Internal stabilization for the wave equation
Talk at Chengdu (China) 2004
1
Motivation
The purpose of this talk is to give some appliations of the quantiation of unique ontinuation
property in the framework of stabilization for the wave equation in bounded domain.
Here, we deal with the following unique ontinuation property: if the solution of the wave equation with
homogeneous Dirihlet boundary ondition is null in a subdomain then the initial data are identially
zero.
The more strong quantiation of unique ontinuation property is given by the following observability
estimate:
Let be a bounded onneted C 3 don
main in R , n 1, and ! be a non-empty open subset of . Let us onsider the following wave
equation in R, with initial data and homogeneous Dirihlet boundary ondition :
8
t2 u u = 0 in R
<
u = 0 on R
: u (; 0) = u , u (; 0) = u in .
0 t
1
Under geometrial hypothesis, there exist a onstant > 0 and a time T > 0 suh that for all initial
data (u0 ; u1 ) 2 H01 (
) L2 (
), the solution u satises the following estimate :
Theorem, Observabilty estimate [ BLR℄, [ Bu℄ .-
k(u0 ; u1 )k2L2 (
)H 1 (
) Z TZ
0
!
ju (x; t)j2 dxdt .
whih implies that the energy E (w; t) = 12 R
jt w (x; t)j2 + jrw (x; t)j2 dx of a damped wave equation w dened below, satises
E (w; T ) + 1 E (w; 0) < E (w; 0)
More preisely, we have the following stabilization result :
Let be a bounded onneted C 3 domain in Rn ,
n 1, and ! be a non-empty open subset of . Let 2 C01 (! ) be a non-negative funtion. Let
us onsider the following damped wave equation in (0; +1), with initial data and homogeneous
Theorem, Exponential deay [ BLR℄ .-
Dirihlet boundary ondition:
8 2
< t w w + (x) t w = 0 in (0; +1)
w = 0 on (0; +1)
:
w (; 0) = w , w (; 0) = w in .
0
1
t
1
Under geometrial hypothesis, there exist two onstants ; > 0, suh that for all initial data
(w0 ; w1 ) 2 H01 (
) L2 (
), the solution w satises the following deay estimate :
kw (; t) ; t w (; t)k2H01 (
)L2 (
) e
t
k(w0 ; w1 )k2H01 (
)L2 (
) , 8t 0 .
Both above theorems are equivalent.
The geometrial hypothesis are that, rst we an dene a unique generalized geodesi (x (t) ; (t)) and
then, every generalized geodesi (x (t) ; (t)) 2 Rn nf0g meets ! Rn nf0g at time T .
These geometrial hypothesis are automatially satised if ! = or if n = 1.
Furthermore, these geometrial hypothesis are neessary to get an uniform and exponential deay rate.
Indeed, we an use the onstrution of gaussian beams to prove that :
Let be a smooth bounded onneted domain in Rn , n 1, and ! be a
non-empty open subset of . Let 2 C01 (! ) be a non-negative funtion. If for all T > 0, there exists
a ray x (t) whih never meets ! for t 2 [0; T ℄, then for all " > 0, there exists a solution w" of
8 2
< t w" w" + (x) t w" = 0 in (0; +1)
w" = 0 on (0; +1)
:
w" (; 0) = w"0 , t w" (; 0) = w"1 in ,
Theorem [ BLR℄ .-
suh that
E (w" ; 0) = 1 and E (w" ; t) > 1
" for all t 2 [0; T ℄
.
Also, this means that we an not hope to get an observability estimate. Other kind of estimates have
been introdued by F. John and we alled it Holder estimate or logarithmi estimate. Suh estimates
still tradue the unique ontinuation property.
Without any hypothesis on the geometry, we will prove the following logarithmi estimate :
Let be a bounded onneted C 2 domain in Rn , n > 1,
and ! be a non-empty open subset of . Let us onsider the following wave equation in R, with
Theorem, Logarithmi estimate .-
initial data and homogeneous Dirihlet boundary ondition :
8
t2 u u = 0 in R
<
u = 0 on R
: u (; 0) = u , u (; 0) = u in
0 t
1
.
Then, there exist a onstant > 0 and a time T > 0 suh that for all initial data (u0 ; u1 ) 2 H01 (
) L2 (
), (u0 ; u1 ) 6= 0, the solution u satises the following estimate :
k(u0 ;u1 )k2H 1 (
)L2 (
) Z Z
0
T
k(u ;u )k2
u0 ; u1 k2L2 (
)H 1 (
) e 0 1 L2 (
)H 1 (
)
ju x; t j2 dxdt
0 !
k(
)
( )
.
Remark that this estimate is equivalent to
k(u0 ; u1 )k2L2 (
)H 1 (
) k(u0 ;u1)k 1 2 k(u0 ; u1 )k2H01 (
)L2 (
) ,
ln 2 + kukL2H(!0(
)(0;TL)) (
)
2
or equivalently,
Z TZ
k(u0 ; u1 )k2L2 (
)H 1 (
) e="
ju (x; t)j2 dxdt + " k(u0 ; u1 )k2H01 (
)L2 (
) , 8" > 0 ,
0 !
where the value of the onstant > 0 may hanged from line to line but not its dependene.
Now let us give the assoiate stabilization result :
Let be a bounded onneted C 2 domain in Rn , n > 1, and
! be a non-empty open subset of . Let 2 C01 (! ) be a non-negative funtion. Let us onsider the
following damped wave equation in (0; +1), with initial data and homogeneous Dirihlet boundary
Theorem, Logarithmi deay .-
ondition:
8 2
< t w w + (x) t w = 0 in (0; +1)
w = 0 on (0; +1)
:
w (; 0) = w , w (; 0) = w in .
0
1
t
Then, there exists a onstant > 0 suh that for all initial data (w0 ; w1 ) 2 H01 (
) L2 (
), the
solution u satises the following L2 deay estimate :
kw (; t) ; t w (; t)k2L2 (
)H 1 (
) ln (2+ t) k(w0 ; w1 )k2H01 (
)L2 (
) , 8t 0 .
The proof of the logarithmi deay rate of the damped wave equation is easy if we have the logarithmi
estimate for the wave equation. Now, we will fous our attention on the proof of the logarithmi
estimate. We divide it into two steps:
step 1 : redution to an unique ontinuation for the ellipti operator via a gaussian transform
step 2 : Holder estimate for the ellipti operator with Dirihlet boundary ondition
Let be a bounded onneted C 2
1
0 and Æ 2 (0; T=2). Let us onsider the following ellipti
equation of seond order in (0; T ) with the Dirihlet boundary ondition :
8 2
< t v + x v = 0 in (0; T )
v = 0 on (0; T )
: v = v (x; t) 2 H 2 (
(0; T )) .
Theorem, Holder estimate for Laplae operator .-
domain in Rn , n > . We hoose T >
Then, for all ' 2 C01 (
(0; T )), ' 6= 0, there exist > 0 and 2 (0; 1) suh that for all solution v,
we have :
! Z T Z
!1 Z T ÆZ
Z TZ
2
2
2
jv (x; t)j dxdt jv (x; t)j dxdt
j'v (x; t)j dxdt
.
Æ
0 0 Remark that this estimate is equivalent to
Z T ÆZ
Æ
jv (x; t)j2 dxdt e="
Z TZ
0
j'v (x; t)j2 dxdt + e
1="
Z TZ
0
jv (x; t)j2 dxdt , 8" > 0 ,
where the value of the onstant > 0 may hanged from line to line but not its dependene.
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