On self similar solutions well posedness and

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Fabrice Planchon

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On self-similar solutions, well-posedness and the conformal wave equation Fabrice Planchon Abstract We prove that the initial value problem for the conformally in- variant semi-linear wave equation is well-posed in the Besov space _ B 1 2 ;1 2 (R n ). This induces the existence of (non-radially symmetric) self-similar solutions for homogeneous data in such Besov spaces. Introduction We are interested in the Cauchy problem for the conformal semi-linear wave equation (1) 8 < : u = juj n+3 n1 ; u(x; 0) = u 0 (x); @ t u(x; 0) = u 1 (x): As usual, scaling plays an important role when looking for the lowest possible regularity on the data: it reads (2) 8 > < > : u 0 (x) ! u 0; (x) = n1 2 u 0 (x) u 1 (x) ! u 1; (x) = n+1 2 u 1 (x) u(x; t) ! u (x; t) = n1 2 u(x; t): Well-posedness

wave equation

equation has

similar solution

strichartz estimates

conformal semi-linear

super-conformal

handle when dealing

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Planchon Wave equation

On
self-similar
solutions,
w
2
0
place
ell-p
)
osedness
the
and
u
the
x
conformal
t
w
lo
a
d'Analyse
v
it
e
;
equation
(
F
1
abrice
=
Planc
wn
hon
2

data)
Abstract
([13]).
W
ersit
e
Cedex
pro
>
v
)
e
=
that
x
the
u
initial
n
v
u
alue
(
problem
2
for
ell-p
the
initial
conformally
)
in-
2
v
(or
arian
direct
t
estimates
semi-linear
er-conformal
w
URA
a
et
v
75
e
the
equation
(2)
is
>
w
(
ell-p
u
osed
x
in
n
the
0
Beso
u
v
)
space
;
_
=
B
2
1
x
2
x;
;
u
1
t
2
n
(
(
R
:
n
is
).
hold
This
(
induces
u
the
H
existence
H
of
Indeed,
(non-radially
in
symmetric)
for
self-similar
is
solutions
quence
for
ersiv
homogeneous
Stric
data
the
in
Lab
suc

h
189,
Beso
e
v
Curie,
spaces.
BC
In
P
tro
on
duction
data:
W
reads
e
8
are
<
in
:
terested
0
in
x
the
!
Cauc
0
h
(
y
)
problem

for
1
the
u
conformal
(
semi-linear
)
w
1
a
x
v
!
e
1
equation
(
(1)
)
8

<
+1
:
u

(
u
)
=
(
j
t
u
!
j

n
x;
+3
)
n

1
1
;
u
u
x;
(
)
x;
W
0)
osedness
=
kno
u
to
0
for
(
data
x
u
)
;
;
1
@
2
t
1
u

(
1
x;
([6]).
0)
this
=
cal
u
time
1
global
(
small
x
result
)
a
:
conse-
As
of
usual,
disp
scaling
e
pla
of
ys
hartz
an
In
imp
sup
ortan

t
oratoire
role
Num
when
erique,
lo
CNRS
oking
Univ
for

the
Pierre
lo
Marie
w
4
est
Jussieu
p
187,
ossible
252
regularit
aris
y
1range,
i.e.
a
non-linearit
)
details),
as
y
complicated
with

a
dealt
p
adapted
o
least
w
j
er
e
p
One
>
are
n
p
+3
w
n
[12]
1
Beso
,
2
w
j
ell-p
b
osedness
The
holds
j
in
of
H
w
s
ected
p
+
with
N
s
a
p
in
=
Beso
n
y
2
o
2
terizations
p
are
1
that
,
for
and
(2
w
S
as
)
obtained
B
later
(
in

[9])
elongs
b
the
y
h
dev
10]
eloping
with
suitable
h
generalizations
generic
of
with
Stric
e
hartz
n
estimates.
w
In
text
[10],
The
w
non
e
range,
generalized
to
these
dev
results
pro
to
in
w
of
ell-p
w
osedness
through
in
calization
the
of
Beso
Definition
v
n
space
1
_
b
B
>
s
=
p
,
;
,
1
j
2
0
,
n
allo
b
wing
and
self-similar
m
solutions
v
for
distribution.
suitable
j
(small)
k
initial
.
data.
remark
Ho
yp
w
y
ev
b
er,
in
the
p
pro
u
of
2
breaks
although
do
are
wn
generalize
for
(
p
u
=
2
n
Here
+3
deal
n
n
1
2
,
a
while
y
it
the
is
the
nev
e
ertheless
tec
the
apply
easiest
teger
case
sup
to
ypassing
handle
hniques
when
osition
dealing
spaces
with
ed
data
references
in
the
Sob
F
olev
sense
spaces.
regularit
Our
data.
purp
this
ose
recall
is
spaces
to
c
generalize
frequency
the
]
results
the
of
w
[10
terested
]
Let
to
(
this
suc
limiting

case,
j
and
1
b
=
y

an
,
in
x
teresting
nj
t
x
wist
j
w

e
j
will
+1
ha
Let
v
in
e
R
to
s
rely
.
on
y
the
to
full
p
range
if
of
sum
Stric

hartz
)
estimates
to
[5,
temp
9,
The
7]
=
for
k
frequency
f
lo
p
calized
l
functions,
]).
while
nal
our
concerns
results
t
in
e
[10
non-linearit
]
whic
w
can
ere
e
only
with:
using
[9,
[13].
only
W
o
e
ers
refer
p
to
p
[10]
N
for
treated,
a
suc
more
results
ex-
exp
tensiv
do
e
to
discussion
F
on
u
the

connection
p
with
p
self-similar
R
solutions.
.
W
w
e
will
simply
with
recall
=
that
+3
_
1
H
=
1
in
2
suitable
is
a
the
,
scale-in
to
v
con
arian
of
t
w
space
v
with
equation.
resp
same
ect
hnique
to
ould
(1),
for
and
in
relaxing
p
the
the
data
er-conformal
f
b
(
more
x
tec
)
related
to
comp
b
in
e
v
in
(as
_
elop
B
in
1
and
2
therein),
;
vided
1
regularit
2
of
allo
is
ws
some
for
at
scale
the
in
y
v
the
arian
T
t
end
data
section
and
e
therefore
what
leads
v
to
are
self-similar
their
solutions
harac-
for
via
a
lo
large
([1
class
for
of
for
data,
range
whic
indices
h
e
do
in
not
in.
need
1
to

b
S
e
R
radially
)
symmetric.
h
Recall
b
suc
=
h
for
an

homogeneous

data
and
of

degree
0
1
j
f
j
(
2
x

)
(
=
)

2
(

x=
j
j
)
x
S
j
=
)
j
=

j

x
=
j
j
is
S
c
.
haracterized
f
b
e
y
S

(
2
n
_
,
H
<
1
p
2
W
(
sa
S
f
n
elongs
1
_
)
s;q
([3]).
if
This
only
is

to
partial
b
P
e
m
compared
j
with
f
requiremen
con
ts
erges
lik
f
e
a

ered
2

C
sequence
n
j
(
2
S
s
n

1
(
)
)
in
L
previous
b
w
to
orks
q
([11
2Another
closely
related
t
akly
Theorem
a
yp
<
e
(4)
of
x;
space
j
will
getting
b
2
e
k
of
1
help:
(6)
Definition
p
2
2
Let
n
u
eral
(
u
x;
2
t
2
)
<
2
(5)
S
1
0
_
(
!
R
unique
n
)
+1
t
)
p
,
0

b
j
class"
b
are
e
(
a
(
frequency
;
lo
)
calization
k
with
+
resp
2
ect
Then
to
of
the
t
x
B
v
@
ariable.
L
W
;
e
t
will
(
sa
over,
y
assumption
that
B
u
j
2
u
L
x

j
t
q
(
wher
_
p
B
(
s;q
attempt
p
largest
)
that
i
c
(3)
p
2
L
j
0
s
)
k
B