数理解析研究所講究録
1009 巻 1997 年 121-131

121
NONLINEARDINGER EQUATIONSIN FRACTIONAL ORDER SOBOLEV SPACES$\mathrm{S}\mathrm{C}\mathrm{H}\mathrm{R}\tilde{\mathrm{O}}$T. OZAWA$(\prime 5^{\iota}’/\mathrm{E}\sim$$r_{6}^{\gamma\wedge \mathrm{t}}\mathrm{L}$)

Department of Mathematics, Hokkaido University

In this note I describe some recent work on nonlinear$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{r}\tilde{\mathrm{o}}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}$equations, done

jointly with M. Nakamura $[27, 28]$ . We consider the nonlinear Schr\"odinger equationsof the form
$i\partial_{t}\tau\iota+\Delta u=f(u)$where$\tau\iota$

is a complex-valued function of,$(t, x)\in \mathrm{R}\cross \mathrm{R}^{n},$(1)$\partial_{\ell}=\partial/\partial t,$$\Delta$

is the, and is a complex-valued function, a typical form of which is the
single power interaction
Laplacian in$\mathrm{R}^{n}$

$f$

(2)

$f(u)=\lambda|u|p-1\tau\iota$

with$\lambda\in \mathrm{R}$and $1<p<\infty$ .

There is a large literature on the Cauchy problem for the equation (1) and on

the asymptotic behavior in time of the global solutions [2, 4, 5-9, 12-17, 22, 25,
and references therein]. The Cauchy problem for (1) has been studied mainly in
the Sobolev spaces$H^{m}$

of integral order $m$ , especialy $m=0,1,2$ , while there

arises a new interest in the treatment of the Cauchy problem in the Sobolev spaces$H=(1-\Delta)^{-\cdot/2}L^{2}(\mathrm{R}^{n})$ of
fractional order with $0\leq s<n/2$ . In [5], Cazenave
and Weissler proved that the Cauchy problem for (1) with (2) has global solutionsin $H$ for the data $\phi\in H$ with $||(-\Delta)/2\phi;L^{2}||$ sufficiently smal, provided that
$p=1+4/(n-2_{S})$ and $[s]<p-1$ , whereis the greatest integer that is less thanor equal to . In [14], Kato generalized the results in [5] in some directions. In [7],Ginibre, Ozawa, and Velo proved the existence and asymptotic completeness of the
wave operators for (1) with a class of interactions including (2) on smal asymptotic
states in $H$ , provided that $1+4/n\leq p\leq 1+4/(n-2_{S})$ and $\epsilon<\min(2,p)$ . In[22], Pecher proved that the Cauchy problem for (1) with (2) has global solutionsin $H$ for small data in $H$ , provided that $1+4/n\leq p<1+4/(n-2_{S})$ and$1<s< \min(4,p+1)$ or $4\leq s<p+2$ . In connection with the $H$ theory for(1) with (2), a homogeneity argument indicates that the power in (2) is critical[resp. subcritical] at the level ofif and only if $p=1+4/(n-2s)$ [resp.
$s$$[\epsilon]$

$s$

$p$

$H^{\cdot}$

122$p<1+4/(n-2\iota)]$. To sum up with this definition, the critical case is studied in[5, 7, 14] and the subcritical case is studied [7, 14, 22].

The purpose of this paper is to study the $H$ theory for (1) with a class ofinteractions including (2) in more detail both in the critical and subcritical casesin the Eamework of low energy scattering. We prove the existence and asymptotic
completeness of the wave operators for (1) on smal asymptotic states in $H$ in thecritical case with $s< \min(n/2,p)$ as wel as in the subcritical case with $s<p$ .
Moreover, smallness assumption is shown to be necessary only for thenorm of, where $\epsilon_{0}\equiv n/2-2/(p-1)$ .
the fractional derivativeof the data$L^{2}$$(-\Delta)^{\prime 0/2}\phi$$\phi\in H^{\iota}$

is not an odd integer, an additio $na1$ assumption such as $s<p$
is required to keep the smoothness of compatible with the behavior at zero.
Concerning the number , we notice the folowing simple facts: (1)in thecritical case. (2) $s_{0}<s$ in the subcritical case. (3) is critical at the level of $H0$ .
(4) $0\leq\epsilon_{0}<n/2$ .
Here, when

$p$

$f$

$s=\epsilon_{0}$$s_{0}$$p$-theory with $0\leq s<n/2$ , the power behavior
theof the nonlinearity determines the order of the Sobolev space where the smallness of

As we see above, as$\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{s}\backslash$$H^{\cdot}$

the data is imposed to ensure the existence an un iqueness of global $H$ -solutions.
This is the right phenomenon, as is usual with other nonli ear evolution equations$\mathrm{d}$$n$with dilation structure, such as the heat equation with single power interaction and

the Navier-Stokes equations.
In contrast, when $s>n/2$ , no specific behavior of the nonlinearity is requiredof the H.-theory for (1) at least locally in time. In fact, when $s>n/2$ , forthe existence and uniqueness of local
solutions one has only to assume that$H^{\iota_{-}}$, where differentiability refers to the real sense andis the smalest integer greater than or equal to . The proof depends on the usual
for $s>n/2$ in an essential way.
Sobolev embedding$f\in C^{h}(\mathrm{C};\mathrm{c})$with$f(\mathrm{O})=0$

$k$

$s$

$H^{\cdot}\subset L^{\infty}$

The case $s=n/2$ may therefore be regarded as the borderline in two as pects:(1) No power behavior of interactio amounts to the critical nonlinearity at the$n$level of $H^{n/2}$ . (2) Poinwise control of solutions falls beyond the scope of thetheory, so that any argument similar to that of the$H^{n/2_{-}}$.-theory with $s>n/2$ breaks$H^{l}.$

down even for local theory without specific behavior of interaction.
In addition to the critical phenomena described above, $H^{n/2}$ -solutions deserve

attention as finite energy solutions for $n=2$ and as strong solutions for $n=4$ .
-solutions to (1) with
We prove the existence and uniqueness of global
small Cauchy data under the nonlinearity of exponential type. This is reminiscent$H^{n/2}$

123
of Trudinger’s inequality, which replaces the Sobolev embedding in the limiting caseon the basis of the exponential estimates in terms of functions in the critical order

Sobolev space$H^{n/2}$.

To state the results precisely, we use the folowing notation. For any$1\leq r\leq\infty,$

and any$r$$L’=L^{f}(\mathrm{R}^{n})$denotes the Lebesgue space onwith 1 $<r<\infty,$ $H,$$=(1-\Delta)^{-\cdot/2}L$’defined in terms of Bessel potentials. For any$\infty,$$Bi,m$$s\in \mathrm{R}$$\mathrm{R}^{n}$. For any

and anywith, where$\{\phi_{j}\}$

is

$a$

and any$r,$$m$$s\in \mathrm{R}$with $1\leq r,$ $m\leq$dyadic decomposition on

en otes thewithdenotes the Sobolev spacedenotes the Besov space defined as the space of distributions$\{2 j||\phi_{j}*u;L’||\}_{j=0}^{\infty}\in t^{m}$

$r$

$u$

$\mathrm{R}^{n}$such that. For any

Sobolev space
defined as the space of classes of distributions modulo polynomials such that$(-\Delta)^{*/2}u\in L’$ . For any
and any $m$ with $1\leq r,$ $m\leq\infty,\dot{B}i,m$ denotes
the homogeneous Besov space defined as the space of classes of distributions$s\in \mathrm{R}$$r$$1<r<\infty,\dot{H}i\mathrm{d}$$\mathrm{h}_{\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{u}}\mathrm{e}\mathrm{n}e\mathrm{S}$$u$$s\in \mathrm{R}$

$r,$

$u$

modulo polynomials such that

, whereis a dyadic
decomposition on. We refer to [1, 10, 24] for general information on Besov
an Thiebel-Lizorkin spac and th homogeneous versions. For simplicity, we put$H^{\cdot}=H_{2},\dot{H}\cdot=H_{2},$For any interval
an any Banach
space $X$ we denote by $C(I;X)$ the space of strongly continuous functions from to$X$ and by;
the space of strongly measurable functions from to $X$ suchthat $||u(\cdot);x||\in L^{q}(I)$ . Let $U(t)=\exp(il\Delta)$ be the free propagator, namely theone parameter group which solves the free Schr\={o}dinger equation. For any with
$2\leq r\leq\infty$ , we define $\delta(r)=n/2-n/r$ . Concerning the space-time
integrability
properties with respect to, it is convenient to call a pair of exponents $(q, r)$admissible if $0\leq 2/q=\delta(r)<1$ , which is understood to be $0\leq 2/q=\delta(r)\leq 1/2$
when $n=1$ . The Cauchy problem for the equation (1) with data $u(t_{0})=U(t_{0)\phi}$
at time will be treated in the form of the integral equation$\{2 j||\psi_{j}*u;L’||\}_{j}^{\infty}=-\infty\in f^{m}$$\{\psi_{j}\}$$\mathrm{R}^{n}\backslash \{0\}$$es$$\mathrm{d}$$e\mathrm{i}\mathrm{r}$$B^{l},=Bi_{2},’\dot{B}_{f}^{l}=\dot{B}i_{2},\cdot$$I\subset \mathrm{R}$$\mathrm{d}$$I$$L^{q}(I$

$u$

$I$

$r$

$U(\cdot)$$t_{0}$($)$u(t)=U(t-t_{0})u(t_{0})-i \int_{t_{0}}^{\ell_{U(-T}}\ell)f(u(\tau))d_{T}$$=U(t)\phi-i(G_{t}f\mathrm{o}(u))(t)$,where the second line is understood to define the integral operator
line of (3) is formally equivalent to (1) with Cauchy data$u(t_{0})$$G_{t_{0}}$. The firstgiven at finite time

, while the second li $ne$ will be used to describ the Cauchy problem for (1) withdata at time $t_{0}=0$ as wel as at $t_{0}=\pm\infty$ . The integral equation (3) will be$e$$t_{0}$$\phi$

124
studied in the spaces

and$X^{\cdot}$$\mathrm{Y}^{\cdot}$with $s\geq 0$ defined as,$X^{l}=C( \mathrm{R};H\iota)\mathrm{n}\bigcap_{0\leq 2/-^{\delta}(’)<1}L^{q}(\mathrm{R};B’,)$..

Note that $X$

Y. For the nonlinear interaction$\subset$stant$\mathrm{c}\mathrm{o}n$$(\mathrm{A})_{h}$behaving as a power atandwith integer with
$f$

$p$

$k$

$(\mathrm{B})_{h}$.

$f\in C^{h}(\mathrm{C};\mathrm{c})$$(\mathrm{A})_{h}$.

$0\leq 2/-^{\delta()1},<$zero, we introduce the folowing assumptions$0\leq k\leq p$$L^{q}(\mathrm{R};H, )$$\cap$$\mathrm{Y}^{\cdot}=C(\mathrm{R};H^{\cdot})\cap$$C$

an $\mathrm{d}f^{(j)}(0)=0$ for all

such that for all$z_{1},$

$j$

with

$0\leq j\underline{<}k$$C(|_{Z_{1}1^{p}+||)}-h-1Z_{2}p-h-1|z1-z_{2}|$$C|z_{1^{-z}2}|^{pk}-$$f\in C^{h}(\mathrm{C};\mathrm{c})$exists a constant$C$and $f^{(j)}(0)=0$ for

such that for all$\mathrm{a}1j$$f^{(j)}$$s$

if $p\geq k+1$ ,if $p<k+1$ .

with $0 \leq j\leq\max(k-1, \mathrm{o})$ . There$z\in \mathrm{C}$$|f^{(h)}(z)|\leq^{c}|z|p-h$

Here

$e$

$z_{2}\in \mathrm{C}$$|f^{(h)}(Z_{1})-f^{()}h(Z_{2})|\leq\{$$(\mathrm{B})_{h}$. There xi ts a.

den otes any of the j-th order derivatives of$f$with respect to

$z$

an

$\mathrm{d}\overline{z}$denotes the maximum of the moduli of thos derivatives. Note that
[resp.
] implies
an that[resp.
] for all with
implies$0\leq j\leq k$ . Single power interaction (2) satisfies
with $0\leq k<p$ (see [11]).
and$|f^{(j)}|$$(A)_{h}$$e$$(B)_{h}$$(B)_{h}$$(A)_{h}$$\mathrm{d}$$(B)_{j}$$(A)_{j}$$j$$(A)_{h}$With the notation above we now state the main results in this paper. Theorem
1 is devoted to the critical case and Theorem 2 is devoted to the subcritical case.
, we define
For any $s,p,$ with $s\geq s_{0}\equiv n/2-2/(p-1)\geq 0,$$\epsilon>0$$\epsilon$$\dot{B}_{\epsilon}=\{\psi_{\in H}\cdot ; ||\psi;\dot{H}\cdot 0||<\epsilon\}$

Theorem 1.

(I) Let$\epsilon$

and

$p$

.

satisfy$0<s<n/2$ ,$\iota<p=1+4/(n-2_{S})$sa tisfy(I) For any $da$ ta

Let

$f$

$u\in X^{\iota}$$(A)_{[\iota]}$.

. Then there exists$\phi\in\dot{B}_{\epsilon}$$\epsilon>0$.

with th followin property.
$e$

$g$

at time $t_{0}=0$ th equation $(S)$ has a unique solu tion$\mathrm{e}$125(2) For any data$u\in X^{\iota}s\mathrm{u}\mathrm{c}\mathrm{A}$at time $t_{0}=+\infty$ the equation $(S)$ has a nique solu tion$\phi_{+}\in\dot{B}_{\epsilon}$$\mathrm{u}$th at$||u(t)-\sigma(t)\phi+;H||arrow 0$$(S)$

For any data$u\in X^{\cdot}$(4)$tarrow+\infty$

at time $t_{0}=-\infty$ the equa tion $(S)$ has$\phi_{-}\in\dot{B}_{\epsilon}$$a$$\mathrm{u}n\mathrm{i}q$ue solu tionsuch th at$||u(t)-U(t)\phi_{-;}H||arrow 0$(4) For any$\phi_{\pm}\in H$

as$\phi\in\dot{B}_{\epsilon}$at$\mathrm{t}\mathrm{i}\mathrm{m}et_{0}=0$

sa tisfying (4)(II) Let an integer

and

$s$

where

$,$

$p$

$tarrow-\infty$.

(4)

there exists a unique air of asymp totic states$p$is the uniq

$u$

as

$\mathrm{u}e$solution given by Part (1).

satisfy$0\leq s<n/2$ ,$s\leq p=1+4/(n-2_{S})$Let$\mathrm{Y}$$f$.

sa tisfy $(B).$ . Then all the onclusion of Part (I) hold if $X$ is replaced by$c$throughout the statement

Remark 1[2, 6, 15].

$of\mathrm{p}_{a\mathrm{r}}\mathrm{t}(I)$$\mathrm{s}$.

Th power $p=1+4/(n-2_{S})$ comes out as
$e$

$a$

critical one
if and only

in $H$ in the sense thatis invariant under the dilation
if $\epsilon=n/2-2/(p-1)$ , wherewith $\lambda>0$ andthe dilation above leaves (1) with (2) invariant. Another characterization is given
as the power which makes the estimates of the form$||u;\dot{H}||$$u\mapsto u_{\lambda}$$u_{\lambda}(t, X)\equiv\lambda^{-2/(p)}-1u(\lambda-2t,$ $\lambda-1_{\mathrm{Z})}$$||G_{0}f(u);L^{q}(\mathrm{R};\dot{H},l)||\leq C||u;L^{q}(\mathrm{R};\dot{H}i)||^{p}$with any admissible pair $(q, r)$ invariant under the dilation$u(\lambda^{-2}t, \lambda-1ae)$with$\lambda>0$$u\mapsto u_{\lambda}$, where$u_{\lambda}(t, x)\equiv$.

Remark 2. In part (I) of Theorem 1, the assumption $s<n/2$ is required to’is required to keep thesmoothness of the non linearity compatible with a power behavior such as (2) atzero when is not an odd integer. The condition
keep the critical power finite, while the assumption$\epsilon<p$

$f$

$p$

$0<s< \min(n/2,1+4/(n-2\epsilon))$is equivalent to:(a) $\epsilon\in(0, n/2)$ if $n\leq 7$ ,(b) $s\in(\mathrm{O}, s_{-(}n))\cup(\epsilon_{+}(n),n/2)$ if $n\geq 8$ , where.

$\epsilon_{\pm}(n)=(n+2\pm(n^{2}-4n-28)^{1}/2)/4$

126Compare those two conditions (a) and (b) with those given in [14]. The$\mathrm{r}\mathrm{e}s$triction$s<p$ may be partially removed by taking into account the regularity in time.
direction in more$\mathrm{d}e\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}(\mathrm{S}\mathrm{e}e[22])$

Remark 3. Theorem 1 shows the existence and asymptotic completeness of theonas the maps
. The scatteringwave operators$\dot{B}_{\epsilon}$$\phi_{\pm}rightarrow u(\mathrm{O})=\phi$$W\pm \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$operator operator is then defin onasNote that smallness
assumption is imposed on the data only through the fractional derivative of critical$S$$S=W_{+}^{-1}\mathrm{o}W-\cdot$$\dot{B}_{\epsilon}$$e\mathrm{d}$order $n/2-2/(p-1)$ , which is equal to$s$in the critical cas .
$e$Remark 4. The existence and asymptotic completeness of the wave operators$\mathrm{h}$as been proved in [25] for (1) with (2) with $p=1+4/(n-2),$ $n\geq\,$ on small

asymptotic states in. A part of the result in [25] was then reproduced in [17].
Part (1) of Theorem 1 (I) is proved for (2) with $[s]+1<p=1+4/(n-2_{S})$ and$0<\iota<n/2$ . Related results were proved by Pecher $[19, 20]$ for the nonlinearwith $p=1+4/(n-2)$ and n\geq $.
Klein-Gordon equation in$H^{1}$$H^{1}$

Theorem 2.

(I) Let $\iota>0$ and $p>1+4/n$ satisfy$s<p<\{$
Let$f$satisfy$(A)_{[\cdot]}$(1) For any data$u\in X^{\iota}$$1+4/(n-2_{S})$. Then there exists$\phi\in\dot{B}_{\epsilon}$$\epsilon>0$$a$$s\geq n/2$,$\iota<n/2$.

with the following property.

at time $t_{0}=0$ the. Moreover, there exists.
satisfying(2) For any $da$ ta

if
if$\infty$uation $(S)$ has a unique solu tion
uniq $ue$ pair of asympto states$e\mathrm{q}$$\mathrm{t}\mathrm{i}c$$\phi_{\pm}\in H^{\iota}$$\{4)_{\pm}$at time $t_{0}=+\infty$ [resp.
$eq$ uation $(S)$ A as a unique solution
satisfying (4)(II) Let $s>0$ be an integer and let $p>1+4/n$ satisfy$\phi_{+}\in\dot{B}_{\epsilon}$$\phi_{-}\in\dot{B}_{\epsilon}$$u\in X^{\iota}$$\infty$$s\leq p<\{$Let$\mathrm{Y}^{\iota}$satisfy. Then$1+4/(n-2_{S})$at time $t_{0}=-\infty$] the

fresp. (4) ].

if $f\geq n/2$ ,if $s<n/2$ .

the conclusions of Part (I) hold if $X$ is replaced byth roughout the statement ofPart (I).
$f$$(B)_{\iota}$$d\mathrm{J}$Remark 5. Theoren 2 shows the existence and asymptotic completeness of the

wave operators on for (1) in the subcritical case. Note that smallness assumption
is imposed $on$ the dat only through the &actional derivative of critical order $n/2-$$2/(p-1)$ , which is less than in the subcritical case. For the Cauchy Problem inboth critical and subcritical cases the observation of this kind is made by.
$\dot{B}_{\epsilon}$

$a$

$s$

$\mathrm{K}\mathrm{a}\mathrm{t}_{0}[14]$127
Bemark 6. The assumptions of Theorem 2 cover for instance the case where$s_{0}=1[25]$ , and therefore the result of Theorem
n=$, $p=1+4/(n-2)=5,$
2 gives partial answer to Question 4 of Kenig, Ponce, andunder thesmallness assumption on. Relat results were proved byforthe nonlin ear Klein-Gordon equation with $n=3,p=5,$.
$\epsilon=2,$$a$$\mathrm{V}\mathrm{e}\mathrm{g}\mathrm{a}[15]$$||\phi;\dot{H}^{1}||$$e\mathrm{d}$$\mathrm{R}a\mathrm{u}\mathrm{c}\mathrm{h}[23]$$\epsilon=2,$ $\epsilon_{0}=1$

To describe the nonli near interaction
as well as with a vanishing behavior aswith an exponential growth at infinity$f$power at zero, for
we introduce thefollowing assumptions $(C)_{m}$ with $m\geq 1$ anwith $m\geq 0$ .
$\lambda>0$$a$$\mathrm{d}(D)_{m}$$(C)_{1}$:$f\in C^{1}(\mathrm{C};\mathrm{c})$and $f(\mathrm{O})=0$ . There exist a constant
$s$

$C$

such that for all$z\in \mathrm{C}$.

$|f’(z)|\leq c_{e^{\lambda 1^{z}}}|^{2}|z|^{2}$$(C)_{m}$for

for $m\geq 2:f\in C^{m}(\mathrm{C};\mathrm{C})$ and $f(\mathrm{O})=0$ . There exists a constant $C$ such that$\mathrm{a}.11z\in \mathrm{C}$and $2\leq k\leq m$$|f’(Z)|\leq Ce^{x|}|z|^{\mathrm{a}}z|$,$|f^{(h)}(z)|\leq Ce\lambda|z|^{2}$$(D)_{0}$$z_{1},$:$f\in C(\mathrm{C};\mathrm{C})$and $f(\mathrm{O})=0$ . There exists a constant $C$ such that for all$z_{2}\in \mathrm{C}$$|f(z_{1})-f(Z2)|\leq o(ex\mathrm{I}z_{1}|^{2}|z1|4+ex|z_{2}|^{2}|z2|4)|z1^{-}z_{2}|$$(D)_{m}$for $m\geq 1$ : In addition to $(C)_{m},$$f^{(m})$satisfies the estimate for all$|f(m)(z1)-f(m)(Z_{2})|\leq c(e^{\lambda|}z11^{2}+e^{x|z}2|^{2})|Z_{1}-z_{2}|$We solve the equation (3) in the Banach space

$Z$

if

$f$

.

is even,$n$$Z=c( \mathrm{n};Hn/2)\mathrm{n}\bigcap_{/0\leq 2^{\mathit{5}(}-\prime)<1}Lq(\mathrm{R};\dot{B}^{0},\cap B^{n/2},)$

Let $n\geq 1$ . $Hn$ is even let$z_{1},$ $z_{2}\in \mathrm{C}$defined by$Z=C(\mathrm{R};H^{n/}2)\cap 0\leq 2/q=\mathrm{n}\cdot L^{q}(\mathrm{R};H,n/2)s(’)<1$

Theorem 3.

.

if

is odd.

$n$satisfy $(C)_{n/2}$ for$so\mathrm{m}e\lambda>0$

. If

$n$

satisfy $(D)_{(n-}1)/2$ for some
. Then there exists
with theproperty.
(1) For any dataat time $t_{0}=0$ the $eq$ ua tion
un ique solution$u\in Z$ , whereis the all inwith center and radius . Moreover there
sa tisfying (4)exists un ue air
replaced by $n/2$ .

is odd let$\lambda>0$$f$$\epsilon>0$$fo\mathrm{J}low\mathrm{i}ng$$\phi\in B_{\epsilon}$$B_{\epsilon}$$a$$\mathrm{i}q$

$p$

$b$

$\phi_{\pm}\in H^{n/2}$$(S)\mathrm{A}\mathrm{a}S$$H^{n/2}$$0$$wi\mathrm{t}l\iota s$$\epsilon$

$a$

128

at time $t_{0}=+\infty$ th equation $(S)$ A as a uniq solu tion
witA replac $ed$ by $n/2$ .
$(S)$ For any datanique
at time $t_{0}=-\infty$ the $eq$ ua tion $(S)$ A as
solution $u\in Z$ satisfying (4)-with a replaced by $n/2$ .

(2) For any data$u\in Z$ satisfying (4)$e$$\phi_{+}\in B_{\epsilon}$$\mathrm{u}e$$\iota$$a$$\phi_{-}\in B_{\epsilon}$$u$Remark 7. The assumptions of the theorem above cover for instance the nonlinearities of the form$f(u)=\pm(e^{\lambda \mathrm{I}u|}-12-\lambda|u|^{2})u$$f(u)=\pm(e^{\lambda||^{2}}-u1)u$$f(u)=\pm(e^{\lambda|\tau\iota|^{\mathrm{a}}}-1)$

with$\lambda>0$$n=1$ ,

$n=2$ , $,

for
for

for

$n\geq 4$,, which need not be the same as that of $(C)_{m}$ or of $(D)_{m}$ .

-theory the nonlinearity is required toRemark 8. In the framework of pureat least $p\geq 1+4/n$ at the origin. On the other hand,behave as a power
the nonlinearity is required to have the differentiabihty of order greater than orequal to $n/2$ at the origin. To take those requirement into account, it is sufficientfor
for $n=1,$
to suppose that the nonlinearity should behave as a power$n=2$ , $, andfor $n\geq 4$ to keep everything smooth. This is the reason why wehave imposed additional power behavior at the origin of the nonlinearity. Although$H^{\iota}$$u^{p}$$s$$u^{3}$$u^{5}$$u^{2}$

there is a room to reduce the order of power behavior at the origin to the minim
value $1+4/n$ , that is outside the purpose of this paper since we intend to keep the$a1$

exposition not too technical.
Remark 9.

To our knowledge there is $no$ other work to treat the Schr\"odinger

equation with nonlinearity of exponential growth in the $H$ -theory with $s\leq n/2$ .
at infinity seems to be
In view of budinger’s in equality the growth rate as. Note that the -norm is out of control of the
optimal at the level of-norm even when the latter is infinitesimally small.
$e^{\lambda|z|^{2}}$$H^{n/2}$$L^{\infty}$$H^{n/2}$

Remark 10. The theorem above proves the existence an asymptotic completeon the small asymptotic states:
ness of the wave operators$\mathrm{d}$$W_{\pm}$in$H^{n/2}$$\phi_{\pm}\mapsto u(\mathrm{O})=\phi$$\phi_{\pm}$.

We now give a brief sketch of the proofs. As usual the method depends on afor $an$or with metric oncontraction argument on (3) in$X=X,\mathrm{Y}^{\iota}$$Z$$L^{q}(\mathbb{R};L^{\tau})$129
admissinle pair of exponents $(q, r)$ in which the Strichartz type estimates for the freepropagator fit naturally. For that purpose we prove thatthe norms appearing
in the definition of $X$ are reproduced by the right hand side of (3) and that the
metric on
is contracted. At a technical level we need the following key
estimates. We use Lemma 1 for the proof of Theorems 1 and 2, while Lemma 2 is.
required to estimate the exponen.tial functions bom Theorem 3.
$\mathrm{a}1$$L^{q}(\mathrm{R};L^{f})$Lenmla 1.

Let

$p$

and

$\iota$satisfy $1\leq p<\infty$ and $0\leq s\leq p$ . Let$1/t=1/r+(p-1)/m$ . Let$(l)When\delta$ is not an integer, assume in addition th at $r,m\geq 2$ and$1<t\leq\gamma<\infty,$ $1<m\leq\infty,$$f$$t,$ $r,$ $m$$f\in C^{[\cdot]}(\mathrm{c};\mathrm{C})$satisfy.

$\epsilon<p$and thatsa $tisRes(A)_{[_{l}}]$ . Then$||f(u);\dot{B}i||\leq C||u;\dot{B}_{m}^{0}||^{p-\mathrm{l}}:||u;\dot{B},$$||$$||f(u);\dot{B}i||\leq C(||u;\dot{B}_{\infty}^{0}||+||u;L^{\infty}||)p-1||‘ u;\dot{B}i||$(2) $WhenS$ is an int eger, assume that

$fsat\mathrm{i}_{S}{\rm Res}(B)_{[\cdot]}$

any

$q$

$m<\infty$,

if$m=\infty$.

. Then$||f(u);\dot{H}i||\leq C||u;L^{m}||p-1||u;\dot{H}i||$

Lemma 2.

if.

Let $1<r<\infty$ . Then th $ere$ exists a constant $C_{0}>0$ such that forwith $r\leq q<\infty$$the\vee f.o\mathrm{J}lowing$

estimates hold.

$||u;L^{q}||\leq c_{0}q^{1/(-}’|2+2)/(2q)|u;\dot{H}n/2||^{1}-,/q||u;L^{\mathrm{f}}||$” $q$,$||u;\dot{B}_{q..\prime}^{0}||\leq C_{0}q^{1/}’-2)/(2q)|\sim 2+(|u;\dot{H}n/2||^{1-}’/q||u;\dot{B}0||^{\prime,q}$.

The proof of Lemma 1 follows closely that of [7; Lemma 3.4] in the se nse that we

make se of $an$ equivalent norm on Besov spaces in terms of modulus of continuity$\mathrm{u}$with the second differences, though actual proof is rather involved because of higher
derivatives of functions. Lemma 2 follows$\mathrm{i}\mathrm{n}e$$\mathrm{h}\mathrm{o}\mathrm{m}$qualities between Besov and Sobolev spaces.

[ $18$ ; Inequality (2.6)] and convexity$\mathrm{S}\mathrm{e}e-[17,18]$

for details.

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