Logarithmi L2 deay rate
for the damped wave equation
Kim Dang PHUNG
1
Introdution and main result
The purpose of this note is to prove the following result:
Theorem .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 ondition:
8
< t2 u u + (x) t u = 0 in (0; +1)
u = 0 on (0; +1)
(1.1)
:
u (; 0) = u0 , t u (; 0) = u1 in .
Then, there exists a onstant C > 0 suh that for all initial data (u0 ; u1 ) 2 H01 (
) L2 (
), the
solution u of ( 1.1) satises the following L2 deay estimate :
ku (; t) ; t u (; t)k2L2 (
)H
1
(
) 2
ln (2 + t) k(u0; u1)kH01 (
)L2(
) , 8t 0 .
C
(1.2)
Similar kind of deay estimate already appears in [ LR℄, [ A℄ and [ L℄ in the framework of boundary
stabilization for hyperboli equations and for the H01 deay rate of the solution of damped wave equation
with H 2 (
) \ H01 (
) H01 (
) initial data (whih means that our result is not new).
2
Proof of Theorem
We begin to prove the following result:
Proposition 1 .Let be a bounded onneted C 2 domain in Rn , n > 1, and ! be a nonempty open subset of . Let 2 C01 (! ) be a non-negative funtion. Let us onsider the following
damped wave equation in (0; +1), with smooth initial data and homogeneous Dirihlet boundary
ondition:
8
< t2 u
:
u + (x) tu = 0 in (0; +1)
u = 0 on (0; +1)
u (; 0) = u0 , t u (; 0) = u1 in .
1
(2.1)
Then, there exists a onstant C > 0 suh that for all initial data
the solution u of ( 2.1) satises the following deay estimate :
(u0; u1) 2 H 2 (
) \ H01 (
) H01 (
),
ku (; t) ; t u (; t)k2H01 (
)L2 (
) ln (2C+ t) k(u0 ; u1 )k2H 2 (
)\H01 (
)H01 (
) , 8t 0 .
(2.2)
Indeed, we know the following result.
Proposition 2 .Let be a bounded onneted C 2 domain in Rn , n > 1, and !o ! be a
non-empty open subset of . Let us onsider the following wave equation in R, with initial data
(u0; u1) 2 H01 (
) L2 (
) and homogeneous Dirihlet boundary ondition :
8
t2 z z = f in R
<
z = 0 on R
(2.3)
:
z (; 0) = u0 , t z (; 0) = u1 in .
Then, there exist a onstant > 0 and a time T > 0 suh that for all f 2 L2 (
(0; T )), for all
(u0; u1) 2 H 2 (
) \ H01 (
) H01 (
), the solution z of ( 2.3) satises the following estimate :
R
R
R
R
k(u0 ; u1)k2H01 (
)L2(
) e=" 0T !o jt z (x; t)j2 dxdt + 0T jf (x; t)j2 dxdt
+" k(u0; u1)k2H2 (
)\H01 (
)H01 (
) , 8" > 0 .
(2.4)
Now, we apply it with z = u and f = tu. We obtain that there exist a onstant > 0 and a
time T > 0, suh that for all (u0; u1) 2 H 2 (
) \ H01 (
) H01 (
), the solution u of the damping wave
equation (2.1) satises the following estimate :
)k2 1
Z
T
Z
( ) j ( )j2 dxdt + " k(u0; u1)k2H2 (
)\H01 (
)H01 (
) , 8" > 0 .
0 (2.5)
Clearly, reproduing the proof, we also have that there exist a onstant > 0 and a time T > 0, suh
that for all (u0; u1) 2 H 2 (
) \ H01 (
) H01 (
), the solution u of the damping wave equation (2.1)
satises the following estimate : for all 0,
R
R
k(u (; ) ; t u (; ))k2H01 (
)L2 (
) e=" +T (x) jt u (x; t)j2 dxdt
(2.6)
+" k(u0; u1)k2H2 (
)\H01 (
)H01 (
) , 8" > 0 .
that is
R
k(u (; ) ; t u (; ))k2H01 (
)L2 (
) e=" +T dtd k(u (; t) ; t u (; t))k2H01 (
)L2 (
) dt (2.7)
+" k(u0; u1)k2H2 (
)\H01 (
)H01 (
) , 8" > 0 .
k(u0 ; u1
H0 (
) L2 (
)
e
="
x t u x; t
Let us denote for some onstant o > 0, only depending on ,
k(u (; ) ; t u (; ))k2H01 (
)L2 (
)
L ( ) =
1
(2.8)
2
o k(u0 ; u1 )kH 2 (
)\H 1 (
)H 1 (
)
0
0
Consequently from (2.7), there exist a onstant > 0 and a time T > 0, suh that for all 0,
.
L ( + T ) (2.9)
ln 3 + L() L1 (+T )
2
Let t > 0, we write t = nT + s with 0 s < T , so nT t < (n + 1) T . From the non-inreasing
property of L, we hek that L (t) L (nT ). Now we hoose = (n 1) T 0 and we get the following
estimate:
.
L (nT ) (2.10)
ln 3 + L((n 1)T1) L(nT )
Suppose that 0 < L ((n 1) T ) L (nT ) n1 , then
L (nt) ln (3+ n) ,
Suppose that L ((n 1) T ) L (nT ) > n1 , then
L (nT ) 1 < L ((n 1) T ) L (nT ), so L (nT ) <
(2.11)
+ 1 L ((n 1) T ) ,
n)
for n > 1, we obtain
and using the dereasing property of n 7 ! ln(2+
n
n
n
ln(3+n) L (nT )
<
<
Then we have for all n > 1, either
ln(3+n) n L ((n
n+1
ln(3+(n 1)) L ((n
n
n
1) T ) ln(3+(n
1) T ) .
ln(3+n) L (nT ) 1
ln(3+
n)
L (nT ) < ln(3+(n 1)) L ((n
or Then indutively, we onlude that for all n > 1,
ln (3 + n) L (nT ) 1 ,
1))
ln(3+(n 1))
1) T ) .
whih gives for all t T
k(u (; t) ; t u (; t))k2H01 (
)L2 (
)
= L (t) L (nT ) ln (3+ n) ln (2 + t=T ) ,
2
o k(u0 ; u1 )kH 2 (
)\H 1 (
)H 1 (
)
0
0
that is for all t 1
k(u (; tT ) ; t u (; tT ))k2H01 (
)L2(
) o ln (2+ t) k(u0 ; u1 )k2H 2 (
)\H01 (
)H01 (
) ,
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
that is for all t T 2
k(u (; t) ; t u (; t))k2H01 (
)L2(
) o ln (22+ t) k(u0 ; u1 )k2H 2 (
)\H01 (
)H01 (
) .
(2.18)
Now we onludeR tthe proof of the L2 deay as follows.
let v (x; t) = 0 u (x; `) d` 1 (u1 (x) + (x) u0 (x)), then v (x; t) = R0t u (x; `) d` + u1 (x)+
(x) u0 (x) and t v (x; t) = u (x; t),
t2 v (x; t) + (x) t v (x; t) = t u (x; t) + (x) u (x; t)
= tu (x; 0) + R0t t2u (x; `R) td` + (x) u (x; 0) + (x) R0t tu (x; `) d`
= u1 (x) + (x) u0 (x) + 0 u (x; `) d`
= v (x; t)
(2.19)
3
Also, v (x; 0) =
1 (u1 (x) + (x) u0 (x)), tv (x; 0) = u0 (x), and the homogeneous Dirihlet
boundary ondition is satised. Consequently, there exists a onstant C > 0 suh that for all t 0,
ku (; t) ; t u (; t) + u (; t)k2L2 (
)H 1 (
) k(v (; t) ; t v (; t)) k2H01 (
)L2 (
) C
1 (u1 + u0 ) ; u0 2H2 (
)\H01 (
)H01 (
) ,
ln(2+
t)
(2.20)
but 1 (u1 + u0)H2 (
) is bounded by k(u0; u1)kH01 (
)L2(
).
This ompletes the proof.
Referenes
[ A℄ K. Ammari, Dirihlet boundary stabilization of the wave equation, Asymptoti Analysis 30 (2002)
117-130.
[ L℄ G. Lebeau, Equation des ondes amorties, in Algebrai and Geometri Methods in Mathematial
Physis, Math. Phys. Stud. 19, Kluwer Aad. Publ., Dordreht, 1996,p.73-109.
[ LR℄ G. Lebeau and L. Robbiano, Stabilisation de l'equation des ondes par le bord, Duke Math. J.
86 (1997) 465-491.
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