Advanced Studies in
Pure
Mathematics
23, 1994
Spectral
and
Scattering
Theory
and
Applications
pp. 83-103
Scattering
Theory
in
the
Energy
Space
for a
Class
of
Nonlinear
Wave
Equations
J.
Ginibre
Dedicated
to
Professor
ShigeToshi
Kuroda
on
his
simtieth
birthday
§1.
Introduction
The
purpose
of
this
talk
is
to
present a survey of
the
theory
of
scattering
for a class of nonlinear wave
equations
of
the
form
(1.1)
in a space
of
initial
data
and
asymptotic
states
as large
as
the
energy
space associated
with
that
equation.
The
exposition will follow
the
treat-
ment
given
in
[12].
Here
<p
is a complex valued function defined
in
space
time
JRn+l,
a is
the
Laplace
operator
in
!Rn,
and
f is a nonlinear suit-
ably
regular complex valued function satisfying polynomial
bounds
at
zero
and
at
infinity. A large
amount
of work
has
been
devoted
to
the
theory
of
scattering
for
the
equation
(1.1)
and
for several
other
equa-
tions,
and
we shall devote most of
this
introduction
to
a
partial
review
of nonlinear
scattering
in
order
to
put
the
subsequent
treatment
of
(1.1)
into
perspective.
The
general
setting
is
the
following.
One
considers a semilinear
equation
(1.2)
8tu
=Lu+
F(u)
where L is a linear antiselfadjoint
operator
in
some Hilbert space
1£,
and
generates a one
parameter
unitary
group U(t) =
exp(tL)
in
1£.
One
is
interested in
situations
where
the
global
Cauchy
problem
for (1.2) is well
understood
in some space X ( which
may
or
may
not
coincide
with
1£).
In
particular
any
initial
data
u
0
E X should generate a unique global
X valued solution of (1.2)
with
u(O)
= u
0
and
with
suitable regularity
Received
February
2, 1993.
84
J.
Ginibre
in time.
One
is
then
interested
in
studying
the
asymptotic
behaviour
in
time
of
the
solutions
of
{1.2)
by comparison
with
the
solutions
of
the
linear
equation
{1.3)
hereafter referred
to
as
the
free equation.
That
study
gives rise
to
the
following two questions.
(1)
Given
U±
EX,
does
there
exist a {unique) solution u of
the
equation
{1.2)
that
behaves
at
t-+
±oo as
the
solution
U(•)u±
of
the
free
equation
{1.3)
generated
by
u±,
for instance in
the
sense
that
{1.4)
llu(t) -
U(t)u±;
XII
-t
0
as
t-+
±oo
or
(1.4')
IIU(-t)u(t)
- U±i XII
-t
0
as
t-+
±oo.
If
that
is
the
case, one defines
the
wave
operators
n±
as
the
maps
U±
-+
u(O)
thereby
obtained.
This
first question is referred
to
as
that
of
the
existence
of
the
wave operators. Actually, one
may
be
interested
in
comparing solutions
of
{1.2)
and
{1.3)
in
a sense different from
and
in
fact stronger
than
{1.4) {1.4
1
).
For
instance
one
may
require
that
{1.5)
llu
-
U(-)u±;
X([T, ±oo))II
-t
0
as
T-+
±oo
where X ( I) is a space
of
X valued functions defined
in
a
time
interval J
with
prescribed behaviour in time. Such a convergence is
in
fact needed
in
order
to
develop a consistent
theory
of scattering.
The
second question is somehow
the
converse
of
the
first one.
(2)
Given a generic X valued solution
of
{1.2)
generated
by
initial
data
u(O)
=
uo
E
X,
does
there
exist
U±
E X such
that
u behaves asymp-
totically as
U(·)u±
as t
-+
±oo
in
the
same
sense as above.
If
that
is
the
case for all u
0
E
X,
one says
that
asymptotic
completeness {AC)
holds
in
X.
Note
that
this
notion
of
asymptotic
completeness is very
restrictive, since
the
only
asymptotic
evolution which is
used
is
the
free
evolution. In
the
linear
quantum
mechanical
many
body
problem,
this
would correspond
to
the
case where
asymptotic
completeness is achieved
by
the
completely free channel, a
situation
typical
of
purely repulsive in-
teractions.
A general
method
to
prove
the
existence
of
the
wave operators,
and
the
one
to
be
used
in
all
the
examples
to
follow, consists
in
solving
the
Cauchy problem for
{1.2)
with
infinite initial time.
In
fact
the
Cauchy
Scattering Theory for Nonlinear Wave Equations
85
problem for (1.2)
with
initial
data
uo
at
time
to is equivalent
to
the
integral
equation
(1.6)
u(t)
=
U(t
- t
0
)u
0
+
{tdrU(t
-
r)F(u(r)).
lto
The
solution u expected
to
behave
as
U(•)u±
at
t
---+
±oo
should
then
be
obtained
by
taking
uo
=
U(t
0
)u±
and
letting
t
0
---+
±oo.
Re-
stricting
one's
attention
to
positive
times
for definiteness, one
obtains
the
equation
(1.7)
u(t)
= U(t)u+ - [
00
drU(t
-
r)F(u(r))
to
be
solved for u for given u+·
One
can
then
try
to
solve (1.7)
by
a
contraction
method
in
a
time
interval
[T,
oo) for T sufficiently large,
and
then
continue
the
solution u
thereby
obtained
to
all times by using
the
known results
on
the
Cauchy
problem
at
finite times.
The
contraction
step
requires
the
use
of
a space
X([T,
oo))
of
X valued functions of
time
with
a suitable
time
decay,
in
order
to
control
the
integral
in
(1.7).
That
time
decay
has
to
be
satisfied
by
the
solutions U(-)u+
of
the
free equa-
tion. As a
standard
by
product
of
the
previous
method,
one
obtains
a
proof
of
the
existence
of
global solutions
and
of
asymptotic
completeness
for small
data.
The
method
also requires
that
F(u)
exhibit a suitable
decay
in
time
for u
in
the
relevant space X (
[T,
oo)).
This
in
turns
re-
quires
that
the
function F
tend
to
zero sufficiently fast when u
tends
to
zero.
In
the
case where F satisfies power
bounds
in
u
as
u
---+
O,
that
condition reduces
to
lower
bounds
on
the
associated exponents.
Asymptotic
completeness for general
data,
once
the
previous results
are
available, reduces
to
proving
that
generic solutions of (1.2)
with
initial
data
in
X exhibit
the
time
decay
that
is used in
the
definition
of
the
space X(·) used
to
solve
the
Cauchy
problem
at
infinity.
The
question
of
AC therefore reduces
to
the
derivation
of
a priori
estimates
and
depends
in
a specific way
on
the
invariances
and
conservation laws
of
the
equation
at
hand.
As should
be
clear from a previous remark,
it
always requires a repulsivity condition
on
the
interaction
term
F.
We now review briefly some
of
the
available results for
the
most
studied
equations, namely
the
nonlinear Schrodinger (NLS)
equation
(1.8)
i8t'P =
-(1/2)~cp
+
f(cp),
the
nonlinear wave (NLW)
equation
(1.1),
and
the
nonlinear Klein Gor-
don
(NLKG)
equation
(1.9)
86
J.
Ginibre
which differs from (1.1)
by
the
presence
of
a mass
term
m
2
1.p.
For clarity
we
restrict
our
attention
ot
the
case where
the
nonlinear
interaction
term
is a single power
(1.10)
For
those
three
equations,
the
global Cauchy problem is well
understood
in
the
energy space X
0
,
to
be
defined below, for>. 2 0
and
1
Sp<
p*
=
1 +
4/(n
-
2)
in
space dimension n 2
2.
For
the
NLS equation, one
takes
u =
'P
and
F(u) =
-if('P),
the
free
evolution
group
is U(t) =
exp(i(t/2)~),
the
conserved energy is
(1.11)
E(1.p)
=
(1/2)Jlv7'PII~
+ JdxV(1.p)
where
11
·
llr
denotes
the
norm
in
U =
Lr(JR.n)
and
(1.12)
in
the
special case (1.10).
Furthermore
the
L
2
norm
of
'P
is also con-
served,
and
the
energy space is
the
standard
Sobolev space X
0
= H
1
.
For
the
NLW
and
NLKG
equations, one
takes
u = (
'P,
8t'P)
and
F(u) = (O,-f(1.p)).
The
solution
of
the
free
equation
generated
by
the
initial
data
u
0
=
('Po,
'¢
0
)
at
time
t = 0 is
(1.13)
where
K(t)
=
w-
1
sin wt,
K(t)
= cos wt, w =
FE
for NLW
(w
✓-~
+ m
2
for NLKG), so
that
the
free evolution
group
is
(1.14)
U(t) = (
k(t)
~(t))
.
-w
2
K(t) K(t)
The
energy is
(1.15)
for NLW(NLKG),
and
is conserved
in
the
sense
that
E(1.p,
8t'P)
=
Const.
for solutions
of
the
equation.
The
energy space is X
0
= (iI1
n£P+
1
) ffiL
2
for NLW
and
X
0
= H
1
ffi
L
2
for NLKG, where H
1
is
the
homogeneous
Sobolev space associated
with
H
1
.
We now
summarize
the
main
results available
on
the
existence
of
the
wave
operators
for
the
NLS, NLW
and
NLKG
equations
with
power
Scattering Theory for Nonlinear Wave Equations
87
nonlinearity (1.10). For
the
NLS
equation
(5,
6, 7, 9, 17, 25, 42, 45, 46],
the
wave
operators
are
known
to
exist
in
the
energy space X
0
= H
1
for
4/n
< p - l <
4/(n
- 2)
[7].
In
the
smaller space X = I: defined by
(1.16)
the
wave
operators
are
known
to
exist for
4/(n
+ 2) <
p-1
<
4/(n
- 2)
[5].
Finally for O < p -
l::;
2/n,
the
wave
operators
do
no
exist even in
the
£
2
-sense, namely (1.4)
with
X = L
2
implies
U±
= 0
and
u = 0
(41].
There
is a huge
literature
on
the
theory
of
scattering
and
related
problems (including global existence for small
data)
for
the
NLW equa-
tion
[10-12, 14-16, 18, 19, 22-24, 28,
29,
33-36, 38-42]. For
that
equa-
tion,
the
wave
operators
are
known
to
exist
in
the
space
(1.17)
for p
1
(n) < p < p*, where p
1
(n) is
the
larger
root
of
the
equation
(28,
29]
(1.18)
n(n
-
l)p
2
- (n
2
+ 3n - 2)p + 2 = 0.
That
lower
bound
on
p is
not
expected
to
be
optimal
however.
One
expects
the
same
result
to
hold (possibly
in
a smaller space) for p
0
(n) <
p < p*, where p
0
(n) is
the
larger
root
of
the
equation
(1.19)
(n
-
l)p(p
-
1)
= 2(p + 1).
That
result is proved only
in
dimensions n = 2
and
3
and
on
special
sets of regular
asymptotic
states
(15,
18, 36]. For p ::; p
0
(n),
the
wave
operators
are
expected
not
to
exist,
in
view of
the
existing finite
time
blow
up
results for small solutions
(14,
18, 19, 39].
In
the
energy space
X
0
,
the
wave
operators
exist
under
assumptions
on
f which barely fail
to
include (1.10)
with
p = p*,
the
reason being
that
the
lower limit
on
p
required for
the
existence
of
the
wave
operators
turns
out
to
be
p > p*
in
that
case
and
conflicts
with
the
condition p < p* required
to
solve
the
global Cauchy problem
at
finite times
(12].
It
is one
of
the
purposes of
this
talk
to
present
that
theory.
For
the
NLKG
equation
(3,
4, 9, 31],
the
wave
operators
are
expected
to
exist for
4/n
<
p-l
<
4/(n-2)
in
the
energy space
and
for p
0
(n+l)
<
p <
p*
in
a
suitably
smaller space,
but
the
available
treatments
of
the
problem
in
the
literature
do
not
seem
to
be
optimal.
We
next
summarize
the
main
results available
on
the
question
of
asymptotic
completeness (AC) for
the
same
equations. As mentioned
above,
the
proof
of
AC requires a repulsivity condition, namely
>.
~
0 in
