J. Math. Anal. Appl. 295 (2004) 527–538
www.elsevier.com/locate/jmaa
Note on the cost of the approximate controllability
for the heat equation with potential
K.-D. Phung
Received 2 September 2003
Available online 28 May 2004
Submitted by R. Triggiani
Abstract
We prove the approximate controllability for the heat equation with potential with a cost of order
ec/ε when the target is in H01 (Ω) with a precision in L2 (Ω) norm. Also a quantification estimate of
the unique continuation for initial data in L2 (Ω) of the heat equation with potential is established.
 2004 Elsevier Inc. All rights reserved.
1. Introduction and main results
Throughout this paper, Ω is a bounded domain in Rn , n 1, with a boundary ∂Ω of
class C 2 , ω is a non-empty open subset of Ω and T > 0 is a real number. Further, we
denote · ∞ the usual norm in L∞ (Ω × (0, T )) and we consider a = a(x, t) a function
in L∞ (Ω × (0, T )).
In this paper we study the following heat equation with a potential a in L∞ (Ω × (0, T )):

∂t u − ∆u + au = f · 1|ω in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
(1.1)

u(·, 0) = u0 in Ω,
where f ∈ L2 (ω × (0, T )), u0 ∈ H01 (Ω) and 1|ω denotes the characteristic function of the
set ω.
It is well known from [6] or [2] that we can act through f ∈ L2 (ω × (0, T )) when
u0 ∈ L2 (Ω) in order to get the null controllability result u(·, T ) = 0 and furthermore, the
following estimate holds [3]: there exists a constant c0 > 0 such that
E-mail address: kim-dang.phung@cmla.ens-cachan.fr.
0022-247X/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2004.03.059
528
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
1
2/3
f L2 (ω×(0,T )) exp c0 1 + + T a∞ + a∞
u0 L2 (Ω) .
T
(1.2)
(1.2) is an explicit estimate with respect to both quantities T > 0 and a∞ 0, of the
control function f and may be viewed as a measure of the cost of the null controllability
for the heat equation with potential.
Here we ask whether the following steering property for the heat equation with potential
holds when u0 = 0: there exist two constants D > 1 and c = c(T , a∞ ) > 1 depending
on both quantities T > 0 and a∞ 0 such that for all ε > 0, for all ud ∈ H01 (Ω), there
exists a suitable approximate control function f depending on ε such that
Du 1 d H (Ω)
0
(1.3)
f L2 (ω×(0,T )) c exp
ud L2 (Ω)
ε
and
u(·, T ) − ud 2
ε.
L (Ω)
(1.4)
Our goal is to measure the cost of the approximate control function f and furthermore to
give an explicit estimate with respect to ε, T and a∞ 0.
This problem has received a particular attention from [3] where it is proved that the cost
of the approximate controllability for the heat equation with potential is of order ec/ε if
2
ud ∈ H 2 (Ω) ∩ H01 (Ω), and of order ec/ε if ud ∈ H01 (Ω). But when a is a constant or more
generally√of the form a(x, t) = a1 (x) + a2 (t), where a1 ∈ L∞ (Ω) and a2 ∈ L∞ (0, T ), the
order ec/ ε if ud ∈ H 2 (Ω) ∩ H01 (Ω) is optimal [3, Theorem 6.2].
Let us make the first observation: by simple changes of variables, we are reduced to
check that for all ε > 0, for all T > 0, for all wd ∈ H01 (Ω) there exists a function ϑ such
that
ϑL2 (ω×(0,T )) ceD/ε wd L2 (Ω) ,
and such that the following steering property holds:
w(·, 0) − wd 2
εwd H 1 (Ω) ,
L (Ω)
0
where w is the solution of the following backward heat equation with potential:

∂t w + ∆w − aw = ϑ · 1|ω in Ω × (0, T ),
w = 0 on ∂Ω × (0, T ),

w(·, T ) = 0 in Ω.
(1.5)
(1.6)
(1.7)
Now let us consider ϕ the solution of the heat equation without control when the initial
data ϕ0 ∈ L2 (Ω):

∂t ϕ − ∆ϕ + aϕ = 0 in Ω × (0, T ),
(1.8)
ϕ = 0 on ∂Ω × (0, T ),

ϕ(·, 0) = ϕ0 in Ω.
Then by classical integrations by parts, we get
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
529
T ϑ(x, t)ϕ(x, t) dx dt = −
0 ω
w(x, 0)ϕ(x, 0) dx
Ω
(−w(·, 0) + wd )ϕ0 dx −
=
Ω
wd ϕ0 dx.
(1.9)
Ω
Also suppose that the solution w of (1.7) exists with (1.6) and (1.5) then for all ε > 0,
for all wd ∈ H01 (Ω), we have using Cauchy–Schwarz inequality
wd ϕ0 dx εwd H 1 (Ω) ϕ0 L2 (Ω) + ceD/ε wd L2 (Ω) ϕL2 (ω×(0,T )) .
(1.10)
0
Ω
Consequently, we obtain choosing wd = (−∆)−1 ϕ0 that
ϕ0 H −1 (Ω) εϕ0 L2 (Ω) + ceD/ε ϕL2 (ω×(0,T )) ,
∀ε > 0,
(1.11)
if ϕ0 = 0,
(1.12)
or equivalently
ϕ0 L2 (Ω)
ϕ0 L2 (Ω) c exp D
ϕL2 (ω×(0,T ))
ϕ0 H −1 (Ω)
or equivalently
ϕ0 H −1 (Ω) ln 2 +
D
1 ϕ0 L2 (Ω)
c ϕL2 (ω×(0,T ))
ϕ0 L2 (Ω) ,
(1.13)
where the values of the constants c = c(T , a∞ ) > 1 and D > 1 may changed from line
(1.11) to line (1.13) but not their dependence.
(1.13) is a quantitative estimate for unique continuation for initial data in L2 (Ω) of
the heat equation with potential from an interior observation. This kind of logarithmic
estimate already appears in the context of the cost of approximate control and stabilization
for hyperbolic equation [5,7]. Here we will follow the strategy in [7] (see also [4]) to prove
that (1.13) implies an approximate result with an estimate (1.3) of the cost function.
The first main result of this paper is as follows.
Theorem 1. There exist two constants c1 , c2 > 1 such that for all ε > 0, for all T > 0,
for all a ∈ L∞ (Ω × (0, T )), for all ud ∈ H01 (Ω), there exists a control function fε ∈
L2 (ω × (0, T )) such that
D∇ud L2 (Ω)
fε L2 (ω×(0,T )) E exp
ud L2 (Ω) ,
ε
u(·, T ) − ud 2
ε,
(1.14)
L (Ω)
where u ∈ C([0, T ]; L2 (Ω)) is the unique solution of the heat equation with potential and
control function

∂t u − ∆u + au = fε · 1|ω in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
(1.15)

u(·, 0) = 0 in Ω,
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K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
and E > 1, D > 1 are given by
2
D = D(T , a∞ ) = c1 T ec1 T a∞ + T1 ,
2
2/3
E = E(T , a∞ ) = exp(c2 (1 + T a∞ (1 + ec2 T a∞ ) + a∞ )).
(1.16)
Of course we will also need to prove the estimate (1.13). Our second main result is as
follows.
Theorem 2. There exist two constants c1 , c2 > 1 such that for all T > 0, for all a ∈
L∞ (Ω × (0, T )), for all initial data u0 ∈ L2 (Ω) such that u0 = 0, the solution u of the
homogeneous heat equation with potential

∂t u − ∆u + au = 0 in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
(1.17)

u(·, 0) = u0 in Ω,
satisfies
u0 H −1 (Ω) ln 2 +
D
1 u0 L2 (Ω)
E uL2 (ω×(0,T ))
u0 L2 (Ω) ,
(1.18)
where E > 1, D > 1 are given by
2
D = c1 T ec1 T a∞ + T1 ,
2
2/3
E = exp(c2 (1 + T a∞ (1 + ec2 T a∞ ) + a∞ )).
Remarks. (1) Note that E(T , 0) is a constant not depending on T > 0.
(2) An application of Theorem 1 to get a space of exact controllable target data may be
possible following [7] based on properties of Riesz basis (see [8]).
The plan of the paper is as follows. In Section 2 we prove Theorem 1 as an application
of Theorem 2. Section 3 contains the proof of Theorem 2. Finally in the last section some
comments are added.
2. Proof of Theorem 1
Theorem 1 is easily deduced from the following result.
Theorem 3. There exist two constants c1 , c2 > 1 such that for all ε > 0, for all T > 0,
for all a ∈ L∞ (Ω × (0, T )), for all wd ∈ H01 (Ω), there exists a control function ϑε ∈
L2 (ω × (0, T )) such that
D
ϑε L2 (ω×(0,T )) exp
wd L2 (Ω) ,
ε
w(·, 0) − wd 2
εwd H 1 (Ω) ,
(2.1)
L (Ω)
0
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
531
where w ∈ C([0, T ]; L2 (Ω)) is the unique solution of the backward heat equation with
potential and control function

−∂t w − ∆w + aw = Eϑε · 1|ω in Ω × (0, T ),
w = 0 on ∂Ω × (0, T ),
(2.2)

w(·, T ) = 0 in Ω,
and E > 1, D > 1 are given by
2
D = c1 T ec1 T a∞ + T1 ,
2
2/3
E = exp(c2 (1 + T a∞ (1 + ec2 T a∞ ) + a∞ )).
Proof. We will use the logarithmic estimate (1.18) from Theorem 2.
Let us introduce the operator
C : ϑ ∈ L2 ω × (0, T ) → w(·, 0) ∈ L2 (Ω),
where w is the solution of

∂t w + ∆w − aw = −Eϑ · 1|ω
w = 0 on ∂Ω × (0, T ),

w(·, T ) = 0 in Ω.
(2.3)
in Ω × (0, T ),
(2.4)
It is well known that if ϑ ∈ L2 (ω × (0, T )), then w ∈ C([0, T ]; H01(Ω)) and in particular w(·, 0) ∈ H01 (Ω) ⊂ L2 (Ω) with compact injection. Thus, the operator C is linear,
continuous and compact from L2 (ω × (0, T )) to L2 (Ω). We define F = Im C the space of
exact controllability initial data with the following norm:
(2.5)
w0 F = inf ϑL2 (ω×(0,T )) | Cϑ = w0 .
We will need to construct the dual operator of C. Let u0 ∈ L2 (Ω), we consider the
unique solution u ∈ C([0, T ]; L2(Ω)) of

∂t u − ∆u + au = 0 in Ω × (0, T ),
(2.6)
u = 0 on ∂Ω × (0, T ),

u(·, 0) = u0 in Ω,
and we know from Theorem 2 that
D
u0 H −1 (Ω) u0 L2 (Ω)
ln 2 + Eu 2
u0 L2 (Ω) .
(2.7)
L (ω×(0,T ))
Let us introduce the operator
K : u0 ∈ L2 (Ω) → Eu|ω ∈ L2 ω × (0, T ) ,
(2.8)
is linear, continuous, compact from L2 (Ω)
where u is the solution of (2.6). The operator K
to L2 (ω × (0, T )). Remark that the operator K is the adjoint of C and we denote C ∗ = K.
Indeed, by multiplying (2.6) by w the solution of (2.4) where ϑ ∈ L2 (ω × (0, T )) and by
applying the Green formula, we obtain the following duality relation:
T ϑ(x, t)Eu(x, t) dx dt =
0 ω
w(x, 0)u(x, 0) dx,
Ω
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K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
i.e., for all ϑ ∈ L2 (ω × (0, T )), for all u0 ∈ L2 (Ω), we have
T ϑK(u0 ) dx dt =
0 ω
u0 C(ϑ) dx.
(2.9)
Ω
We define F = (Im C) with the norm
T 1/2
2
Eu(x, t) dx dt
u0 F = K(u0 )L2 (ω×]0,T [) =
.
(2.10)
0 ω
Note that (2.9) and Holmgren theorem imply that F = Im C is dense in L2 (Ω).
From the duality relation (2.9), we have by choosing ϑ = K(u0 ) ∈ L2 (ω × (0, T )) that
K(u0 )2 2
= u0 C K(u0 ) dx.
(2.11)
L (ω×(0,T ))
Ω
We use the notation (·, ·) to describe the scalar product on L2 (Ω).
Let B = C ◦ K. Then (2.11) becomes: for all u0 ∈ L2 (Ω), K(u0 )2L2 (ω×(0,T )) =
(B(u0 ), u0 ).
The operator B is non-negative, compact from L2 (Ω) to L2 (Ω) and self-adjoint on
2
L (Ω). We deduce that it has a discrete spectrum and we associate in L2 (Ω) the Hilbert
basis with eigenfunctions ξn of B and eigenvalues µn , where µn > 0, is non-increasing
and tends to zero.
element g ∈ L2 (Ω) we have the Fourier expansion g =
Consequently, for2 every 2
(g,
ξ
)ξ
,
g
=
n n
n>0
n>0 |(g, ξn )| < +∞ and also
L2 (Ω)
2
K(g)2 2
= (Bg, g) =
µn (g, ξn ) .
(2.12)
L (ω×(0,T ))
n>0
Note that from (2.10), (2.12) and by duality, we deduce that
1 (g, ξn )2 .
g2F =
µn
n>0
Let us introduce the sets Sn = {m > 0 | αn+1 < µm αn }, where αn = eµ1 +e
e−e for
2
all n > 0. Then each function g ∈ L (Ω) can be represented in the form g = n>0 gn ,
where
gn =
(g, ξm )ξm .
(2.13)
n
m∈Sn
Also, we have g2L2 (Ω) = n>0 gn 2L2 (Ω) , where gn 2L2 (Ω) = m∈Sn |(g, ξm )|2 .
Further, g2F = n>0 K(gn )2L2 (ω×(0,T )) , where
2
K(gn )2 2
=
µm (g, ξm ) .
(2.14)
L (ω×(0,T ))
m∈Sn
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
533
From (2.14), we have, if gn = 0, that
gn 2L2 (Ω)
1
1
<
.
αn K(gn )2 2
αn+1
L (ω×(0,T ))
(2.15)
From (2.7) and (2.15), we obtain, if gn = 0, that
gn H −1 (Ω) D
gn L2 (Ω) .
√
[ln(2 + 1/ αn )]
(2.16)
Let us introduce, with gn given by (2.13),
2
2
gn gI =
αn+1 gn L2 (Ω) < +∞ .
I = g=
n>0
(2.17)
n>0
Then the dual space of I is defined by
1
2
2
I = g=
gn gI =
gn L2 (Ω) < +∞ .
αn+1
n>0
(2.18)
n>0
From (2.15) and more exactly F ⊂ I , we have that
I ⊂ F.
(2.19)
z ∈ H01 (Ω).
Now, let N > 0. Let
zn +
zn
z=
nN
Then we can write, in
with zn =
(z, ξm )ξm .
n>N
L2 (Ω),
(2.20)
m∈Sn
The properties of the sequence (αn )n>0 and (2.19), (2.16) imply that there exists a constant c3 > 0 such that the following relations hold:
2 2
1
zn zn c3
F
I
nN
nN
2
1 =
z
,
ξ
p m αn+1
n>0
m∈Sn pN
2
1 =
(z,
ξ
)(ξ
,
ξ
)
q
q m αn+1
n>0
=
nN
m∈Sn pN q∈Sp
1
αn+1
1
αN+1
1
αN+1
m∈Sn q∈S1 ∪···∪SN
2
(z, ξq )(ξq , ξm )
m∈S1 ∪···∪SN q∈S1 ∪···∪SN
m∈S1 ∪···∪SN
2
(z, ξq )(ξq , ξm )
(z, ξm )2 1
αN+1
z2L2 (Ω) ,
(2.21)
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K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
2
zn − z
2
nN
=
L (Ω)
zn 2L2 (Ω) =
n>N
n>N
zH 1 (Ω) zn H −1 (Ω)
0
n>N
zH 1 (Ω)
0
n>N
D
zn L2 (Ω)
√
[ln(2 + 1/ αn )]
zH 1 (Ω) 0
(z, zn )
n>N
D2
√
[ln(2 + 1/ αn )]2
zn 2L2 (Ω)
n>N
1
D
zH 1 (Ω) zn − z
√
2
2
0
[ln(2 + 1/ αn )]
L (Ω)
n>N
nN
D
zH 1 (Ω) c3
zn − z
(2.22)
√
2 .
0
ln(2 + 1/ αN+1 )
L (Ω)
nN
We conclude the proof of Theorem 3 by choosing N > 0 such that
1
D
≈ ln 2 + √
ε
αN+1
in order to have, with z = wd ,
Cϑ − wd L2 (Ω) c3 εwd H 1 (Ω) ,
0
D
wd L2 (Ω) .
ϑL2 (ω×(0,T )) c3 exp
ε
2
(2.23)
3. Proof of Theorem 2
We proceed in three steps.
Step 1. We apply the observability estimate [3, Theorem 1.2] with a simple change of
variable in time.
Then there exists a constant c0 > 0 such that for any solution u of (1.17), for any L > 0,
one has
L 1
2/3
u(·, L)2 2
u(x, t)2 dx dt.
exp c0 1 + + La∞ + a∞
L (Ω)
L
0 ω
Step 2. We look for an estimate of the form
2
u(·, 0)2 −1
constantu(·, L)L2 (Ω) .
H (Ω)
(3.1)
(3.2)
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
535
This is a backward estimate for the heat equation. To do this, we apply the ideas in [1].
Let us consider for almost all t ∈ [0, L] such that u(x, t) = 0 the following quantity:
Ψ (t) =
u(·, t)2L2 (Ω)
u(·, t)2H −1 (Ω)
.
(3.3)
We first prove that
2
d
1 au(·, t)H −1 (Ω)
Ψ (t) c4 a2∞ Ψ (t),
dt
2 u(·, t)2 −1
H (Ω)
(3.4)
where c4 > 0 is a constant only depending on the geometry and may change of value in the
lines below.
Indeed we have the two following energy equalities: posing f = −au,
2
1 d
u(·, t)2 2
+ u(·, t)H 1 (Ω) = f (·, t), u(·, t) ,
(Ω)
L
0
2 dt
2
2
1 d
u(·, t) −1
+ u(·, t)L2 (Ω) = f (·, t), (−∆)−1 u(·, t) .
H (Ω)
2 dt
Then from (3.5), (3.6) and (3.3), we get
2
d
Ψ (t) =
−u2H 1 (Ω) + (f, u) u2H −1 (Ω)
4
dt
0
uH −1 (Ω)
+ u4L2 (Ω) − u2L2 (Ω) f, (−∆)−1 u .
(3.5)
(3.6)
(3.7)
Also,
u4L2 (Ω) − u2L2 (Ω) f, (−∆)−1 u
2 2
f
f
, (−∆)−1 u − = ∆u + , (−∆)−1 u
2
2
2
2
f
f
−1 2
−1
(−∆) u H 1 (Ω) − , (−∆) u ∆u + 0
2 H −1 (Ω)
2
2
2
f f
2
2
−1
, (−∆) u .
uH 1 (Ω) + − (f, u) uH −1 (Ω) − 2 H −1 (Ω)
2
0
Finally, (3.7) and (3.8) imply
2
2 f f
2
d
2
−1
.
Ψ (t) ,
(−∆)
u
−
u
−1
H (Ω)
dt
2 H −1 (Ω)
2
u4 −1
H
(3.8)
(3.9)
(Ω)
Replacing f = −au, we get the desired inequalities (3.4).
Consequently, from (3.4) we obtain for all t ∈ (0, L),
2
Ψ (t) ec4 La∞ Ψ (0).
Secondly, remark that from (3.6) and (3.10), we have
(3.10)
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K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
2
1 d
u(·, t)2 −1
+ Ψ (t)u(·, t)H −1 (Ω) + a(·, t)u(·, t), (−∆)−1 u(·, t)
(Ω)
H
2 dt
2
2
1 d
u(·, t)2 −1
+ Ψ (t)u(·, t)H −1 (Ω) + c4 a∞ Ψ (t)u(·, t)H −1 (Ω)
H
(Ω)
2 dt
1 d
c4 La2∞
u(·, t)2 −1
Ψ
(0)
+
c
+
e
a
Ψ (0) u(·, t)2H −1 (Ω) .
4
∞
H (Ω)
2 dt
(3.11)
0=
Consequently, integrating (3.11) on (0, L), we get the desired estimate
2
2 u(·, 0)2 −1
exp 2ec4 La∞ Ψ (0) + c4 a∞ Ψ (0) L u(·, L)H −1 (Ω) .
H (Ω)
(3.12)
Step 3. We choose an adequate L > 0 to conclude the proof of Theorem 2. Let us denote
λ1 > 0 a constant such that
λ1 Ψ (0).
(3.13)
Now, we choose
λ1
L=T
T,
Ψ (0)
(3.14)
then we obtain from (3.1) and (3.12) that the solution u of (1.17) satisfies if u(x, t) = 0 for
almost all t ∈ [0, T ],
u(·, 0)2 −1
E1 exp D1 Ψ (0)
H (Ω)
T u(x, t)2 dx dt,
(3.15)
0 ω
where E1 and D1 are two constants such that

√
c0
D1 = 2ec4 T a2∞ T λ1 + √
,
T λ1
√
E = c exp(c λ T a ec4 T a2∞ + c (1 + T a + a2/3 )).
1
4
4
1
∞
0
∞
∞
Multiplying (3.15) by Ψ (0), we get
u(·, 0)2 2
E
Ψ
(0)
exp
D1 Ψ (0) u2L2 (ω×(0,T )) .
1
L (Ω)
(3.16)
(3.17)
So, if u0 = 0, we have
1 u0 L2 (Ω) E1 Ψ (0) exp D1 Ψ (0) uL2 (ω×(0,T ))
2
√
2 E1
exp D1 Ψ (0) uL2 (ω×(0,T )) ,
D1
(3.18)
and also clearly for all c1 > 1,
u0 L2 (Ω)
D1
√
exp c1 D1 Ψ (0) .
2 E1 uL2 (ω×(0,T ))
(3.19)
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
We choose c1 > 1 large enough to get
u0 L2 (Ω)
D1
exp c1 D1 Ψ (0) ,
2+ √
4 E1 uL2 (ω×(0,T ))
537
(3.20)
and finally we have
u0 H −1 (Ω) ln 2 +
c1 D1
u0 L2 (Ω) .
u0 L2 (Ω) D
1
√
4 E1 uL2 (ω×(0,T ))
That completes the proof of Theorem 2.
(3.21)
2
4. Further comments
A quantitative estimate of unique continuation for initial data in H01 (Ω) of the heat
equation with potential, solution u of (1.17), may be realized by using properties of the
quantity
Ψ (t) =
∇u(·, t)2L2 (Ω)
u(·, t)2L2 (Ω)
,
(4.1)
which are well described in [1].
We find that there exist two constants c1 , c2 > 1 such that for all T > 0, for all a ∈
L∞ (Ω × (0, T )), for all initial data u0 ∈ H01 (Ω) such that u0 = 0, the solution u of (1.17)
satisfies
D1
u0 L2 (Ω) (4.2)
∇u0 L2 (Ω) ,
1 ∇u0 L2 (Ω)
ln 2 + E1 u 2
L (ω×(0,T ))
where E1 and D1 are two constants such that
D1 = c1 1 + T1 ,
2/3
E1 = exp(c2 (1 + T 2 a2∞ + T a∞ + a∞ )).
(4.3)
This improves the finite dimensional observability estimate of [3, Theorem 1.5] in the
sense that if u0 = 0,
2/3 u0 H 1 (Ω) exp c2 1 + T 2 a2∞ + T a∞ + a∞
0
1 u0 H01 (Ω)
× exp c1 1 +
uL2 (ω×(0,T )) .
T u0 L2 (Ω)
(4.4)
Acknowledgments
This work was written while author had a post-doctoral position at the University Autonoma de Madrid. He
thanks the departments of mathematics of the University of Autonoma de Madrid and the University of Valencia
and specially Prof. E. Zuazua and Prof. Rosa Donat for their kind hospitality and fruitful exchanges. Support by
the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282, is acknowledged.
538
K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538
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