$$p,$ $q,$$\beta$$\gamma$

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