Differential and Integral Equations
Volume 19, Number 6 (2006), 627–668
QUANTITATIVE UNIQUENESS FOR TIME-PERIODIC
HEAT EQUATION WITH POTENTIAL AND ITS
APPLICATIONS
K-D. Phung
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China
G. Wang
Department of Mathematics, Wuhan University, Wuhan 430072, China
The Center for Optimal Control and Discrete Mathematics
Huazhong Normal University, Wuhan 730079, China
(Submitted by: Viorel Barbu)
Abstract. In this paper, we establish a quantitative unique continuation property for some time-periodic linear parabolic equations in
a bounded domain Ω. We prove that for a time-periodic heat equation with particular time-periodic potential, its solution u(x, t) satisfies u(·, 0)L2 (Ω) ≤ C u(·, 0)L2 (ω) where ω ⊂ Ω. Also we deduce
the asymptotic controllability for the heat equation with an even, timeperiodic potential. Moreover, the controller belongs to a finite dimensional subspace and is explicitly computed.
1. Introduction and main results
Throughout this paper, Ω is a connected bounded domain in Rd , d ≥ 1,
with a boundary ∂Ω of class C 2 , ω is a non-empty open subset of Ω. Let
1 > 0 be the first eigenvalue of the operator −Δ with the Dirichlet boundary
condition (i.e., the smallest strictly positive eigenvalue of −Δ in H01 (Ω)). Let
T > 0 be a real number; we denote by .∞ the usual norm in L∞ (Ω×(0, T )).
In this paper, we study an infinite-dimensional system generated by the
following parabolic equation
∂t y − Δy + ay = f in Ω × (0, +∞),
(1.1)
y = 0 on ∂Ω × (0, +∞),
Accepted for publication: February 2006.
AMS Subject Classifications: 35B10, 93B07, 93B60.
The first author thanks the support of the NCET od China under grant NCET-04-0882,
and the NSF of China under grants 10371084 and 10525105.
627
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K-D. Phung and G. Wang
with an even, T -periodic, bounded potential a = a(x, t) ∈ L∞ (Ω × R) satisfying
1 ≤ a∞
and
a(x, t + T ) = a(x, t) = a(x, −t) in Ω × R.
(1.2)
We assume that y(·, 0) = y o ∈ L2 (Ω) and f = f (x, t) ∈ L1loc (0, +∞; L2 (Ω)),
so that (1.1) admits a unique solution y = y(x, t) ∈ C([0, +∞); L2 (Ω)). Let
{G(t, s)}0≤s≤t<+∞ be the T -periodic evolutionary process on L2 (Ω), such
that
t
G(t, r)f (·, r)dr,
(1.3)
y(·, t) = G(t, s)y(·, s) +
s
for all 0 ≤ s ≤ t, where the T -periodicity of G(t, s) means G(t + T, s + T ) =
G(t, s).
Notice that any bounded function in L∞ (Ω × (0, T /2)) can be extended
to be an even, T -periodic potential a ∈ L∞ (Ω × R). Here, we are only
interested in the case where (a∞ − 1 ) ≥ 0 and a is not positive, because
if (a∞ − 1 ) < 0 or a ≥ 0, then the system (1.1) is clearly stable when
f = 0.
The Poincaré map is usually defined by G(T + t, t). We restrict our attention on the operator G(T, 0)
G(T, 0) : L2 (Ω) −→ L2 (Ω)
y(·, 0) −→ y(·, T ).
(1.4)
It is well known that G(T, 0) is compact by the smoothing action of the
diffusion. Since a is even and T -periodic, we see that G(T, 0) is self-adjoint.
Consequently, there is a complete orthonormal set in L2 (Ω) formed of eigenfunctions (zjo )j≥1 of G(T, 0) corresponding to eigenvalues (λj )j≥1 , where
λj = λj (T ) depends on T , λj ∈ R, λj → 0 as j → +∞. Thus, we may
arrange the eigenfunctions so that the sequence {|λj |}j≥1 is non-increasing,
· · · ≤ |λm+1 | ≤ |λm | ≤ · · · ≤ |λ1 |. Clearly, λj = 0 from the backward
uniqueness property for the linear parabolic equation. Furthermore, we will
see that for all T > 0 and for all even, T -periodic, bounded potentials a, any
eigenvalue λj (T ) of G(T, 0) satisfies
1 +
ln |λj (T )|
≤ a∞ .
T
(1.5)
The first result of the paper is as follows.
Theorem 1. There exist two constants co ∈ (0, 1) and C > 0, both only
depending on ω and Ω, such that if we choose the time-periodicity T ∈ (0, co ]
Quantitative uniqueness for time-periodic heat equation
629
and an even, T -periodic potential a ∈ L∞ (Ω × R) such that
1 ≤ a∞ ≤ 1
co 1/4
,
T
then any eigenfunction zjo of G(T, 0) in L2 (Ω), corresponding to the eigenvalue λj (T ) with |λj (T )| ≥ 1, satisfies
Ω
G(t, 0)zjo (x)2 dx ≤ eCa4/3
∞
ω
G(t, 0)zjo (x)2 dx
for all t ≥ 0.
The knowledge of the eigencouple of the Poincaré map plays a key role in
the study of periodic parabolic systems (see [14], [15]). Theorem 1 implies
clearly that the eigenfunctions zjo of the Poincaré map corresponding to the
eigenvalues λj with |λj | ≥ 1 have the unique continuation property.
It is classical in control theory for linear partial differential equations that
the unique continuation property is linked with the approximate controllability and, more precisely, a quantitative uniqueness result yields an estimate
of the cost of the approximate control (see e.g. [12]). Here, our second theorem establishes an asymptotic controllability (or open loop stabilizability)
for the heat equation with time-periodic potential.
Theorem 2. There exist two constants co ∈ (0, 1) and C > 0, both only
depending on Ω and ω, such that if we choose the time-periodicity T ∈ (0, co ]
and an even, T -periodic potential a ∈ L∞ (Ω × R) such that
1 ≤ a∞ ≤ 1
co 1/4
,
T
and if there exists m > 0 such that
· · · ≤ |λm+1 (T )| < 1 ≤ |λm (T )| = · · · = |λ2 (T )| = |λ1 (T )| ,
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K-D. Phung and G. Wang
then for each initial data y o ∈ L2 (Ω), the control function fc ∈ C([0, T ]; L2 (Ω))
satisfying
⎧
(i)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(ii)
⎪
⎪
⎨
fc (x, T − t) =
σj (t)G(t, 0)zjo (x),
j=1,··,m
⎛
⎞
σ1 (t)
−1
⎜ . ⎟
o
o
⎜ . ⎟ = −1
G(t,
0)z
(x)G(t,
0)z
(x)dx
κ
j
T
ω
⎝ . ⎠
1≤κ,j≤m
σm (t)
⎛
⎞
⎪
⎪
y o (x)G(T, 0)z1o (x)dx
⎪
Ω
⎪
⎪
⎜
⎟
⎪
..
⎪
⎟,
⎪
×⎜
⎪
.
⎝
⎠
⎪
⎪
⎪
o
o
⎪
⎪
y
(x)G(T,
0)z
(x)dx
m
⎪
Ω
⎪
T
4/3
⎩
(iii) 0 fc (·, s)L2 (ω) ds ≤ eCa∞ e(a∞ −1 )T y o L2 (Ω) ,
implies that the solution y = y(x, t) ∈ C([0, +∞); L2 (Ω)) of the following
heat equation with potential a and control function fc ,
⎧
⎨ ∂t y − Δy + ay = fc · 1|ω×(0,T ) in Ω × (0, +∞),
y = 0 on ∂Ω × (0, +∞),
⎩
y(·, 0) = y o in Ω,
satisfies, for all t ≥ T ,
y(·, t)L2 (Ω) ≤ Cm e−γt y o L2 (Ω) ,
where
4/3
2eC (a∞ +a∞ T )
− ln |λm+1 (T )|
.
> 0 and Cm =
γ=
T
|λm+1 (T )|2
The notion of asymptotic controllability is standard in the nonlinear control theory of finite-dimensional systems (see e.g. [7]). Here, (i) says that
our control has a finite-dimensional structure (see also [4], [5]). (ii) implies
that the operator associating the initial data with the control function is
linear which gives a kind of robustness property of our control. (iii) gives
us an explicit expression of the cost of the control, from which we see easily
that we can act a control on the equation in a very short time T , but as a
payment, evenness and T -periodicity for the potential a are required.
From the following property of the Poincaré map,
G(nT, 0)zjo = (λj )n zjo
(1.6)
for all n ∈ N, it is clear that if all eigenvalues λj satisfy |λj | < 1, then the
system (1.1) is stable without control. On the other hand, if the modulus of some eigenvalues of G(T, 0) are bigger than one, then the equation
Quantitative uniqueness for time-periodic heat equation
631
(1.1) with f = 0 is unstable. We call such eigenvalues unstable eigenvalues.
Theorem 2 amounts to saying that if all unstable eigenvalues have the same
absolute value then the equation (1.1) with f = fc · 1|ω×(0,T ) can be stabilized by a control fc in a finite-dimensional subspace of L2 (Ω) spanned by
(zjo )j=1,··· ,m which are the eigenfunctions of G(T, 0) corresponding to the unstable eigenvalues. So it is important to study the unique continuation of the
eigenfunctions of G(T, 0) corresponding to the unstable eigenvalues. This is
why we restrict our attention in Theorem 1 to the case where |λj | ≥ 1.
The unique continuation property or more precisely the number of zeros
of a one-dimensional parabolic equation is well described in [1] which leads
to the study of the asymptotic behaviour of the periodic 1d quasilinear parabolic equation [3], or to the consideration of the structural stability of the
periodic 1d semilinear heat equation [6].
In order to get Theorem 1 and Theorem 2, we need a more general unique
continuation result in bounded domains Ω ⊂ Rd , d ≥ 1, described as follows.
Theorem 3. There exist two constants co ∈ (0, 1) and C > 0, both only
depending on Ω and ω, such that for all T ∈ (0, co ], β ≥ 0 and q ∈ L∞ (Ω ×
(0, T )) satisfying
co 1/4
,
1 + β ≤ q∞ ≤ 1
T
we have
4/3
|u(x, 0)|2 dx ≤ eCq∞
Ω
where u ∈
tion
C([0, T ]; L2 (Ω))
|u(x, 0)|2 dx,
ω
solves the following T -periodic linear heat equa-
⎧
⎨ ∂t u − Δu + βu + qu = 0 in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
⎩
u(·, 0) = u(·, T ) in Ω.
Strong unique continuation for the heat equation with potential in bounded domains Ω ⊂ Rd , d ≥ 1, has been widely studied, (see for instance [10],
[11] and references therein). In [17], one obtains that any solution v of the
linear heat equation with time-independent Lipschitz continuous potential
such that v(·, to ) = 0 in ω, for some to ∈ (0, T ), satisfies v = 0 in Ω × (0, T ).
The idea of the proof in [17] is to transform the solution v to a solution
of an elliptic equation where the unique continuation property for elliptic
operators can be used (see also [8]). Here we are interested in the unique
continuation and its quantification of the linear time-periodic heat equation
with time-dependent potential. The basic technique used here is to combine
632
K-D. Phung and G. Wang
some properties of time-periodic solutions of the heat equation with a kind
of global Carleman estimate for an elliptic operator.
This paper is organized as follows. In Section 2, we give an introduction
of the Poincaré map and some related preliminary results, then we prove
Theorem 1. In Section 3 and Section 4, we give the proofs of Theorem 2 and
Theorem 3 respectively. In Section 5, we point out some related results for
a particular case where the potential depends only on time. Finally, in the
Appendix, we give the proof of the Global Carleman inequality we need.
2. The Poincaré map
Let a be an even, T -periodic, bounded function in L∞ (Ω × R); more
precisely a(x, t + T ) = a(x, t) = a(x, −t) in Ω × R. We consider the following
heat equation with the potential a = a(x, t),
∂t y − Δy + ay = f in Ω × (s, +∞),
(2.1)
y = 0 on ∂Ω × (s, +∞),
where s ≥ 0 and f ∈ L1loc (0, +∞; L2 (Ω)). Let y ∈ C([s, +∞); L2 (Ω)) be
the mild solution of (2.1) and G(t, s) be the T -periodic evolutionary process
generated by −Δ + aI in L2 (Ω). More precisely, {G(t, s)}0≤s≤t<+∞ is a
family of bounded linear operators from L2 (Ω) to itself, such that
⎧
⎪
(i)
G(t, t) = I for all t ≥ 0,
⎪
⎪
⎪
⎪
⎪
⎨(ii) G(t, r)G(r, s) = G(t, s) for all 0 ≤ s ≤ r ≤ t,
(iii) The map (t, s) → G(t, s)g is continuous for every fixed g ∈ L2 (Ω),
⎪
⎪
⎪
(iv) G(t + T, s + T ) = G(t, s) for all 0 ≤ s ≤ t,
⎪
⎪
⎪
⎩(v)
G(t, s) ≤ e(a∞ −1 )(t−s) for all 0 ≤ s ≤ t.
(2.2)
Thus, the mild solution y of equation (2.1) can be written in the following
form
t
y(·, t) = G(t, s)y(·, s) +
G(t, r)f (·, r)dr
(2.3)
s
for all 0 ≤ s ≤ t.
Now, we are going to focus our attention on the Poincaré map G(T, 0) and
obtain some properties in our case. Since a is even and T -periodic, we get
that G(T, 0) is self-adjoint. Indeed, let pT ∈ L2 (Ω) and p ∈ C([0, T ]; L2 (Ω))
be the solution of
⎧
⎨ −∂t p − Δp + ap = 0 in Ω × (0, T ),
p = 0 on ∂Ω × (0, T ),
(2.4)
⎩
T
in Ω,
p(·, T ) = p
Quantitative uniqueness for time-periodic heat equation
then for all y o ∈ L2 (Ω),
o
y (x)p(x, 0)dx =
[G(T, 0)y o (x)] pT (x)dx;
Ω
633
(2.5)
Ω
that is,
(2.6)
G∗ (T, 0)pT (x) = p(x, 0).
On the other hand, because a(x, T − t) = a(x, t), the solution w(x, t) =
p(x, T − t) solves
⎧
∂t w − Δw + aw = 0 in Ω × (0, T ),
⎪
⎪
⎨
w = 0 on ∂Ω × (0, T ),
(2.7)
w(·, 0) = pT in Ω,
⎪
⎪
⎩
w(·, T ) = p(·, 0) in Ω.
Finally, for all pT ∈ L2 (Ω),
G∗ (T, 0)pT (x) = G(T, 0)pT (x).
(2.8)
L2 (Ω),
we deduce that there
As G(T, 0) is a compact self-adjoint operator on
is a complete orthonormal set in L2 (Ω) formed by the eigenfunctions (zjo )j≥1
of G(T, 0) with corresponding eigenvalues λj = λj (T ) ∈ R, which converge
to 0 as j → +∞. Thus, we can arrange the eigenfunctions so that the
sequence {|λj |}j≥1 is non-increasing.
Now we are able to give some properties of the eigencouple (zjo , λj (T ))
associated to the compact self-adjoint operator G(T, 0).
Lemma 2-1. For all s ∈ [0, T ] and φ ∈ L2 (Ω),
o
φ(x)G(T − s, 0)zj (x)dx =
G(T, s)φ(x)zjo (x)dx.
Ω
Ω
It follows immediately from Lemma 2-1 that for all s ∈ [0, T ],
G(T − s, 0)zjo (x)
G(s, 0)zio (x)
dx = δij .
λj
Ω
(2.8)
Lemma 2-2. As soon as the time-periodicity T > 0 and an even, T -periodic
potential a ∈ L∞ (Ω × R) are chosen, then any eigenvalue λj (T ) of G(T, 0)
satisfies
ln |λj (T )|
1 +
≤ a∞ .
T
Note that in particular, under the hypothesis of Theorem 1, the eigenvalues λj (T ) have to satisfy
co 1/4
ln |λj (T )|
1 +
.
(2.9)
≤ a∞ ≤ 1
T
T
634
K-D. Phung and G. Wang
2.1. Proof of Lemma 2-1. Let g = g(x, t) ∈ C([s, T ]; L2 (Ω)) be the solution of
⎧
⎨ ∂t g − Δg + ag = 0 in Ω × (s, T ),
g = 0 on ∂Ω × (s, T ),
(2.10)
⎩
g(·, s) = φ in Ω,
and zj = zj (x, t) ∈ C([0, T ]; L2 (Ω)) be the solution of
⎧
⎨ ∂t zj − Δzj + azj = 0 in Ω × (0, T ),
zj = 0 on ∂Ω × (0, T ),
⎩
zj (·, 0) = zjo in Ω.
(2.11)
Now, from the relation a(·, T − t) = a(·, t), we obtain that the function
t → Ω g(x, t)zj (x, T − t)dx is constant. Indeed,
d
g(x, t)zj (x, T − t)dx
dt Ω
∂t g(x, t)zj (x, T − t)dx +
g(x, t) − (∂t zj )(x, T − t) dx
=
Ω
Ω
−(−Δ + a(x, t))g(x, t)zj (x, T − t)dx
=
Ω
g(x, t)(−Δ + a(x, t))zj (x, T − t)dx = 0.
(2.12)
+
Ω
Consequently, for all t ∈ [s, T ],
g(x, s)zj (x, T − s)dx =
g(x, t)zj (x, T − t)dx =
g(x, T )zjo (x)dx.
Ω
Ω
Ω
(2.13)
That completes the proof of Lemma 2-1.
2.2. Proof of Lemma 2-2. Let
ln |λj (T )|
β =
∈R
T
and u = u(x, t) ∈ C([0, 2T ]; L2 (Ω)) be the solution of
⎧
+ au = 0 in Ω × (0, 2T ),
⎨ ∂t u − Δu + βu
u = 0 on ∂Ω × (0, 2T ),
⎩
u(·, 0) = zjo in Ω.
By an energy method, we have the following equality
1d
2
2
|u(x, t)| dx +
|∇u(x, t)| dx + β
|u(x, t)|2 dx
2 dt Ω
Ω
Ω
(2.14)
(2.15)
Quantitative uniqueness for time-periodic heat equation
635
a(x, t) |u(x, t)|2 dx = 0.
+
(2.16)
Ω
Also, notice that u(x, t) = e−βt zj (x, t) = e−βt G(t, 0)zjo (x) = 0 and
u(·, 2T ) = u(·, 0)
From (2.16) and (2.17), we get
1 + β − a
∞
0
2T
in Ω.
(2.17)
|u(x, t)|2 dx ≤ 0.
(2.18)
Ω
Consequently,
ln |λj (T )|
1 + β = 1 +
≤ a∞ .
T
This is the desired inequality.
(2.19)
2.3. Proof of Theorem 1. For |λj (T )| ≥ 1, we have that
ln |λj (T )|
≥0
T
and u(x, t) = e−βt zj (x, t) ∈ C([0, 2T ]; L2 (Ω)) satisfies
⎧
⎨ ∂t u − Δu + βu + au = 0 in Ω × (0, 2T ),
u = 0 on ∂Ω × (0, 2T ),
⎩
u(·, 2T ) = u(·, 0) in Ω.
β=
(2.20)
(2.21)
Now, we apply Theorem 3 for the 2T -periodic solution u of (2.21) with
q = a ∈ L∞ (Ω × (0, 2T )) satisfying
co 1/4
ln |λj (T )|
(2.22)
≤ a∞ ≤ 1
1 +
T
2T
to obtain
2
Ca4/3
∞
|u(x, 0)| dx ≤ e
|u(x, 0)|2 dx.
(2.23)
Ω
ω
It follows by translation in time and by the 2T -periodicity of u that for all
t ∈ [0, 2T ],
2
Ca4/3
∞
|u(x, t)| dx ≤ e
|u(x, t)|2 dx.
(2.24)
Ω
ω
Consequently, we obtain that for all t ∈ [0, 2T ],
G(t, 0)zjo (x)2 dx ≤ eCa4/3
G(t, 0)zjo (x)2 dx.
∞
Ω
Hence, if we choose co in Theorem 1 to be
we complete the proof of Theorem 1.
(2.25)
ω
co
2
in Theorem 3 and the same C,
636
K-D. Phung and G. Wang
3. The asymptotic controllability problem
In this section, we prove Theorem 2. Recall that if (a∞ − 1 ) < 0, then
the system (1.1) is already stable with f = 0. We consider the solution of the
following heat equation with a potential a = a(x, t) ∈ L∞ (Ω × R) satisfying
a(x, t + T ) = a(x, t) = a(x, −t) in Ω × R and (a∞ − 1 ) ≥ 0,
⎧
⎨ ∂t y − Δy + ay = fc · 1|ω×(0,T ) in Ω × (0, +∞),
(3.1)
y = 0 on ∂Ω × (0, +∞),
⎩
y(·, 0) = y o in Ω,
where y o ∈ L2 (Ω) and fc ∈ C([0, T ]; L2 (Ω)). It is clear that for all integers
n ≥ 1, we have
nT
o
y(·, nT ) = G(nT, 0)y +
G(nT, s) fc (·, s) · 1|ω×(0,T ) ds.
(3.2)
0
(zjo , λj (T ))
Let
G(T, 0), then
be the eigencouple of the compact self-adjoint operator
o
y =
Ω
j≥1
y o (x)zjo (x)dx zjo .
(3.3)
It follows from Lemma 2-1 that for any s ∈ [0, T ],
G(T, s) fc (x, s) · 1|ω×(0,T )
G(T, s)(fc (x, s) · 1|ω×(0,T ) )zjo (x)dx zjo (x)
=
=
j≥1
Ω
j≥1
Ω
fc (x, s) · 1|ω×(0,T ) G(T − s, 0)zjo (x)dx zjo (x).
(3.4)
By (3.3), (3.4) and by the periodicity of G(t, s), we infer
o
G(nT, 0)y =
y o (x)zjo (x)dx G(nT, 0)zjo
Ω
j≥1
=
j≥1
and
λnj
Ω
y o (x)zjo (x)dx zjo
(3.5)
G(nT, s) fc (x, s) · 1|ω×(0,T ) = G(nT, T )G(T, s) fc (x, s) · 1|ω×(0,T )
=
fc (x, s) · 1|ω×(0,T ) G(T − s, 0)zjo (x)dx G(nT, T )zjo (x)
j≥1
Ω
Quantitative uniqueness for time-periodic heat equation
=
j≥1
=
j≥1
1|(0,T ) ·
fc (x, s)G(T −
ω
λn−1
1|(0,T ) ·
j
ω
s, 0)zjo (x)dx
637
G((n − 1)T, 0)zjo (x)
fc (x, s)G(T − s, 0)zjo (x)dx zjo (x).
(3.6)
We conclude from (3.5) and (3.6) that y(x, nT ) = j≥1 λnj Yj (T )zjo (x) for all
integers n ≥ 1, where
T
1
o
o
y (x)zj (x)dx +
fc (x, s)G(T − s, 0)zjo (x)dxds. (3.7)
Yj (T ) =
λj 0 ω
Ω
Recall that we have arranged the eigenfunctions so that {|λj |}j≥1 is a nondecreasing sequence. Let m > 0 be such that |λj | ≤ |λm+1 | < 1 for all
j ≥ m + 1 and 1 ≤ |λm | ≤ |λj | for all j = 1, · · · , m. Now we decompose
y(x, nT ) as follows
y(x, nT ) = y1,m (x, nT ) + y2,m (x, nT ),
where
y1,m (x, nT ) =
(3.8)
λnj Yj (T )zjo (x)
(3.9)
λnj Yj (T )zjo (x).
(3.10)
j=1,··· ,m
and
y2,m (x, nT ) =
j≥m+1
Note that y1,m is the projection of y in the subspace of L2 (Ω) spanned
by the eigenfunctions (zjo )j=1,··· ,m of G(T, 0) corresponding to the unstable
eigenvalues (λj )j=1,···,m while y2,m is the projection of y in the subspace
of L2 (Ω) spanned by the eigenfunctions (zjo )j≥m+1 of G(T, 0) corresponding to the stable eigenvalues (λj )j≥m+1 . Next, we estimate respectively
2
2
Ω |y2,m (x, nT )| dx and Ω |y1,m (x, nT )| dx.
It follows first from (3.10) that
|y2,m (x, nT )|2 dx =
|λj |2n |Yj (T )|2
Ω
j≥m+1
=
j≥m+1
zjo (x) 2
|λj | y(x, T )
dx .
λj
Ω
2n Indeed, because of the evenness and T -periodicity of a, we have
T
G(T − s, 0)zjo (x)
fc (x, s)
dxds
λj
ω
0
(3.11)
638
K-D. Phung and G. Wang
T
=
Ω
0
G(T − s, 0)zjo (x)
dxds
λj
∂t y(x, s) − Δy(x, s) + a(x, T − s)y(x, s)
zjo (x)
=
y(x, T )
dx −
λj
Ω
Ω
y o (x)zjo (x)dx.
(3.12)
As |λj | ≤ |λm+1 | < 1 for all j ≥ m + 1, we deduce from (3.11) that
2
|y2,m (x, nT )|2 dx ≤ |λm+1 |2(n−1)
y(x, T )zjo (x)dx
Ω
j≥m+1
≤ |λm+1 |2(n−1)
Ω
|y(x, T )|2 dx.
(3.13)
Ω
On the other hand, we have
|y1,m (x, nT )|2 dx =
Ω
|λj |2n |Yj (T )|2
j=1,··· ,m
by (3.9) where
T 1
1
o
o
y (x)zj (x)dx +
fc (x, s)G(T − s, 0)zjo (x)dx ds.
Yj (T ) =
T Ω
λj ω
0
(3.14)
Now we are in position to construct a control function fc = fc (x, s) ∈
C([0, T ]; L2 (Ω)) such that for all x ∈ Ω, for all s ∈ [0, T ],
σi (s)G(s, 0)zio (x),
(3.15)
fc (x, T − s) =
i=1,···,m
where σi = σi (s) ∈ C([0, T ]) and
1
1
o
o
y (x)zj (x)dx +
fc (x, s)G(T − s, 0)zjo (x)dx = 0
(3.16)
T Ω
λj ω
for all j = 1, · · · , m, which implies Ω |y1,m (x, nT )|2 dx = 0.
By (3.15) and (3.16), it suffices to obtain {σi (s)}i=1,··· ,m by solving the
following linear system
λj
σi (s) zi (x, s)zj (x, s)dx = −
y o (x)zjo (x)dx;
(3.17)
T Ω
ω
i=1,··· ,m
that is,
⎡ ···
⎢
..
..
⎣
.
.
ω z1 (x, s)zm (x, s)dx · · ·
ω z1 (x, s)z1 (x, s)dx
⎞
σ1 (s)
⎥⎜
⎟
..
..
⎦⎝
⎠
.
.
σm (s)
ω zm (x, s)zm (x, s)dx
ω zm (x, s)z1 (x, s)dx
⎤⎛
Quantitative uniqueness for time-periodic heat equation
⎛ =−
1⎜
⎝
T
Ωy
o (x)z
1 (x, T )dx
..
.
639
⎞
⎟
⎠
(3.18)
o
Ω y (x)zm (x, T )dx
where zj = zj (x, t) ∈ C([0, +∞); L2 (Ω)) is the solution of
⎧
⎨ ∂t zj − Δzj + azj = 0 in Ω × (0, +∞),
zj = 0 on ∂Ω × (0, +∞),
⎩
zj (·, 0) = zjo in Ω.
(3.19)
The existence of suchσi (s) ∈ C([0, T ]) can be proved
if and only if for all
o
o
s ∈ [0, T ], the matrix
is invertible.
ω G(s, 0)zi (x)G(s, 0)zj (x)dx
1≤i,j≤m
G(s, 0)zio (x)G(s, 0)zjo (x)dx
. By contra!1≤i,j≤m
diction, we assume that M (s) = ω zi (x, s)zj (x, s)dx 1≤i,j≤m is a not invertible matrix for some s ∈ [0, T ] fixed. Then the lines of M (s) are linearly
in
dependent which implies that
{zi (x, s)}i=1,··· ,m
are linearly dependentm−1
2
L (ω). Hence there exists μ1 (s), · · · , μm−1 (s) a non-zero vector in R
such that
3.1. Invertibility of
ω
μ1 (s)z1 (·, s) + · · · + μm−1 (s)zm−1 (·, s) − zm (·, s) = 0
in ω.
(3.20)
However, by the assumption that |λm+1 | < 1 ≤ |λm | = · · · = |λ2 | = |λ1 |, it
follows that the function
κs (x, t) = μ1 (s)z1 (x, t) + · · · + μm−1 (s)zm−1 (x, t) − zm (x, t)
satisfies
⎧
⎨ ∂t κs − Δκs + aκs = 0 in Ω × (0, 2T ),
κs = 0 on ∂Ω × (0, 2T ),
⎩
κs (·, 2T ) = |λ1 |2 κs (·, 0) in Ω.
(3.21)
(3.22)
ln|λ1 |
Let u(x, t) = e− T t κs (x, t), then u ∈ C([0, 2T ]; L2 (Ω)) solves
⎧
1|
⎨ ∂t u − Δu + ln|λ
T u + au = 0 in Ω × (0, 2T ),
u = 0 on ∂Ω × (0, 2T ),
⎩
u(·, 2T ) = u(·, 0) in Ω.
(3.23)
By Theorem 3, there exists a constant c1 > 0 such that if we choose a ∈
L∞ (Ω × R) such that
c1 1/4
1 ≤ a∞ ≤ 1
,
(3.24)
2T
640
K-D. Phung and G. Wang
and noting that
1 +
ln |λ1 |
≤ a∞
T
and
ln |λ1 |
≥ 0,
T
(3.25)
which follows from Lemma 2-2 and the fact that |λ1 | ≥ 1, then the solution
u = u(x, t) of (3.23) satisfies
u(·, 0) = 0 in ω =⇒ u(·, 0) = 0 in Ω.
(3.26)
It follows by translation in time and by the 2T -periodicity of u that
u(·, s) = 0 in ω =⇒ u(·, s) = 0 in Ω.
(3.27)
So κs (·, s) = 0 in Ω and
o
o
− zm
= 0 in Ω.
(3.28)
μ1 (s)z1o + · · · + μm−1 (s)zm−1
!
This contradiction shows that the matrix ω zi (x, s)zj (x, s)dx 1≤i,j≤m is
invertible for all s ∈ [0, T ] and
⎞ ⎡ ⎤−1
⎛
z
(x,
s)z
(x,
s)dx
·
·
·
z
(x,
s)z
(x,
s)dx
σ1 (s)
1
1
m
1
ω
ω
⎟ ⎢
⎥
⎜
..
..
..
..
⎠=⎣
⎦
⎝
.
.
.
.
z1 (x, s)zN (x, s)dx · · ·
zm (x, s)zm (x, s)dx
σm (s)
⎞ ω
⎛ ω1 o
− T Ω y (x)z1 (x, T )dx
⎟
⎜
..
(3.29)
×⎝
⎠.
.
1
o
− T Ω y (x)zm (x, T )dx
Consequently, if we take the control function fc ∈ C([0, T ]; L2 (Ω)) in equation (3.1) as
fc (x, t) =
σi (T − t)zi (x, T − t),
(3.30)
i=1,··· ,m
then the corresponding solution y ∈ C([0, +∞); L2 (Ω)) satisfies
y(·, nT )L2 (Ω) = y2,m (·, nT )L2 (Ω) ≤ |λm+1 |(n−1) y(·, T )L2 (Ω)
≤ |λm+1 |(n−1) y o L2 (Ω) e(a∞ −1 )T
(3.31)
T
+ |λm+1 |(n−1)
fc (·, s)L2 (ω) e(a∞ −1 )(T −s) ds,
0
for all integers n ≥ 1, where λm+1 is the first eigenvalue such that |λm+1 | < 1.
Quantitative uniqueness for time-periodic heat equation
641
3.2. Estimate of fc (·, s)L2 (ω) . It follows from (3.31) that we would like
T
to compute 0 fc (·, s)L2 (ω) e(a∞ −1 )(T −s) ds. We have
⎞
⎛
σ
(s)
1
⎟
⎜
..
σ1 (s) · · · σm (s) ·
zi (x, s)zj (x, s)dx
⎠
⎝
.
1≤i,j≤m
ω
σm (s)
⎞
⎛ o
y (x)z1 (x, T )dx
Ω
⎜
1
⎟
..
σ1 (s) · · · σm (s) · ⎝
=−
(3.32)
⎠,
.
o
T
Ω y (x)zm (x, T )dx
or equivalently
2
1
σi (s)zi (x, s) dx = −
y o (x)
T
ω
Ω
i=1,··· ,m
which implies
"2
" "
"
σi (s)zi (·, s)" 2
"
i=1,··· ,m
σi (s)zi (x, T )dx, (3.33)
i=1,··· ,m
"
" 1 o
"
"
σi (s)zi (·, T )" 2
y L2 (Ω) "
T
L (ω)
L (Ω)
i=1,··· ,m
"
"
σi (s)G(T, s)zi (·, s)" 2
≤
"
1 o
"
≤ y L2 (Ω) "
T
i=1,··· ,m
"
1 o
"
≤ y L2 (Ω) "G(T, s)
T
L (Ω)
"
"
σi (s)zi (·, s)"
i=1,··· ,m
≤
"
1 o
"
y L2 (Ω) e(a∞ −1 )(T −s) "
T
L2 (Ω)
"
"
σi (s)zi (·, s)"
i=1,··· ,m
L2 (Ω)
Now we claim that there exists a constant c > 0 such that
" "
" "
"
"
"
"
σi (s)zi (·, s)" 2
≤ c"
σi (s)zi (·, s)"
"
i=1,··· ,m
L (Ω)
i=1,··· ,m
.
L2 (ω)
(3.34)
.
Indeed, since |λm+1 | < 1 ≤ |λm | = · · · = |λ2 | = |λ1 |, the function
κs (x, t) =
σi (s)zi (x, t)
(3.35)
(3.36)
i=1,··· ,m
satisfies
⎧
⎨ ∂t κs − Δκs + aκs = 0 in Ω × (0, 2T ),
κs = 0 on ∂Ω × (0, 2T ),
⎩
κs (·, 2T ) = |λ1 |2 κs (·, 0) in Ω.
(3.37)
642
K-D. Phung and G. Wang
ln|λ1 |
Let u(x, t) = e− T t κs (x, t), then u ∈ C([0, 2T ]; L2 (Ω)) satisfies
⎧
1|
⎨ ∂t u − Δu + ln|λ
T u + au = 0 in Ω × (0, 2T ),
u = 0 on ∂Ω × (0, 2T ),
⎩
u(·, 2T ) = u(·, 0) in Ω.
(3.38)
By Theorem 3, there exists a constant c1 > 0, which is the same as that
in (3.24), such that if we choose a ∈ L∞ (Ω × R) satisfying (3.24) and note
(3.25) then the following estimate holds
2
Ca4/3
∞
|u(x, 0)| dx ≤ e
|u(x, 0)|2 dx,
(3.39)
Ω
ω
where C > 0 is a constant which depends only on ω and Ω. It follows by
translation in time and by the 2T -periodicity of u solution of (3.38) that for
all s ∈ [0, T ],
2
Ca4/3
∞
|u(x, s)| dx ≤ e
|u(x, s)|2 dx,
(3.40)
Ω
which implies
ω
2
Ca4/3
∞
|κs (x, s)|2 dx
|κs (x, s)| dx ≤ e
Ω
or equivalently
"
" "
"
σi (s)zi (·, s)"
"
i=1,··· ,m
(3.41)
ω
L2 (Ω)
"
4/3 "
≤ eCa∞ "
i=1,··· ,m
"
"
σi (s)zi (·, s)"
L2 (ω)
,
(3.42)
as desired. By (3.42) and (3.34) we get
" "
4/3
1
"
"
σi (s)zi (·, s)" 2 ≤ y o L2 (Ω) e(a∞ −1 )(T −s) eCa∞ . (3.43)
"
T
L (ω)
i=1,··· ,m
Now we conclude that if we take co =
all a ∈ L∞ (Ω × R) satisfying
c1
2
where c1 is given in (3.39) then for
co 1/4
,
(3.44)
T
we may take the control function fc ∈ C([0, T ]; L2 (Ω)) of the form (3.15)
with estimates
T T
1 Ca4/3
a∞ −1 (T −s)
o
∞ y 2
fc (·, s)L2 (ω) ds ≤ e
e
ds
L (Ω)
T
0
0
4/3
(3.45)
≤ eCa∞ e a∞ −1 T y o L2 (Ω)
1 ≤ a∞ ≤ 1
Quantitative uniqueness for time-periodic heat equation
and
0
T
643
4/3
fc (·, s)L2 (ω) e(a∞ −1 )(T −s) ds ≤ eCa∞ e2(a∞ −1 )T y o L2 (Ω) ,
(3.46)
so that the corresponding solution y ∈ C([0, +∞); L2 (Ω)) of (3.1) satisfies
(3.31).
3.3. Exponential decay of the solution. Now we turn to obtaining the
4/3
exponential decay of the solution. Let CT,a = 1 + eCa∞ e(a∞ −1 )T . By
(3.31) and (3.46), it follows that
y(·, nT )L2 (Ω) ≤ |λm+1 |(n−1) CT,a e(a∞ −1 )T y o L2 (Ω)
(3.47)
for all integers n ≥ 1.
For any t ≥ T , we may write t = nT + σ where n ≥ 1 and 0 ≤ σ < T .
Since y(·, t) = G(t, nT )y(·, nT ), it follows from (3.47) that
y(·, t)L2 (Ω) = y(·, nT )L2 (Ω) e(a∞ −1 )(t−nT ) ≤ y(·, nT )L2 (Ω) e(a∞ −1 )T
1
1
≤ exp − (n + 1) ln
CT,a e2(a∞ −1 )T y o L2 (Ω) ,
|λm+1 (T )| |λm+1 (T )|2
(3.48)
which implies
CT,a e2(a∞ −1 )T o
t 1
y L2 (Ω)
y(·, t)L2 (Ω) ≤ exp − ln
T
|λm+1 (T )|
|λm+1 (T )|2
(3.49)
for all t ≥ T , and completes the proof of Theorem 2.
4. The quantitative unique continuation for time-periodic heat
equation with potential
In this section, we shall show Theorem 3 by dividing the proof into three
steps. In the first step, we study the solutions u locally in ω×{t = 0} then, in
the second step, we apply a global Carleman inequality for elliptic operators
where time t ∈ (0, T ) is seen as a parameter. In the last step, we conclude
the proof of Theorem 3 by choosing an adequate time T .
4.1. Step 1: Local property of u near t = 0. The goal of this subsection
is to prove the following result:
644
K-D. Phung and G. Wang
Lemma 4-1. There exist a non-empty open subset ω
of ω and a constant
Cω > 1 depending only on ω and Ω such that for all T > 0 and integers
M ≥ 2, we have
M d
2
2q∞ t
2+ M
|u(x, t)| dx ≤ e
t
|u(x, 0)|2 dx
Cω (M + 2)
ω
Ω
1 − C (M + 2)2 tM d
ω
2q∞ t
(M + 2) M
|u(x, 0)|2 dx
+e
1 − Cω (M + 2)2 t
ω
for all t ∈ (0, T ), where the solution u solves the following linear heat equation with potential
⎧
⎨ ∂t u − Δu + βu + qu = 0 in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
⎩
u(·, 0) ∈ L2 (Ω).
Proof of Lemma 4-1. Let xo ∈ Rd and ro > 0; we denote
$
#
B(xo , r) = #x ∈ Rd : |x − xo | ≤ ro ,
$
D(xo , r) = x ∈ Rd : |xi − xoi | ≤ ro , for all i = 1, · · · , d .
(4.1)
Without loss of generality, we suppose that there exists r > 0 such that
D(0, 2r) ⊂ ω. Then B(xo , r) ⊂ ω for all xo ∈ [−r, r]d . Let N ≥ 4 be an
arbitrary but fixed integer; we have
B(xo , r/N ) ⊂ B(xo , 2r/N ) ⊂ · · · ⊂ B(xo , r) ⊂ ω.
(4.2)
Let us define a sequence of smooth functions Φn ∈ C0∞ (B(xo , nr/N )) for
n = 2, · · · , N such that
⎧
0 ≤ Φn ≤ 1,
⎨
Φn = 1 in B(xo , (n − 1)r/N ),
(4.3)
⎩
|∇Φn | ≤ 2N/r.
The construction of such a sequence Φn is standard.
Let Φ ∈ C0∞ (B(xo , ro )); multiplying the equation in Lemma 4-1 by Φ2 u,
we get the usual energy equality
1d
2
2
|Φu(x, t)| dx +
|Φ∇u(x, t)| dx + β
|Φu(x, t)|2 dx
(4.4)
2 dt Ω
Ω
Ω
= − q(x, t) |Φu(x, t)|2 dx − 2 Φ(x)∇u(x, t) · ∇Φ(x)u(x, t)dx,
Ω
Ω
Quantitative uniqueness for time-periodic heat equation
645
from which, after some calculation involving the Cauchy-Schwarz inequality,
we infer
1d
|Φu(x, t)|2 dx
(4.5)
2 dt B(xo ,ro )
≤ q∞
|Φu(x, t)|2 dx + ∇Φ2L∞ (B(xo ,ro ))
|u(x, t)|2 dx.
B(xo ,ro )
B(xo ,ro )
Multiplying (4.5) by e−2q∞ t , we obtain
d
|Φu(x, t)|2 dx e−2q∞ t
dt B(xo ,ro )
2
|u(x, t)|2 dx e−2q∞ t .
≤ 2 ∇ΦL∞ (B(xo ,ro ))
(4.6)
B(xo ,ro )
Taking Φ = Φn and ro = nr/N in (4.6), using (4.3), it follows that
d
gn (t) ≤ 8(N/r)2 gn+1 (t)
dt
for all n = 2, · · · , N − 1, where
|Φn u(x, t)|2 dx e−2q∞ t .
gn (t) =
(4.8)
Integrating (4.7) over (0, t), we obtain that for all n = 2, · · · , N − 1,
t
gn+1 (τ )dτ + gn (0).
gn (t) ≤ 8(N/r)2
(4.9)
(4.7)
B(xo ,nr/N )
0
By induction, we deduce that for n = 0, · · · , N − 3,
t
j
2 n+1 n
t
gn+3 (τ )dτ +
g2 (t) ≤ 8(N/r)
8(N/r)2 t gj+2 (0). (4.10)
0
j=0,··· ,n
By (4.8), (4.3) and (4.10), after some computation, we have
2
−2q∞ t
|u(x, t)| dx e
≤
|u(x, t)|2 dx e−2q∞ t
√
o
o
D(x ,r/(N d))
B(x ,r/N )
≤
|Φ2 u(x, t)|2 dx e−2q∞ t ≤ g2 (t)
B(xo ,2r/N )
≤ (8(N/r)2 )N −2 tN −3
t
gN (τ )dτ +
0
≤ (8(N/r)2 )N −2 tN −3
t
0
B(xo ,r)
(8(N/r)2 t)j gj+2 (0)
j=0,··· ,N −3
|ΦN u(x, τ )|2 dxe−2q∞ τ dτ
646
K-D. Phung and G. Wang
+
2
j
(8(N/r) t)
j=0,··· ,N −3
2 N −2 N −3
≤ (8(N/r) )
B(xo ,(j+2)r/N )
t
|u(x, τ )|2 dxe−2q∞ τ dτ
t
0
1 − (8(N/r)2 t)N −2 +
1 − 8(N/r)2 t
|Φj+2 u(x, 0)|2 dx
Ω
|u(x, 0)|2 dx.
(4.11)
ω
On the other hand, it is easy to check that for all τ ∈ (0, T ),
2
−2q∞ τ
|u(x, τ )| dx e
≤
|u(x, 0)|2 dx,
Ω
(4.12)
Ω
which together with (4.11) implies
2
2q∞ t
2 N −2 N −2
|u(x, t)| dx ≤ e
(8(N/r) )
t
|u(x, 0)|2 dx
√
D(xo ,r/(N d))
+ e2q∞ t
(8(N/r)2 t)N −2 1 −
1 − 8(N/r)2 t
Ω
|u(x, 0)|2 dx.
(4.13)
ω
Now, we break the rectangle [0, √rd ]d ⊂ Rd into N d small rectangles Dj ,
j = 1, · · · , N d , having the same shape, by dividing each interval [0, √rd ] into
%
N small pieces (i.e., [0, √rd ] =
[k N r√d , (k + 1) N r√d ] and [0, √rd ]2 =
k=0,··,N −1
%
%
k N r√d , (k + 1) N r√d
×
k N r√d , (k + 1) N r√d , ...).
k=0,··,N −1
k=0,··· ,N −1
It is clear that meas Dj1 ∩ Dj2 = 0 for j1 = j2 and also
2
|u
(x,
t)|
dx
=
|u (x, t)|2 dx.
(4.14)
d
0, √r
d
Dj
j=1,··,N d
One can check that for each Dj , j = 1, · · · , N d ,
r Dj ⊂ D xo , k √ ,
N d
(4.15)
√
if xo = (xo1 , · · · , xod ) ∈ [−r, r]d with xoi , j = 1, · · · , d, equals k(r/N d) for
some k = 0, · · · , N − 1, which implies by (4.14) that
2
d 2q∞ t
2 N −2
(8(N/r) t)
|u(x, 0)|2 dx
d |u(x, t)| dx ≤ N e
0, √r
d
+ N d e2q∞ t
1 − (8(N/r)2 t)N −2 1 − 8(N/r)2 t
ω
Ω
|u(x, 0)|2 dx.
(4.16)
Quantitative uniqueness for time-periodic heat equation
647
Let M = N − 2 ≥ 2; we have
M
d
2
2q∞ t 8
2+ M
|u(x, t)| dx ≤ e
(M + 2)
t
|u(x, 0)|2 dx
r2
√r [0,1]d
Ω
d
8
M
(M + 2)2 t d 1 −
r2
+ e2q∞ t (M + 2) M
|u(x, 0)|2 dx.
(4.17)
1 − r82 (M + 2)2 t
ω
Lemma 4-1 follows.
4.2. Step 2: Observability estimate for periodic-parabolic solutions. The goal of this section is to prove the following observability inequality:
Theorem 4-1. Let ω
be a non-empty open subset of ω. There exists a
constant c > 0 such that for all T > 0, β ≥ 0 and q ∈ L∞ (Ω × (0, T )), we
have
T
e2q∞ T
2
4/3 2/3
|u(x, 0)| dx ≤
|u(x, t)|2 dxdt
exp c(1 + β + q∞ )
T
Ω
ω
0
where u ∈ C([0, T ]; L2 (Ω)) solves the following T -periodic in time linear
heat equation with potential
⎧
⎨ ∂t u − Δu + βu + qu = 0 in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
⎩
u(·, 0) = u(·, T ) in Ω.
As we know, for initial condition problems of parabolic equations, the
constant of observability grows in an exponentially fast way of order e1/T
(see [12]). However, Theorem 4-1 shows that the constant of observability
√
explodes for small time only in a polynomially fast way of order 1/ T for
parabolic equations with time-periodic conditions.
In order to prove Theorem 4-1, we need the following propositions.
Proposition 4-1. Let ωo be a non-empty open subset of Ω. There exist a
smooth function ϕ = ϕ(x) ≥ 1 and a constant c > 0, such that for all T > 0,
β ≥ 0, q ∈ L∞ (Ω × (0, T )) and θ ≥ c 1 + β 2/3 + q4/3
∞ , we have
d 1
1
2
2θϕ(x)
e
|∇u(x, t)| dx + β
e2θϕ(x) |u(x, t)|2 dx
dt 2 Ω
2 Ω
e2θϕ(x) |∇ϕ(x)|2 |u(x, t)|2 dx
− θ2
Ω
1
2
2θϕ(x)
2
+
e
|∇u(x, t)| dx + θ
e2θϕ(x) |u(x, t)|2 dx
c Ω
Ω
648
K-D. Phung and G. Wang
≤c θ
2θϕ(x)
e
2
|∇u(x, t)| dx + θ
ωo
3
2θϕ(x)
e
2
|u(x, t)| dx
ωo
for almost every t ∈ (0, T ), where u = u(x, t) solves the following linear heat
equation with potential
⎧
⎨ ∂t u − Δu + βu + qu = 0 in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
⎩
u(·, 0) ∈ H 2 (Ω) ∩ H01 (Ω).
Proposition 4-2. (Global Carleman inequality for the operator Δ − β)
Let ωo be a non-empty open subset of Ω. There exist a smooth function
ϕ = ϕ(x) ≥ 1 and a constant c > 0, such that for all β ≥ 0, q ∈ L∞ (Ω),
4/3
q L∞ (Ω) and w ∈ H 2 (Ω) ∩ H01 (Ω), we have
θ > c 1 + β 2/3 + "2
"
"
"
"2θ∇ϕ · ∇(eθϕ w) − qeθϕ w" 2
L (Ω)
1
2θϕ(x)
2
3
+
e
|∇w(x)| dx + θ
e2θϕ(x) |w(x)|2 dx
θ
c
Ω
Ω
"
"2
" θϕ
"
2θϕ(x)
2
3
≤ "e (Δ − β)w" 2 c θ
e
|∇w(x)| dx + θ
e2θϕ(x) |w(x)|2 dx .
L (Ω)
ωo
ωo
The proof of Proposition 4-2 is given in the Appendix.
Proof of Proposition 4-1. We apply Proposition 4-2 to the solution u of
the linear heat equation with potential, then we deduce that there exist a
smooth function ϕ = ϕ(x) ≥ 1 and a constant c > 0, such that for all β ≥ 0,
4/3 , we have
q ∈ L∞ (Ω × (0, T )), and θ > c 1 + β 2/3 + q∞
"
"
"
"2
"2θ∇ϕ · ∇ eθϕ u(·, t) − q(·, t)eθϕ u(·, t)" 2
L (Ω)
1
+
e2θϕ(x) |∇u(x, t)|2 dx + θ3
e2θϕ(x) |u(x, t)|2 dx
θ
c
Ω
Ω
"
"2
" θϕ
"
≤ "e (Δ − β)u(·, t)" 2
L (Ω)
2θϕ(x)
2
3
+c θ
e
|∇u(x, t)| dx + θ
e2θϕ(x) |u(x, t)|2 dx
(4.18)
ωo
ωo
for almost every t ∈ (0, T ).
On the other hand, we multiply equation −∂t u + (Δ − β)u − qu = 0 by
e2θϕ (Δ − β)u and integrate it over Ω to get
1 d
1d
2
2θϕ(x)
e
|∇u(x, t)| dx + β
e2θϕ(x) |u(x, t)|2 dx
2 dt Ω
2 dt Ω
Quantitative uniqueness for time-periodic heat equation
"
"2
"
"
+ "eθϕ (Δ − β)u(·, t)" 2
L (Ω)
=
q(x, t)u(x, t)(Δ − β)u(x, t)e2θϕ(x) dx
Ω
∂t u(x, t)2θ∇ϕ(x) · ∇u(x, t)e2θϕ(x) dx
−
Ω
q(x, t)u(x, t)(Δ − β)u(x, t)e2θϕ(x) dx
=
Ω
∂t u(x, t)eθϕ(x) 2θ∇ϕ(x) · ∇ eθϕ(x) u(x, t) dx
−
Ω
∂t u(x, t)eθϕ(x) 2θ∇ϕ(x) · (θ∇ϕ(x)eθϕ(x) )u(x, t)dx
+
649
(4.19)
Ω
which implies, with Ξ(t) = Ω e2θϕ(x) |∇u(x, t)|2 dx + β Ω e2θϕ(x) |u(x, t)|2 dx
− 2θ2 Ω |∇ϕ(x)|2 e2θϕ(x) |u(x, t)|2 dx,
"2
"
1d
"
" θϕ
Ξ(t) + "e (Δ − β)u(·, t)" 2
2 dt
L (Ω)
−
q(x, t)u(x, t)(Δ − β)u(x, t)e2θϕ(x) dx
Ω
(4.20)
= − ∂t u(x, t)eθϕ(x) 2θ∇ϕ(x) · ∇ eθϕ(x) u(x, t) dx.
Ω
Replacing ∂t u by (Δ − β) u − qu in the right-hand side of (4.20), we deduce
that
"2
"
1d
"
"
Ξ(t) + "eθϕ (Δ − β)u(·, t)" 2
2 dt
L (Ω)
+
eθϕ(x) (Δ − β)u(x, t) 2θ∇ϕ(x) · ∇ eθϕ(x) u(x, t) dx
Ω
eθϕ(x) (Δ − β)u(x, t) −q(x, t)eθϕ(x) u(x, t) dx
+
Ω
q(x, t)u(x, t)eθϕ(x) 2θ∇ϕ(x) · ∇ eθϕ(x) u(x, t) dx.
(4.21)
=
Ω
By the Cauchy-Schwarz inequality, the right-hand side of (4.21) is bounded
by
q(x, t)u(x, t)eθϕ(x) 2θ∇ϕ(x) · θ∇ϕ(x)eθϕ(x) u(x, t)dx
Ω
650
K-D. Phung and G. Wang
q(x, t)u(x, t)eθϕ(x) 2θ∇ϕ(x) · eθϕ(x) ∇u(x, t)dx
Ω
"
"2
"
"
≤ 2θ2 q∞ ∇ϕ2L∞ (Ω) "eθϕ u" 2
L (Ω)
"
"
"
"
"
" θϕ "
"
+ 2θ q∞ ∇ϕL∞ (Ω) "e u" 2 "eθϕ ∇u" 2 .
+
L (Ω)
L (Ω)
(4.22)
Then, we conclude by (4.21) and (4.22) that
"
"2
1d
"
"
Ξ(t) + "eθϕ (Δ − β)u(·, t)" 2
2 dt
L (Ω)
+
eθϕ(x) (Δ − β)u(x, t) 2θ∇ϕ(x) · ∇ eθϕ(x) u(x, t) dx
Ω
+
eθϕ(x) (Δ − β)u(x, t) −q(x, t)eθϕ(x) u(x, t) dx
Ω
"
"
" θϕ "2
2
2
≤ 2θ q∞ ∇ϕL∞ (Ω) "e u" 2
L (Ω)
"
"
"
"
"
"
"
"
+ 2θ q∞ ∇ϕL∞ (Ω) "eθϕ u" 2 "eθϕ ∇u" 2 .
(4.23)
L (Ω)
L (Ω)
Consequently, there exists a constant c > 0 such for all β ≥ 0 and q ∈
L∞ (Ω × (0, T )), we have
"2
"
1d
"
"
Ξ(t) + "eθϕ (Δ − β)u(·, t)" 2
2 dt
L (Ω)
+
eθϕ(x) (Δ − β)u(x, t) 2θ∇ϕ(x) · ∇ eθϕ(x) u(x, t) dx
Ω
θϕ(x)
θϕ(x)
e
(Δ − β)u(x, t) −q(x, t)e
u(x, t) dx
+
Ω
e2θϕ(x) |∇u(x, t)|2 dx + θ2
e2θϕ(x) |u(x, t)|2 dx
(4.24)
≤
c q∞
Ω
Ω
for almost every t ∈ (0, T ). By (4.18) and (4.24), we get
"2
θϕ
1"
"
"
θϕ
2θ∇ϕ
·
∇
e
u(·,
t)
−
q(·,
t)e
u(·,
t)
"
" 2
2
L (Ω)
1
θ
+
e2θϕ(x) |∇u(x, t)|2 dx + θ3
e2θϕ(x) |u(x, t)|2 dx
2c
Ω
Ω
c
e2θϕ(x) |∇u(x, t)|2 dx + θ3
e2θϕ(x) |u(x, t)|2 dx
−
θ
2
ωo
ωo
Quantitative uniqueness for time-periodic heat equation
"2
1"
"
" θϕ
e
(Δ
−
β)u(·,
t)
" 2
"
2
L (Ω)
"
"2
1d
1"
"
≤−
Ξ(t) − "eθϕ (Δ − β)u(·, t)" 2
2 dt
2
L (Ω)
−
eθϕ(x) (Δ − β)u(x, t) 2θ∇ϕ(x) · ∇ eθϕ(x) u(x, t) dx
Ω
eθϕ(x) (Δ − β)u(x, t) −q(x, t)eθϕ(x) u(x, t) dx
−
Ω
e2θϕ(x) |∇u(x, t)|2 dx + θ2
e2θϕ(x) |u(x, t)|2 dx
+
c q∞
651
≤
Ω
(4.25)
Ω
from which it follows that
"2
1"
" θϕ
"
"e (Δ − β)u(·, t) + 2θ∇ϕ · ∇ eθϕ u(·, t) − q(·, t)eθϕ u(·, t)" 2
2
L (Ω)
1d
1 +
e2θϕ |∇u(x, t)|2 dx + θ2
e2θϕ |u(x, t)|2 dx
Ξ(t) + θ
2 dt
2c
Ω
Ω
2
2θϕ
2
2θϕ
e |∇u(x, t)| dx + θ
e |u(x, t)|2 dx
≤
c q∞
Ω
Ω
c
2
2θϕ
3
e |∇u(x, t)| dx + θ
e2θϕ |u(x, t)|2 dx .
(4.26)
+
θ
2
ωo
ωo
+ 4c
c q∞ in (4.26), we deduce that the
By taking θ ≥ c 1 + β 2/3 + q4/3
∞
inequality
1d
2
2θϕ(x)
2
e
|∇u(x, t)| dx + θ
e2θϕ(x) |u(x, t)|2 dx
Ξ(t) + c
2 dt
Ω
Ω
c
2
2θϕ(x)
3
2θϕ(x)
e
|∇u(x, t)| dx + θ
e
|u(x, t)|2 dx
(4.27)
≤
θ
2
ωo
ωo
holds for all θ ≥ c1 1 + β 2/3 + q4/3
where c1 > 0 is large enough. This
∞
completes the proof of Proposition 4-1.
Proof of Theorem 4-1. Throughout the proof of Theorem 4-1, c denotes
several positive constants only depending on the geometry. By integrating
the estimate in Proposition 4-1 over (0, T ), we get
T
T
2
2θϕ(x)
2
e
|∇u(x, t)| dxdt + θ
e2θϕ(x) |u(x, t)|2 dxdt
(4.28)
0
Ω
T
≤c θ
0
2θϕ(x)
e
ωo
Ω
0
2
|∇u(x, t)| dxdt + θ
3
0
T
ωo
e2θϕ(x) |u(x, t)|2 dxdt ,
652
K-D. Phung and G. Wang
thanks to the T -periodicity of u. With an adequate choice of θ > c 1+β 2/3 +
in (4.28), it follows that, denoting Cβ,q = exp c(1 + β 2/3 + q4/3
q4/3
∞
∞ ) ,
T
T
T 2
2
|u(x, t)| dxdt ≤ Cβ,q
|u(x, t)| dx +
|∇u(x, t)|2 dx .
0
Ω
0
0
ωo
ωo
(4.29)
By the usual energy method (i.e., an equality like (4.4)), we also have
T
T
2
|∇u(x, t)| dxdt ≤ (c + q∞ )
|u(x, t)|2 dx
(4.30)
0
ω
0
ωo
⊂ ω, using the T -periodicity of u.
for some ωo ⊂ ω
Consequently, for all T > 0, β ≥ 0 and q ∈ L∞ (Ω × (0, T )),
T
T
|u(x, t)|2 dxdt ≤ exp c(1+β 2/3 +q4/3
)
|u(x, t)|2 dx. (4.31)
∞
0
Ω
0
ω
where u solves the following T -periodic in time linear heat equation with
potential
⎧
∂ u − Δu + βu + qu = 0 in Ω × (0, T ),
⎪
⎪
⎨ t
u = 0 on ∂Ω × (0, T ),
(4.32)
u(·, 0) = u(·, T ) in Ω,
⎪
⎪
⎩
u(·, 0) ∈ H 2 (Ω) ∩ H01 (Ω).
By density of H 2 (Ω) ∩ H01 (Ω) in L2 (Ω), this last estimate is also true for
solutions u with initial data in L2 (Ω).
On the other hand, we recall that
2
2q∞ t
2q∞ T
e
|u(x, T )| dx ≤ e
|u(x, t)|2 dx
(4.33)
Ω
and
1 2q∞ T
T ≤
−1 =
e
2 q∞
Ω
T
e2q∞ t dt.
(4.34)
0
Finally, it follows from (4.33), (4.34) and (4.31) that
e2q∞ T T
2
2
|u(x, 0)| dx =
|u(x, T )| dx ≤
|u(x, t)|2 dxdt
T
Ω
Ω
Ω
0
T
e2q∞ T
|u(x, t)|2 dxdt.
(4.35)
≤
exp c(1 + β 2/3 + q4/3
∞ )
T
ω
0
This completes the proof of Theorem 4-1.
Quantitative uniqueness for time-periodic heat equation
653
4.3. Step 3: Choice of T . Now we are able to conclude the proof of
Theorem 3 as follows. By integrating over (0, T ) the estimate in Lemma 4-1,
we have
M
d
1 T
|u(x, t)|2 dx ≤ e2q∞ T Cω (M + 2)2+ M T
|u(x, 0)|2 dx
T 0 ω
Ω
M 2
1 − Cω (M + 2) T
d
|u(x, 0)|2 dx. (4.36)
+ e2q∞ T (M + 2) M
1 − Cω (M + 2)2 T
ω
Combining the latter with Theorem 4-1, we deduce that there exists a constant c > 1 such that for all T > 0, q ∈ L∞ (Ω × (0, T )) and integers M ≥ 2,
we have, denoting Cβ,q = exp c(1 + β 2/3 + q4/3
∞ ) ,
M 2
2+d/M
4q∞ T
|u(x, 0)| dx ≤ Cβ,q e
T
|u (x, 0)|2 dx
Cω (M + 2)
Ω
4q∞ T
+ Cβ,q e
d/M
(M + 2)
1 − Cω (M + 2)2 T
M
Ω
1 − Cω (M + 2)2 T
|u (x, 0)|2 dx.
ω
(4.37)
From this we infer that there exists a constant c > 1 such that for all T > 0,
β ≥ 0 and q ∈ L∞ (Ω × (0, T )) satisfying 1 + β ≤ q∞ , the inequality
M
2
c(1+T )q4/3
2+d/M
∞
Cω (M + 2)
|u(x, 0)| dx ≤ e
T
|u(x, 0)|2 dx
Ω
Ω
M 2
c(1+T )q4/3
d/M 1 − Cω (M + 2) T
∞
(M + 2)
|u(x, 0)|2 dx (4.38)
+e
1 − Cω (M + 2)2 T
ω
holds for all integers M ≥ 2.
First we choose an integer M ≥ 2 so that
1 + d
+
2c
≤ M + 1,
(4.39)
M < 2 q4/3
∞
4/3
1
which can be done because q∞ ≥ 1 . By (4.39), it follows that
1 + 2c q4/3
∞ ≤ M.
1 + d ≤ M,
(4.40)
By (4.40) and (4.38), we infer
M 2
3
M −1+c(T −1)q4/3
∞
Cω (2M ) T
|u (x, 0)| dx ≤ e
|u (x, 0)|2 dx
Ω
c(1+T )q4/3
∞
+e
(2M )
1 − Cω (2M )2 T
2
M
1 − Cω (2M ) T
Ω
ω
|u (x, 0)|2 dx.
(4.41)
654
K-D. Phung and G. Wang
Next, we choose T > 0, so that T ≤
1
eCω 4q4/3
∞
1+d
+2c
4/3
1
3 <
1
eCω (2M )3
≤ 1,
which can be done because of (4.39). Then by (4.41), we get
4/3
2
−1
2
2cq∞
2
|u(x, 0)| dx ≤ e
|u(x, 0)| dx + e
(2M )
|u(x, 0)|2 dx, (4.42)
Ω
Ω
ω
which gives
4/3
2 e2cq∞ 4/3 1 + d
|u(x, 0)| dx ≤
+ 2c
|u(x, 0)|2 dx. (4.43)
4 q∞
4/3
1 − e−1
Ω
ω
2
1
Now let co =
eCω 4
1
1+d
+2c
4/3
1
3 , then by (4.43), we conclude that if 1 + β ≤
41
1/4
q∞ and q∞ ≤ 1 cTo
, then there exists a constant C > 0 such that
2
Cq4/3
∞
|u(x, 0)| dx ≤ e
|u(x, 0)|2 dx,
(4.44)
Ω
ω
where the solution u ∈
satisfies the following T -periodic in
time linear heat equation with potential
⎧
⎨ ∂t u − Δu + βu + qu = 0 in Ω × (0, T ),
u = 0 on ∂Ω × (0, T ),
(4.45)
⎩
u(·, 0) = u(·, T ) in Ω.
C([0, T ]; L2 (Ω))
The proof of Theorem 3 is now completed.
5. The autonomous case
Let us denote by i , with 0 < 1 < 2 ≤ 3 ≤ · · · , and by ei for i ≥ 1,
the eigenvalues and eigenfunctions of −Δ in H01 (Ω). Recall here that (ei )i≥1
forms an orthonormal basis of the Hilbert space L2 (Ω).
Let a = a(t) ∈ L∞ (R) be an even, T -periodic potential. From
a Galerkin
− st a(τ )dt Δ(t−s)
method, G(t, s) can be explicitly expressed by G(t, s) = e
e
and one can check easily that the eigencouple (zjo , λj ) of the compact selfadjoint operator G(T, 0) satisfies
λj = e−j T −
T
0
a(τ )dτ
zjo = ej .
Furthermore, we obtain the following asymptotic controllability result.
Quantitative uniqueness for time-periodic heat equation
655
Theorem 5-1. Suppose that a = a(t) ∈ L∞ (R) is a space-independent
potential satisfying
(i)
a(t + T ) = a(t) = a(−t) in R,
T
(ii)
1 ≤ − T1 0 a(t)dt < 2 ;
then for all initial data y o ∈ L2 (Ω) ,the control function f ∈ L1 (0, T ; L2 (Ω))
given by
1 −1 t−TT−t a(τ )dτ Ω y o (x)e1 (x)dx e1 (x)
f (x, t) = − e
2
T
ω |e1 (x)| dx
implies that the solution y = y(x, t) ∈ C([0, +∞); L2 (Ω)) of the following
heat equation with potential a = a(t),
⎧
⎨ ∂t y − Δy + ay = f · 1|ω×(0,T ) in Ω × (0, +∞),
y = 0 on ∂Ω × (0, +∞),
⎩
y(·, 0) = y o in Ω,
satisfies for all t ≥ T ,
1 T
C aα/2
+a∞ T
∞
a(τ )dτ t y o L2 (Ω)
y(t)L2 (Ω) ≤ e
exp − 2 +
T 0
and
0
T
Caα/2
∞
f (s)L2 (ω) ds ≤ e
a∞ −1 T
e
y o L2 (Ω) ,
where C > 0 is a constant which only depends on the geometry and α ≥
0. More precisely, α = 23 for general non-empty subdomains ω ⊂ Ω and
α = 0 under the geometrical control condition of the work of Bardos-LebeauRauch (i.e., Ω ⊂ Rd , d ≥ 1, is of class C ∞ , there is no infinite order of
contact between the boundary ∂Ω and the bicharacteristics of ∂t2 − Δ, and
all generalized bicharacteristic ray of ∂t2 − Δ meets ω × (0, Tc ) for some
0 < Tc < +∞).
Note that no hypotheses
on T > 0 are required with a space- independent
potential which satisfies a∞ − 1 ≥ 0 thanks to (ii). We omit details
of the proof here. For general non-empty subdomains ω ⊂ Ω, we apply
Theorem A (see Appendix) to get a quantification of the unique continuation
property for the eigenfunction of −Δ in H01 (Ω) and deduce α = 23 . When
the geometrical control condition of the work of Bardos-Lebeau-Rauch [2] is
satisfied, then we apply the observability estimate for the wave equation for
a fixed frequency and deduce α = 0.
656
K-D. Phung and G. Wang
If the even, T -periodic bounded potential a = a(t) depends only on the
time t, then G(T, 0) is a positive operator and the first eigenvalue is of
multiplicity one (see the Krein-Rutman theorem in e.g. [15]). We do not
know if such a result is still true for a potential a = a(x, t) depending on
space and time. Observe that here no positivity hypotheses on the potential
or on the initial data are imposed as in [15] or as in [9].
6. Appendix: Global Carleman inequality for the operator
(Δ − β)
Carleman estimates are widely used in applied problems (see [16] for inverse problems or e.g. [13], [12], [18] for controllability problems). In what
follows, we present the proof of the global Carleman inequality for the operator (Δ − β) that we have used (see Proposition 4-2 or Theorem A below). In
our inequality in Theorem A, the presence of the first term in the left-hand
side plays a key role.
Theorem A. Let Ω be a bounded connected domain in Rd , d ≥ 1, with a
boundary ∂Ω of class C 2 , and ωo be a non-empty open subset of Ω. Then
there exist a smooth function ϕ = ϕ(x) ≥ 1 and a constant C > 0, such that
4/3
2/3 + for all β ∈ R, for all q ∈ L∞ (Ω), for all θ > C 1 + |β|
q L∞ (Ω) , for
all w ∈ H 2 (Ω) ∩ H01 (Ω), we have
"
"2
"
"
"2θ∇ϕ · ∇(eθϕ w) − qeθϕ w" 2
L (Ω)
1 3
θ
+
e2θϕ(x) |w(x)|2 dx + θ
e2θϕ(x) |∇w(x)|2 dx
C
Ω
Ω
"2
"
"
" θϕ
"
≤ "e (Δ − β)w
L2 (Ω)
3
2θϕ(x)
2
+C θ
e
|w(x)| dx + θ
e2θϕ(x) |∇w(x)|2 dx .
ωo
ωo
2 1/2
6.1. An equality with weight. We denote . =
the usual
(·) dx
2
norm in L (Ω) associated to the scalar product (·, ·) = (·, ·)dx. From now
on, we shall omit all xs in the functions of x.
Let θ > 0, ϕ = ϕ(x) ∈ C 2 (Rd ) be arbitrary but given. We shall prove
first that the function v(x) = eθϕ(x) w(x) where w ∈ H 2 (Ω) ∩ H01 (Ω) satisfies
Quantitative uniqueness for time-periodic heat equation
the following
⎧ "
"2
"
"
2
⎪
2 (∇ϕ)2 v − βv
⎪
Δv
−
θΔϕv
+
θ
−
q
v
"
" + 2θ∇ϕ∇v + qv
⎪
⎪
⎪
⎪
⎪
+2(boundary terms)
⎪
⎪
2
⎪
⎪
2
3 ∇ϕ∇ (∇ϕ)2 v 2
⎪
⎪
i ϕ(∂i v) + 2θ
⎨ +4θ i ∂
+4θ
ij ϕ∂i v∂j v
i
j=i ∂"
"
⎪
2
⎪
"
"
⎪
" + 2θ2 div(Δϕ∇ϕ)v 2
⎪
= "eθϕ (Δ − β)w
⎪
⎪
⎪
⎪
2 (∇ϕ)2 v − βv
⎪
− qv qv
⎪
Δv
−
θΔϕv
+
θ
−4θ
∇ϕ
q
v∇v
−
2
⎪
⎪
⎪
⎩
+2θβ Δϕv 2 + 2θ Δϕ(∇v)2 − 2θ3 Δϕ(∇ϕ)2 v 2 ,
⎧
⎪
2θβ Δϕv 2+ 2θ Δϕ(∇v)2 − 2θ3 Δϕ(∇ϕ)2 v2
⎨
− qv
= −2θ Δϕv Δv − θΔϕv + θ2 (∇ϕ)2 v − βv
⎪
2 2
⎩
2
2
2
2
q v + θ Δ ϕv
−2θ (Δϕ) v − 2θ Δϕ
and
657
(A1)
(A2)
boundary terms = −θ
(∂n v)2 ∂n ϕ.
(A3)
∂Ω
Proof of (A1). We shall start with computation of all derivatives of v =
as the sum of P1 v and P2 v where
eθϕ w and then write the term eθϕ (Δ − β)w
P1 is a differential operator of order 1 and P2 is a differential operator of
order 2 given below; after that we calculate Ω (P2 v + P1 v)2 dx where the
term Ω (P2 v, P1 v)dx plays a key role.
One can check easily that
Δv = Δeθϕ w + 2∇eθϕ ∇w + eθϕ Δw
= θΔϕv − θ2 (∇ϕ)2 v + 2θ∇ϕ∇v + eθϕ Δw
and
P2 v + P1 v = eθϕ (Δ − β)w,
(A4)
(A5)
where
P1 v = −2θ∇ϕ∇v + qv
(6.1)
− qv.
P2 v = Δv − θΔϕv + θ (∇ϕ) v − βv
2
2
Then we have
" θϕ
"
"
"
"2 = "Δv − θΔϕv + θ2 (∇ϕ)2 v − βv
− qv "2
"e (Δ − β)w
+ −2θ∇ϕ∇v + qv2 + 2(P2 v, P1 v).
(A6)
(A7)
658
K-D. Phung and G. Wang
However, a direct calculation leads us to
− qv, −2θ∇ϕ∇v + qv
(P2 v, P1 v) = Δv − θΔϕv + θ2 (∇ϕ)2 v − βv
2
3
Δϕ∇ϕv∇v − 2θ
∇ϕ(∇ϕ)2 v∇v
(A8)
= −2θ ∇ϕ∇vΔv + 2θ
+ 2θβ ∇ϕv∇v + 2θ ∇ϕ
q v∇v + (P2 v, qv)
2
2
2
= −2θ
∂i ϕ∂i v∂i v − 2θ
∂i ϕ∂i v
∂j v + θ
Δϕ∇ϕ∇(v 2 )
i
i
j=i
2
−θ
q v∇v + (P2 v, qv)
∇ϕ(∇ϕ) ∇(v ) + θβ ∇ϕ∇(v ) + 2θ ∇ϕ
= −θ
∂i ϕ∂i (∂i v)2 − 2θ
∂j (∂i ϕ∂i v∂j v)
3
2
2
i
+ 2θ
i
i
∂ij ϕ∂i v∂j v + θ
j=i
j=i
i
j=i
div(Δϕ∇ϕ)v 2 − θ3 div (∇ϕ(∇ϕ)2 )v 2
3
2 2
2
q v∇v + (P2 v, qv).
div ∇ϕ(∇ϕ) v − θβ Δϕv + 2θ ∇ϕ
+θ
+ θ2
div (Δϕ∇ϕ)v
2
∂i ϕ∂i (∂j v)2
− θ2
From this we get by integration by parts
− qv, −2θ∇ϕ∇v + qv = (boundary terms)
Δv − θΔϕv + θ2 (∇ϕ)2 v − βv
2
2
∂i ϕ(∂i v) + 2θ
∂ij ϕ∂i v∂j v − θ
∂i2 ϕ(∂j v)2
+θ
− θ2
+ 2θ
i
div(Δϕ∇ϕ)v 2 + θ3
i
j=i
div ∇ϕ(∇ϕ)2 v 2 − θβ
i
j=i
Δϕv 2
∇ϕ
q v∇v + (P2 v, qv).
Then by (A7) and (A9), we infer
"
"2 "
"2
"
" θϕ
"
− qv "
"e (Δ − β)w
" − "Δv − θΔϕv + θ2 (∇ϕ)2 v − βv
"
+ −2θ∇ϕ∇v + qv2
− qv, −2θ∇ϕ∇v + qv
= 2 Δv − θΔϕv + θ2 (∇ϕ)2 v − βv
(A9)
Quantitative uniqueness for time-periodic heat equation
= 2(boundary terms) + 2θ
+ 4θ
i
+ 2θ
3
∂i2 ϕ(∂i v)2
(A10)
i
∂ij ϕ∂i v∂j v − 2θ
i
j=i
div ∇ϕ(∇ϕ)2 v 2 − 2θβ
659
∂i2 ϕ(∂j v)2
j=i
− 2θ
2
div(Δϕ∇ϕ)v 2
2
Δϕv + 4θ
∇ϕ
q v∇v + 2(P2 v, qv)
or equivalently
P2 v2 + 2θ∇ϕ∇v − qv2 + 2(boundary terms)
∂i2 ϕ(∂i v)2 + 2θ3 div ∇ϕ(∇ϕ)2 v 2 + 4θ
∂ij ϕ∂i v∂j v
+ 2θ
i
i
j=i
"
"2
" θϕ
"
2
2
= "e (Δ − β)w" + 2θ
q v∇v − 2(P2 v, qv)
div(Δϕ∇ϕ)v − 4θ ∇ϕ
+ 2θ
∂i2 ϕ(∂j v)2 + 2θβ Δϕv 2 .
(A11)
i
By adding 2θ
j=i
2
2
i ∂i ϕ(∂i v)
on both sides of (A11) and noting that
div ∇ϕ(∇ϕ)2 = ∇ϕ∇ (∇ϕ)2 + Δϕ(∇ϕ)2 ,
we get
P2 v2 + 2θ∇ϕ∇v − qv2 + 2(boundary terms)
2
2
3
2 2
∂i ϕ(∂i v) + 2θ
∂ij ϕ∂i v∂j v
∇ϕ∇ (∇ϕ) v + 4θ
+ 4θ
i
i
j=i
"
"2
"
"
= "eθϕ (Δ − β)w
q v∇v − 2(P2 v, qv)
" + 2θ2 div(Δϕ∇ϕ)v 2 − 4θ ∇ϕ
∂i2 ϕ(∂j v)2 + 2θβ Δϕv 2
+ 2θ
i
+ 2θ
j=i
∂i2 ϕ(∂i v)2
− 2θ
3
Δϕ(∇ϕ)2 v 2
i
"
"2
" θϕ
"
2
2
= "e (Δ − β)w" + 2θ
q v∇v − 2(P2 v, qv)
div(Δϕ∇ϕ)v − 4θ ∇ϕ
+ 2θβ Δϕv 2 + 2θ Δϕ(∇v)2 − 2θ3 Δϕ(∇ϕ)2 v 2 ,
(A12)
as desired.
660
K-D. Phung and G. Wang
Proof of (A2). By integration by parts and the fact that v ∈ H 2 (Ω) ∩
H01 (Ω), it follows that
2θ Δϕ∇v∇v = −2θ ΔϕvΔv − 2θ v∇Δϕ∇v
2
= −2θβ Δϕv − 2θ Δϕv(Δ − β)v − 2θ v∇Δϕ∇v
1
2
= −2θβ Δϕv − 2θ ∇Δϕ ∇(v 2 )
2
!
− 2θ Δϕv P2 v + θΔϕv − θ2 (∇ϕ)2 v + qv
(A13)
or equivalently
2θβ Δϕv 2 + 2θ Δϕ(∇v)2 − 2θ3 Δϕ(∇ϕ)2 v 2
2
2 2
2
= −2θ ΔϕvP2 v − 2θ
q v + θ Δ2 ϕv 2 ,
(Δϕ) v − 2θ Δϕ
(A14)
as desired.
Proof of (A3). By (A8) and (A9) we can check that
boundary terms = −θ
∂i ∂i ϕ(∂i v)2
− 2θ
i
∂Ω
j=i
i
j=i
∂Ω
i
i
2
∂i ∂i ϕ(∂j v)2
− θ3 div (∇ϕ(∇ϕ)2 )v 2
2
∂i ϕ(∂i v) ni − 2θ
∂i ϕ∂i v∂j vnj
div (Δϕ∇ϕ)v
∂Ω
+θ
∂j (∂i ϕ∂i v∂j v) + θ
+ θ2
= −θ
i
2
∂i ϕ(∂j v) ni + θ
i
j=i
(Δϕ∇ϕ)v · n − θ
2
2
∇ϕ(∇ϕ)2 v 2 · n.
3
∂Ω
j=i
(A15)
∂Ω
Since v ∈ H 2 (Ω) ∩ H01 (Ω), we deduce from (A15) that
boundary terms = −θ
∂i ϕ(∂i v)2 ni
− 2θ
∂Ω
∂Ω
i
j=i
i
∂i ϕ∂i v∂j vnj + θ
∂Ω
i
j=i
∂i ϕ(∂j v)2 ni
Quantitative uniqueness for time-periodic heat equation
= −2θ
∂Ω
∂Ω
i
∂Ω
= −2θ
∂Ω
= −2θ
i
∂Ω
j=i
∂i ϕ∂i v∂j vnj + θ
j
∇ϕ∇v∂n v + θ
∂Ω
∂i ϕ(∂i v)2 ni
i
∂Ω
∂i ϕ∂i v∂j vnj
j=i
∂i ϕ(∂j v)2 ni + θ
i
i
+θ
∂i ϕ(∂i v) ni − 2θ
2
661
i
∂i ϕ(∂j v)2 ni
j
(∇v)2 ∂n ϕ.
(A16)
∂Ω
Since v|∂Ω = 0, ∇v = (∇v · n)n and then we have
2
2
(∂n v) ∂n ϕ + θ
(∇v) ∂n ϕ = −θ
boundary terms = −2θ
∂Ω
∂Ω
∂Ω
(∂n v)2 ∂n ϕ.
(A17)
This completes the proof of (A3).
6.2. Choice of the weight function. We shall prove that with a particular
choice of ϕ = ϕ(x) ≥ 1 there exists a constant C > 0 such that for all β ∈ R,
4/3
2/3 + q L∞ (Ω) , the estimate
for all q ∈ L∞ (Ω), and θ > C 1 + |β|
"2
"
2 1 3
"
θϕ
θϕ "
θϕ
2
(e w) + θ
θ
∇(eθϕ w)
"2θ∇ϕ∇(e w) − qe w" +
C
Ω
Ω
"
"2
3
2 " θϕ
"
" +C θ
∇(eθϕ w)
≤ "e (Δ − β)w
(eθϕ w)2 + θ
(A18)
ωo
holds for all w ∈
H 2 (Ω)
∩
ωo
H01 (Ω).
Proof of (A18). We study the particular case that ϕ(x) = eρψ(x) , where
ρ > 0 and ψ is a smooth non-negative function. One can check that
Δϕ = ρ2 (∇ψ)2 ϕ + ρΔψϕ
(A19)
and
2ϕ
2 ψϕ
= ρ(∂i ϕ)2 ∂j ψ + ∂i ϕρ∂ij
∂i ϕ∂ij
2
2
2
2
2 ψϕ.
∂i ϕ∂j ϕ∂ij ϕ = ρ (∂j ψ) (∂i ϕ) ϕ + ρ∂i ϕ∂j ϕ∂ij
Next we compute ∇ϕ∇ (∇ϕ)2 = i ∂i ϕ∂i (∇ϕ)2 . Since
2
∂j ϕ∂ij
ϕ
∂i ϕ∂i (∇ϕ)2 = 2∂i ϕ∇ϕ∇∂i ϕ = 2∂i2 ϕ(∂i ϕ)2 + 2∂i ϕ
j=i
(A20)
!
!
2
= 2 ρ2 (∂i ψ)2 ϕ + ρ∂i2 ψϕ (∂i ϕ)2 + 2∂i ϕ
∂j ϕ ρ2 ∂i ψϕ∂j ψ + ρ∂ij
ψϕ
j=i
662
K-D. Phung and G. Wang
!
= 2 ρ2 (∂i ψ)2 ϕ + ρ∂i2 ψϕ (∂i ϕ)2 + 2∂i ϕ
(ρ∂j ψϕ)(ρ∂i ψϕ)ρ∂j ψ
+ 2∂i ϕ
j=i
2
∂j ϕρ∂ij
ψϕ
!
= 2 ρ2 (∂i ψ)2 ϕ + ρ∂i2 ψϕ (∂i ϕ)2
j=i
2
+ 2ρ ∂i ϕ
= 2ρ2
(∂j ψ)2 ϕ∂i ϕ + 2∂i ϕ
j=i
2
∂j ϕρ∂ij
ψϕ
j=i
2
(∂j ψ)2 ϕ(∂i ϕ)2 + 2ρ ∂i2 ψϕ(∂i ϕ)2 + ∂i ϕ
∂j ϕ∂ij
ψϕ ,
j
(A21)
j=i
it follows that
∇ϕ∇ (∇ϕ)2 = 2ρ2 (∇ψ)2 ϕ(∇ϕ)2
2
+ 2ρ
∂j ϕ∂ij
ψ ϕ.
∂i2 ψ (∂i ϕ)2 + ∂i ϕ
i
(A22)
j=i
On the other hand, by (A1) and (A2), we infer
P2 v2 + 2θ∇ϕ∇v + qv2 + 2(boundary terms)
2
2
3
2 2
+ 4θ
∂i ϕ(∂i v) + 2θ
∂ij ϕ∂i v∂j v
∇ϕ∇ (∇ϕ) v + 4θ
i
i
j=i
"
"2
"
2
2
"
= "eθϕ (Δ − β)w
+
2θ
−
4θ
∇ϕ
q v∇v − 2(P2 v, qv)
div(Δϕ∇ϕ)v
"
2
2 2
2
− 2θ ΔϕvP2 v − 2θ
(Δϕ) v − 2θ Δϕ
q v + θ Δ2 ϕv 2
(A23)
which together with (A22) gives
P2 v2 + 2θ∇ϕ∇v − qv2 + 2 (boundary terms)
2
ϕ
∂i ψ∂j ψ∂i v∂j v
(∂i ψ)2 (∂i v)2 +
+ 4θρ
i
+ 4θρ
ϕ
i
+ 4θ3 ρ
i
∂i2 ψ(∂i v)2
+
j=i
∂i2 ψ(∂i ϕ)2 + ∂i ϕ
j=i
2
∂ij
ψ∂i v∂j v
j=i
3 2
+ 4θ ρ
(∇ψ)2 (∇ϕ)2 ϕv 2
2
∂j ϕ∂ij
ψ ϕv 2
"
"2
"
"
= "eθϕ (Δ − β)w
q v∇v − 2(P2 v, qv)
" + 2θ2 div(Δϕ∇ϕ)v 2 − 4θ ∇ϕ
Quantitative uniqueness for time-periodic heat equation
− 2θ
ΔϕvP2 v − 2θ
2
(Δϕ) v − 2θ
2 2
663
2
Δϕ
qv + θ
Δ2 ϕv 2
(A24)
or equivalently
P2 v2 + 2θ∇ϕ∇v − qv2 + 2(boundary terms)
2
2
3 2
+ 4θρ
∂i ψ∂i v + 4θ ρ
ϕ
(∇ψ)2 (∇ϕ)2 ϕv 2
(6.2)
i
"
"2
" θϕ
"
2
2
= "e (Δ − β)w" + 2θ
q v∇v − 2(P2 v, qv)
div(Δϕ∇ϕ)v − 4θ ∇ϕ
2
2 2
2
− 2θ ΔϕvP2 v − 2θ
(Δϕ v − 2θ Δϕ
q v + θ Δ2 ϕv 2
2
− 4θρ
ϕ ∂i2 ψ(∂i v)2 +
∂ij
ψ∂i v∂j v
i
− 4θ3 ρ
i
j=i
∂i2 ψ(∂i ϕ)2 + ∂i ϕ
2
∂j ϕ∂ij
ψ ϕv 2 .
(A25)
j=i
Next we compute the term 2θρ2 (∇ψ)2 ϕ(∇v)2 . To this end, we observe
by (A2) that
2
!
2
2
2θ
ρ (∇ψ) ϕ + ρΔψϕ (∇v) = 2θ Δϕ(∇v)2
2
3
Δϕ(∇ϕ)2 v 2
= −2θβ Δϕv + 2θ
− 2θ ΔϕvP2 v − 2θ2 (Δϕ)2 v 2 − 2θ Δϕ
q v 2 + θ Δ2 ϕv 2
2
!
ρ (∇ψ)2 ϕ + ρΔψϕ (∇ϕ)2 v 2
= −2θβ Δϕv 2 + 2θ3
− 2θ ΔϕvP2 v − 2θ2 (Δϕ)2 v 2 − 2θ Δϕ
q v 2 + θ Δ2 ϕv 2 (A26)
which implies
2
2
2
2
3 2
(∇ψ) ϕ(∇v) = −2θρ Δψϕ(∇v) + 2θ ρ
(∇ψ)2 (∇ϕ)2 ϕv 2
2θρ
3
2
2
2
+ 2θ ρ Δψ(∇ϕ) ϕv − 2θ ΔϕvP2 v − 2θ
(Δϕ)2 v 2
2 + θ Δ2 ϕv 2 .
− 2θ Δϕ(
q + β)v
(A27)
664
K-D. Phung and G. Wang
Finally, it follows from (A25) and (A27) that
P2 v2 + 2θ∇ϕ∇v − qv2 + 2(boundary terms) + 4θρ2
∂i ψ∂i v
2
ϕ
i
+ 4θ3 ρ2 (∇ψ)2 (∇ϕ)2 ϕv 2 + 2θρ2 (∇ψ)2 ϕ(∇v)2
"
"2
" θϕ
"
2
2
= "e (Δ − β)w" + 2θ
q v∇v − 2(P2 v, qv)
div(Δϕ∇ϕ)v − 4θ ∇ϕ
β 2
2
2 2
v − θ Δ2 ϕv 2
(Δϕ) v + 2θ Δϕ q +
− 2 2θ ΔϕvP2 v + 2θ
2
2
∂ij
ψϕ∂i v∂j v
(A28)
− 4θρ
∂i2 ψϕ(∂i v)2 +
i
− 4θ3 ρ
∂i2 ψ(∂i ϕ)2 + ∂i ϕ
i
− 2θρ
j=i
2
∂j ϕ∂ij
ψ ϕv 2
j=i
Δψϕ(∇v)2 + 2θ3 ρ2
(∇ψ)2 (∇ϕ)2 ϕv 2 + 2θ3 ρ
or equivalently
P2 v2 + 2θ∇ϕ∇v − qv2 + 2(boundary terms) + 4θρ2
2
2
2
∂i ψ∂i v
2
ϕ
i
3 2
Δψ(∇ϕ)2 ϕv 2
2
(∇ψ) (∇ϕ) ϕv + 2θρ
(∇ψ)2 ϕ(∇v)2
+ 2θ ρ
"
"2
" θϕ
"
2
2
= "e (Δ − β)w" + 2θ
q v∇v − 2(P2 v, qv)
div(Δϕ∇ϕ)v − 4θ ∇ϕ
β 2
2
2 2
(Δϕ) v + 2θ Δϕ q +
− 2 2θ ΔϕvP2 v + 2θ
v − θ Δ2 ϕv 2
2
2
∂i2 ψϕ(∂i v)2 +
∂ij
ψϕ∂i v∂j v
− 4θρ
i
− 4θ3 ρ
− 2θρ
j=i
∂i2 ψ(∂i ϕ)2 + ∂i ϕ
i
Δψϕ(∇v)2 + 2θ3 η
2
∂j ϕ∂ij
ψ ϕv 2
j=i
Δψϕ(∇ϕ)2 v 2 .
(A29)
Now, we choose ψ such that ψ ∈ C 2 (Ω), ψ(x) > 0 in Ω, ψ(x) = 0 on ∂Ω and
|∇ψ(x)| > 0
∀x ∈ Ω \ωo ,
(A30)
Quantitative uniqueness for time-periodic heat equation
665
where ωo ⊂ ω is an arbitrary fixed subdomain of Ω such that ωo ⊂ ω. The
existence of such a function ψ was proved in [13]. Then the boundary terms
(A3) are non-negative. More precisely, for all ρ > 0, θ > 0,
boundary terms = −θ
(∂n v)2 (ρ∂n ψ)ϕ ≥ 0.
(A31)
∂Ω
By (A30), (A31) and (A29), there exists a constant C > 0, such that for all
ρ > 0 there is a constant Cρ > 0 such that for all θ > 0, we have
1 3 2
2
2
2
2
2
P2 v + 2θ∇ϕ∇v − qv +
(∇ϕ) ϕv + θρ
ϕ(∇v)2
θ ρ
C
Ω\ωo
Ω\ωo
≤ P2 v2 + 2θ∇ϕ∇v − qv2
2 2
+ 2(boundary terms) + 4θρ
∂i ψ∂i v ϕ
i
+ 2θ3 ρ2
(∇ψ)2 (∇ϕ)2 ϕv 2 + 2θρ2 (∇ψ)2 ϕ(∇v)2
"
"2
"
2
2
"
= "eθϕ (Δ − β)w
+
2θ
−
4θ
∇ϕ
q v∇v − 2(P2 v, qv)
div(Δϕ∇ϕ)v
"
β
q + )v 2 − Δ2 ϕv 2
− 2θ 2 ΔϕvP2 v + 2θ (Δϕ)2 v 2 + 2 Δϕ(
2
2
∂ij
ψϕ∂i v∂j v
∂i2 ψϕ(∂i v)2 +
− 4θρ
i
− 4θ3 ρ
− 2θρ
j=i
∂i2 ψ(∂i ϕ)2 + ∂i ϕ
i
Δψϕ(∇v)2 + 2θ3 ρ
2
∂j ϕ∂ij
ψ ϕv 2
j=i
Δψϕ(∇ϕ)2 v 2 .
(A32)
Using the Cauchy-Schwarz inequality in (A32), we get
1 3 2
P2 v2 + 2θ∇ϕ∇v − qv2 +
(∇ϕ)2 ϕv 2 + θρ2
ϕ(∇v)2
θ ρ
C
Ω\ωo
Ω\ωo
"2
"
"
"
q 2L∞ Cρ v 2 + (∇v)2
≤ "eθϕ (Δ − β)w
" + θ2 Cρ v 2 + θ2 1
2
2
2
2
+ 2
q L∞ v + P2 v + θ q L∞ + |β| Cρ v + θCρ v 2
(A33)
2
+ Cθρ ϕ(∇v)2 + Cθ3 ρ (∇ϕ)2 ϕv 2 + Cθρ ϕ(∇v)2 + Cθ3 ρ ϕ(∇ϕ)2 v 2 .
666
K-D. Phung and G. Wang
Finally, we obtain that, (recall that ϕ ≥ 1) for all ρ > 1, for all θ > 1,
1
1 (∇ϕ)2 ϕv 2 + θ
ϕ(∇v)2
P2 v2 + 2θ∇ϕ∇v − qv2 + ρ2 θ3
2
C
Ω\ωo
Ω\ωo
"2
"
"
+ "
q L∞ + q 2L∞ Cρ v 2
≤ "eθϕ (Δ − β)w
" + θ2 1 + |β|
+ Cρ θ3 (∇ϕ)2 ϕv 2 + θ ϕ(∇v)2
"2
"
"
2 2
"
1
+
|
β|
+
q
C
+
2θ
v2
≤ "eθϕ (Δ − β)w
"
ρ
L∞
3
2
2
+ Cρ θ
(∇ϕ) ϕv + θ
ϕ(∇v)2
ωo
ωo
3
2
2
(A34)
(∇ϕ) ϕv + θ
ϕ(∇v)2 ,
+ Cρ θ
Ω\ωo
Ω\ωo
where C > 0 is a constant independent of ρ and θ. Now we may choose and
fix ρ > 1 large enough in (A34) to get
1 3
2
2
2
2θ∇ϕ∇v − qv +
(∇ϕ) ϕv + θ
ϕ(∇v)2
θ
C
Ω\ωo
Ω\ωo
"2
"
"
" θϕ
2
2
2
v +
v2
≤ "e (Δ − β)w" + θ 1 + |β| + q L∞ C
Ω\ωo
ωo
(A35)
(∇ϕ)2 ϕv 2 + θ
ϕ(∇v)2 ,
+ C θ3
ωo
ωo
where C > 0 is a constant independent of θ. Then by (A35), using properties
+ of ψ, we may choose θ large enough so that θ3/2 > C1 1 + |β|
q 2L∞ (Ω)
to get
1 3
2
2
θ
v +θ
(∇v)2
2θ∇ϕ∇v − qv +
C2
Ω\ωo
Ω\ωo
"2
"
"
" θϕ
3
2
2
(A36)
(∇ϕ) ϕv + θ
ϕ(∇v)2 ,
≤ "e (Δ − β)w" + C2 θ
ωo
ωo
where C1 and C2 are two constants independent of θ.
Finally, we have proved that there exists a constant C > 0 such that for
4/3
2/3 + all β ∈ R, q ∈ L∞ (Ω) and θ > C 1 + |β|
q L∞ (Ω) , the following
estimate holds
1 3
2
2
v + θ (∇v)2
θ
2θ∇ϕ∇v − qv +
C
Ω
Ω
Quantitative uniqueness for time-periodic heat equation
"
"2
" θϕ
"
≤ "e (Δ − β)w" + C θ3
2
2
v +θ
ωo
(∇v)
667
.
(A37)
ωo
4/3
2/3 + q L∞ (Ω) ,
Thus, coming back to the function w, for all θ > C 1 + |β|
"
"2
2 1 3
"
"
(eθϕ w)2 + θ
θ
∇(eθϕ w)
"2θ∇ϕ∇(eθϕ w) − qeθϕ w" +
C
Ω
Ω
"
"2
2 " θϕ
"
3
θϕ
2
θϕ
" +C θ
≤ "e (Δ − β)w
(e w) + θ
∇(e w) ,
(A38)
ωo
ωo
which completes the proof of (A18).
Finally, we deduce from (A38),
"2
"
1 3
"
θϕ
θϕ "
θϕ
2
(e w) + θ (eθϕ ∇w)2
θ
"2θ∇ϕ∇(e w) − qe w" +
C
Ω
Ω
"
"2
"
"
(A39)
≤ "eθϕ (Δ − β)w
(eθϕ w)2 + θ
(eθϕ ∇w)2 ,
" + C θ3
ωo
ωo
which implies Theorem A.
This closes the Appendix on the global Carleman inequality for the operator Δ − β where β ∈ R.
Acknowledgment. This work was supported by the grant of Key Lab of
Hubei Province, China.
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