数理解析研究所講究録1102 巻 1999 年 139-147

139

Scattering Theory forNonlinear Klein-Gordon Equation
with Sobolev Critical Power東京大学大学院数理科学研究科

中西賢次(Kenji Nakanishi)

1. INTRODUCTION
In this note I would like to report my result on the scattering of large
energy solutions to the nonlinear Klein-Gordon equation (NLKG) of thefollowing form:$\ddot{u}-\Delta u+m^{2}u+|u|^{p-1}u=0$

where$(t, x)\in \mathbb{R}^{1+n}$,(NLKG)

with $n\geq 3$ and $m\geq 0$ . We will consider the case$p=p^{*}:= \frac{n+2}{n-2}$.

Then $p^{*}+1=2n/(n+2)$ is the Sobolev critical exponent. This equation
has the conserved energy:$E(u;t):= \int_{\mathbb{R}^{n}}\frac{1}{2}(\dot{u}^{2}+|\nabla u|^{2}+(mu)2)+\frac{|u|^{p+1}}{p+1}d_{X=E}(u;0)$.

Denote by $X$ the energy space:$X:= \{(\varphi, \psi)|\int\psi^{2}+|\nabla\varphi|^{2}+|m\varphi|^{2}dX<\infty\}$.

We consider the asymptotic behavior of solutions to (NLKG) with finite
energy, as tends to , compared with solutions to the linear Klein-Gordon:$t$$\infty$$\ddot{v}-\triangle v+m^{2}v=0$

We find from or from , where, such thatsolution of$v$

$u$

$u$

$v$

$u$.

$(\mathrm{K}\mathrm{G})$is a solution of (NLKG) and$(\mathrm{K}\mathrm{G})$$||(u(t),\dot{u}(t))-(v(t),\dot{v}(t))||x$ $arrow 0$.

as$tarrow\infty$.

$v$

is a

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Then, the aim of this study is to show that$(v(\mathrm{O}),\dot{v}(\mathrm{O}))\vdash+(u(\mathrm{O}),\dot{u}(\mathrm{o})):Xarrow X$

homeo.

(S)

This means the asymptotic completeness of the wave operators.
Now I mention the known results on the scattering for (NLKG). First, forthe subcritical case, if $1+4/n<p<p^{*}$ and $m>0$ , then (S) was obtained byBrenner [3] and Ginibre and Velo [5]. In the critical case $p=p^{*}$ , there were
2 results available. If the solutions are radially symmetric, the scattering(S) can be obtained easily from the a priori estimate derived by Ginibre,Soffer and Velo [4]. If $m=0$ , namely for the nonlinear wave, (S) easily

follows from the decay property of the solutions obtained by Bahouri andShatah [1].
But, in the case where is the critical power, the data is nonsymmetric
and $m>0$ , none of the arguments in the above results can be applied, so
the scattering (S) in this case was left open. I have proved the scatteringin that case, which is the main result in this note:$p$

Theorem. Let $p=p^{*}$ and $m\geq 0$ . Then we have (S).
In the rest of this note, I describe the outline of the proof of this theorem.
For a more detailed, rigorous and general proof, see [7].

2. OUTLINE OF THE PROOF
For simplicity, assume that $n=3$ and $m=1$ . Then we have $p=5$ . It is
known that if we have global a priori estimates for certain space-time norms
depending only on the energy:$||u||sT(\mathrm{R})<C(E)$(G),’denotes a certain
then we obtain the desired result (S). Hereafter,
norm, andappropriate space-time norm, which is, in our context,$‘ E$ ’
denotes the energy of the solution . So, our aim is to derive the globala priori estimate (G). It is known that there is a unique global solution of$‘ \mathrm{S}\mathrm{T}$$L_{t}^{8}(L_{x}^{8})$$u$141(NLKG) for any initial data with finite energy such that the-norm of
the solution on any bounded time interval is finite (see, e.g., [6, 8]). But, it
was not known that there are estimates for the -norm depending only on
the energy. lt was unknown even on a finite time interval. This is a typicaldifficulty in the critical case.
$\mathrm{S}\mathrm{T}$$\mathrm{S}\mathrm{T}$In order to prove the a priori estimate (G), first I use Bourgain’s idea,

which he used to solve the nonlinear Schr\"odinger equation with the criticalexponent in the radial case [2]. Roughly speaking, his idea is to relate thedistribution of the-norm in time with the distribution of the energy inspace-time. Remark that the-norm is a Lebesgue norm for . At first, we$\mathrm{S}\mathrm{T}$$\mathrm{S}\mathrm{T}$$t$

do not know how large it is. But we can divide the time interval into small

subintervals such that each subinterval contains the same small ST-norm,say, . Then Bourgain’s lemma below tells us that in each subinterval,
somewhere in the space, there is a certain amount of localized energy. See
Figure 1.
$\epsilon$

Lemma 1 (Bourgain). $Let||u||_{ST()}I=\epsilon<C(E)$ is sufficiently small. Then
we have some subintervaland some ballsuch that diam $D<$$C(E, \epsilon)|J|$ andfor any$J\subset I$$D\subset \mathbb{R}^{n}$$t\in J_{f}$$\int_{D}|\nabla u|^{2}+u^{2}dx>\epsilon^{\alpha}$

where $\alpha>1$ is a certain constant.

,$\int_{D}u^{6}dx>\epsilon^{\alpha}$,142
So, we obtain such an energy lump in $J\cross D$ for each subintervals. If-norm is large, then the number of the subintervals is also large,
the$\mathrm{S}\mathrm{T}$and we obtain a lot of energy lumps correspondingly. Thus, to estimate-norm, it suffices to estimate the number of such energy lumps. Ifthethe-norm in a finite interval is very large, then many energy lumps are
crowding there, and so gathering at some point in space-time, which meansof the energy. I have obtained a new estimate to bound the$\mathrm{S}\mathrm{T}$$\mathrm{S}\mathrm{T}$concen.tration

number of such gathering energy lumps.

Lemma 2. For any finite energy solution
have$u$of (NLKG) and any$\sum_{k\in \mathrm{N}}\sup_{<2^{-k}t<2^{-k+1}}\int_{|x|<\lambda t}u^{6}dX<c(E, \lambda)$.

$\lambda>0$, we(N)From this, the energy lumps in a fat cone $\{|x|<\lambda t\}$ can not be containedin so many of dyadic intervals $(2^{-}k, 2-k+1)$ (see Figure 2). Combinig thisFIGURE 2. Counting the energy concentration

estimate with the finite propagation property and the well-known Morawetz

estimate$\int_{\mathbb{R}^{n+1}}\frac{u^{6}}{|x|}dXdt<C(E)$(M),

we can bound the number of the energy lumps in the time interval $(0,1)$
so that we obtain the time-local a priori estimate for the -norm by the$\mathrm{S}\mathrm{T}$143
energy:$||u||sT(0,1)<o(E)$ .

(L)

In the massless case $m=0$ , we obtain the global estimate (G) from thislocal estimate (L) by a simple scaling argument. However, in the massive
case $m>0$ , there is no scaling which preserves the equation or the energy
space. Moreover, we do not know whether the global version of (N) holds
in the massive case. So, for the global estimate in the massive case, we canuse only the Morawetz estimate (M), the finite propagation property andthe local estimate (L). Again consider the distributed energy lumps given
by Bourgain’s lemma.
$0=T_{0}<T_{1}<\cdots$,$J_{j}\subset I_{j}$$I_{j}:=(T_{j}, T_{j+1})$,$D_{j}$: ball$\int_{D_{\mathrm{j}}}u^{6}dX>\epsilon^{\alpha}$Let$c_{j}$

and$R_{j}$,,$||u||_{S\tau(I_{j})}=\epsilon$$\subset \mathbb{R}^{n}$$(t\in J_{j})$be the center and the radius of,,.
$D_{j}$, and let $t_{j}:=$ inf$J_{j}$.

Consider the truncated cone $K_{j}:=\{(t, x)|t>t_{j}, |x-C_{j}|<t-t_{j}+R_{j}\}$ foreach . Now, in order to employ the Morawetz estimate most effectively forthe-norm estimate, we choose some of the truncated cones, such thatthe bottom of any chosen
does not intersect with the other chosen cones,
and at the same time, every energy lump intersects with some chosen cones(see Figure 3). Then, by the finite propagation property, we can bound the$j$$\mathrm{S}\mathrm{T}$$K_{j}$FIGURE 3.

$\mathrm{C}\grave{\mathrm{h}}.\mathrm{O}\mathrm{S}\backslash \mathrm{e}\mathrm{n}$

cones

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number $N$ of the chosen cones by the total energy. Applying the Morawetz
estimate (M) in each chosen cone, we obtain $N$ inequalities. Then, summingthem up, we obtain an estimate as follows.
$C(E, \epsilon)\geq c(E)N$$\geq$$\int\int\frac{u^{6}}{|x-c_{j}|}dxdt$$\sum$

chosen.
$J$$\geq\sum_{\mathrm{c}\mathrm{h}_{\mathrm{o}\mathrm{s}}\mathrm{e}\mathrm{n}j}j_{k}\mathrm{x}D_{k}\cap K_{j}\sum_{\emptyset\neq}\int_{J}k\int\frac{u^{6}}{|x-c_{j}|}dxdt$.

$\geq\sum_{\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{n}jJk\cross D_{k}}\sum_{\otimes\cap K_{j}\neq}\frac{\epsilon^{\alpha}|J_{k}|}{|t_{k}-t_{j}|+R_{k}+Rj}\geq^{c}(E, \epsilon)\sum_{1\mathrm{a}1k}\frac{R_{k}}{t_{k}+R_{k}}$If the$\mathrm{S}\mathrm{T}$-norm is very large, then there are plenty of energy lumps. So,becomesto make the above quantity $\sum_{k}R_{k}/(t_{k}+R_{k})$ bounded, either
very small or becomes very long. Thus, there are two possibilities. The.
first case is that very highly concentrated energy lumps apper.
-norm appear
Otherwise, very long intervals with small
just after such a long interval
In the latter case, we consider the intervalwith the same initial data as . Since we. Let be the solution tohave global -estimate for by the Strichartz estimate, if there are many$R_{k}$$I_{k}$$(\varliminf R_{k}arrow 0)$$(\varlimsup|I_{k}|arrow\infty)$$\mathrm{S}\mathrm{T}$$I_{k+1}$$I_{k}$$u$$(\mathrm{K}\mathrm{G})$$v_{0}$$\mathrm{S}\mathrm{T}$$v_{0}$

such long intervals , we may choose appropriate such that. Then we use the property of the Klein-Gordon that the lower frequency
part decays faster, to obtain the following lemma.
$I_{k}$$I_{k}$$||v\mathrm{o}||S\tau(I_{k+1})\ll$$\epsilon$Lemma 3. Let $0<T<U<V$ and $||u||ST(T,U)\leq||u||_{S}T(U,V)=\epsilon<C(E)$with the same initial data
be the solution tosufficiently small. Let
as , and assume $||v||_{s}T(U,V)<\epsilon/9$ . Then for any $N>1$ , there exists $L>0$
and $N$ , such that $if|T-U|>L$ , then we havedepending on$v_{0}$$(\mathrm{K}\mathrm{G})$$u$$E_{f}\epsilon$$|| \psi_{N}*u||_{S}\tau(U,V)<\frac{\epsilon}{3}$,

145
where

$\psi_{N}$$\psi\in S(\mathbb{R}^{n})$is a cut-off function in the frequency, defined as follows. Let
be a function such that its Fourier transform satisfies$\tilde{\psi}$$\overline{\psi}(\xi)=$

Then we define$\overline{\psi}_{N}(\dot{\xi}):=\tilde{\psi}(\xi/N)$

Thus, we may deduce that the.

-norm incomes mainly from thehigher frequency part. Since high frequency means high concentration, we
get a highly concentrated energy lump in this interval. So we arrive at
the same situation as in the forrner case.
Now again we use an idea due to Bourgain. We have obtained a very
highly concentrated energy lump. Consider the wave component corresponding to the concentrated energy. Since is also very highly concentrated, it decays very soon. Then its interaction with the remaining part issmall, so that we can separate the concentrated wave from the solution ,and estimate by the remaining part. More precisely, we have the following$I_{k+1}$$\mathrm{S}\mathrm{T}$$I_{k+1}$

$v$

$v$

$v$

$u$

$u$

perturbation lemma.

Lemma 4 (Bourgain). Let $u_{f}W$ be solutions of (NLKG), and let be asolution ofwith the same initial data as $u-W$ . Let $E(u)\leq E_{f}$$E(v)\leq E$ and $||W||sT(0,\infty)=M<\infty$ . Then, there exists $\kappa>0$ depending$E$ andsuch that $if||v||sT(0,\infty)<\kappa$ we have the estimate$v$$(\mathrm{K}\mathrm{G})$$M_{f}$$||u||_{S}T(0,\infty)<c(E, M)$,depending on $E$ and $M$ .

Since the energy of the remaining part $E(W)$ is reduced by the separatedenergy, repeating this argument, the problem comes down to the estimate
for small energy data. Since the global-estimate (G) is well-known forsmall energy data, so we obtain the desired estimate by the induction on
the energy size.
$\mathrm{S}\mathrm{T}$146
Finally, I explain how we can extract the concentrated wave in the above
argument. The idea of the separation of the energy is due to Bourgain, but$v$

the realization of the idea below is quite different from that in [2]. Bourgain’s argument uses essentially the properties of the nonlinear Schr\"odinger,
whereas my argument uses those of (NLKG). Here we use again an estimate
similar to (N):Lemma 5. For any finite energy solution
have$u$of (NLKG) and any$\sum_{k\in \mathrm{N}}<t<2\sup_{2^{-k}}-k+1\int_{|x|<\lambda}tdQ(u)X<c(E, \lambda)$,we$\lambda>0_{f}$$(\mathrm{N}\mathrm{Q})$where we denote$Q(u):=( \dot{u}+\frac{r}{t}ur+\frac{2}{t}u)^{2}+(\frac{r}{t}\dot{u}+ur)^{2}+(1+\frac{r^{2}}{t^{2}})(|u_{\theta}|^{22}+u)+\frac{u^{2}}{r^{2}}$$r=|x|$ ,$\theta=\frac{x}{r}$,$u_{r}=\theta\cdot\nabla u$,$u_{\theta}=\nabla u-\theta u_{r}$In fact, the estimate (N) can be derived from
Hardy-Sobolev type inequality:$(\mathrm{N}\mathrm{Q})$,.

and the following.

$\int_{|x|<\lambda}tdu^{6}X<C(\lambda)||\nabla u||4L^{2}\int_{1}x|<\lambda tuQ()dX$(H)Now we set the space-time origin such that we have the concentrated
energy in $\{(R, x)||x|<R\}$ . We may assume that the radius of the energy, for any $\kappa>0$ we have some timelump $R>0$ is very small. By$(\mathrm{N}\mathrm{Q})$$T\in(R, c(\kappa, E)R)$when $Q(u)$ in the fat cone $\{|x|<5t\}$ becomes small:$\int_{|x|<5}TQ(u;\tau)dx<\kappa$,provided $C(\kappa, E)R<1$ . On the other hand, the energy in the light cone
does not decrease. So, at this time $T$ , we cut off the data by a smooth

which satisfies $\chi=1$ in the light cone and $\chi=0$ out
cut-off function
with the dataof the fat cone. And we define as the solution of$\chi(x)(u(\tau),\dot{u}(T))$ . Then the concentrated energy at $t=R$ is inherited by$\chi(x)$$v$$(\mathrm{K}\mathrm{G})$

147

$v$

. Meanwhile, using the fact thatis contained in the fat cone andis very small, we can deduce that $Q(v)$ remains small forever:$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v(T)$Lemma 6. Let$v$

be a solution to$(\mathrm{K}\mathrm{G})$satisfying$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(v(\tau),\dot{v}(\tau))\subset\{X||x|<\tilde{T}\}$for some$\tilde{T}\geq T>0$

. Then we have for any $t>T_{f}$$\int Q(v;t)dX\leq\int Q(v;\tau)dX+C\tilde{T}^{2}E_{0}(v)$where$E_{0}(v)$,denotes the linear energy of .
$v$

By this lemma and the inequality (H),remains small for $t>T$ .
Then, by the interpolation with the Strichartz estimate, it follows that$||v||sT(T,\infty)$ is also small. Thus, we havesucceeded in extracting the concentrated wave as desired.
$||v(t)||L_{x}6$$v$

REFERENCES
1. H. Bahouri and J. Shatah, Decay estimates for the critical semilinear wave equation,
Ann. Inst. H. Poincar\’e, Anal. Non Lin\’eaire 15 (1998), 783-789.
2. J. Bourgain, Global wellposedness of defocusing critical nonlinear Schr\"odinger equationin the radial case, J. Amer. Math. Soc. 12 (1999), no. 1, 145-171.
3. P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z. 186 (1984), 383-391.
4. J. Ginibre, A. Soffer, and G. Velo, The global Cauchy problem for the critical non-linear
wave equation, J. Fhnct. Anal. 110 (1992), 96-130.
5. J. Ginibre and G. Velo, Time decay of finite energy solutions of the non linear KleinGordon and Schr\"odinger equations, Ann. Inst. Henri. Poincar\’e, 43 (1985), 399-442.
6. L. Kapitanski, Global and uni.que weak solutions of nonlinear wave equations, Math.
Res. Lett., 1 (1994), 211-223.
7. K. Nakanishi, Scattering theory for the nonlinear Klein-Gordon equation with Sobolev
critical power, Internat. Math. Res. Notices 1999, no. 1, 31-60.
8. J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices 1994, no. 7, 303-309.

GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF TOKYO, 3-8-1
KOMABA, MEGURO, TOKYO 153-8914, JAPAN-mail address: nakanisi@ms.u-tokyo.ac jp$E$$\backslash \cdot$