ATH c Vol. 68, No. 6, pp. 1535–1556 · From the standard operator semigroup theory, D(Ak) (k ......

22
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. APPL. MATH. c 2008 Society for Industrial and Applied Mathematics Vol. 68, No. 6, pp. 1535–1556 TIME REVERSAL FOCUSING OF THE INITIAL STATE FOR KIRCHHOFF PLATE KIM DANG PHUNG AND XU ZHANG Abstract. Consider a Kirchhoff plate 2 t u 2 u 2 t Δu = 0 in Ω × (0,T ), with boundary data u u = 0 on Ω×(0,T ) and unknown initial data u(·, 0) = u 0 and t u(·, 0) = u 1 in Ω. We study an inverse problem of determining (u 0 ,u 1 ) from an interior observation u| ω×(0,T ) . Here Ω is a bounded domain, ω a nonempty open subset of Ω, and T> 0 a suitable time duration. By means of an iterative time reversal technique, we derive an asymptotic formula of reconstructing (u 0 ,u 1 ) approximately with a logarithmical convergence rate for smooth initial data. The convergence becomes uniform and exponential when (Ω,ω,T ) satisfies the geometric control condition introduced by Bardos, Lebeau, and Rauch. Key words. Kirchhoff plate, inverse problem, quantitative unique continuation, observability estimate, time reversal technique AMS subject classifications. Primary, 35R30; Secondary, 74K20, 93B07, 35B37 DOI. 10.1137/070684823 1. Introduction and main results. Let Ω R d (d N) be a bounded open set with sufficiently smooth boundary Ω, ω a nonempty open subset of Ω, T > 0 a suitable time duration, and β (0, 1) any fixed parameter. Let M = ( α ij ) 1i,jd C ( Ω; R d×d ) be a symmetric and uniformly positive definite matrix (hence (β ij ) 1i,jd = M 1/2 is well defined). Denote by 1 |ω the characteristic function of ω in Ω. Let Q × (0,T ) and Σ = Ω × (0,T ). Throughout this paper, we shall use C = C,ω,T,d,β, M) to denote a generic positive constant, which may change from line to line. Denote by Δ = d i,j=1 xi (α ij xj ) the “Laplacian” associated to the matrix M. We consider the following Kirchhoff plate equation in an inhomogeneous media: (1.1) 2 t u 2 u 2 t Δu =0 in Ω × R, u u =0 on Ω × R, u(·, 0) = u 0 , t u(·, 0) = u 1 in Ω. Let H = z H 3 (Ω) z z = 0 on Ω × H 2 (Ω) H 1 0 (Ω) . Clearly, H is a Hilbert space with the norm ||(u 0 ,u 1 )|| H = (Δu 0 ,u 1 , Δu 1 ) (L 2 (Ω)) d ×H 1 0 (Ω)×L 2 (Ω) . Received by the editors March 9, 2007; accepted for publication (in revised form) February 8, 2008; published electronically June 6, 2008. This work was supported by the NSF of China under grants 10525105 and 10771149, grant MTM2005-00714 of the Spanish MEC, and the Chinese Post- doctoral Science Foundation. http://www.siam.org/journals/siap/68-6/68482.html Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China (kim dang phung@ yahoo.fr). Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China, and Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China ([email protected]). 1535

Transcript of ATH c Vol. 68, No. 6, pp. 1535–1556 · From the standard operator semigroup theory, D(Ak) (k ......

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SIAM J. APPL. MATH. c© 2008 Society for Industrial and Applied MathematicsVol. 68, No. 6, pp. 1535–1556

TIME REVERSAL FOCUSING OF THE INITIAL STATE FORKIRCHHOFF PLATE∗

KIM DANG PHUNG† AND XU ZHANG‡

Abstract. Consider a Kirchhoff plate ∂2t u+Δ2u−∂2

t Δu = 0 in Ω× (0, T ), with boundary datau = Δu = 0 on ∂Ω×(0, T ) and unknown initial data u(·, 0) = u0 and ∂tu(·, 0) = u1 in Ω. We study aninverse problem of determining (u0, u1) from an interior observation u|ω×(0,T ). Here Ω is a boundeddomain, ω a nonempty open subset of Ω, and T > 0 a suitable time duration. By means of an iterativetime reversal technique, we derive an asymptotic formula of reconstructing (u0, u1) approximatelywith a logarithmical convergence rate for smooth initial data. The convergence becomes uniform andexponential when (Ω, ω, T ) satisfies the geometric control condition introduced by Bardos, Lebeau,and Rauch.

Key words. Kirchhoff plate, inverse problem, quantitative unique continuation, observabilityestimate, time reversal technique

AMS subject classifications. Primary, 35R30; Secondary, 74K20, 93B07, 35B37

DOI. 10.1137/070684823

1. Introduction and main results. Let Ω ⊂ Rd (d ∈ N) be a bounded

open set with sufficiently smooth boundary ∂Ω, ω a nonempty open subset of Ω,T > 0 a suitable time duration, and β ∈ (0, 1) any fixed parameter. Let M =(αij)1≤i,j≤d

∈ C∞ (Ω; Rd×d)

be a symmetric and uniformly positive definite matrix

(hence (βij)1≤i,j≤d = M1/2 is well defined). Denote by 1|ω the characteristic functionof ω in Ω. Let Q = Ω × (0, T ) and Σ = ∂Ω × (0, T ). Throughout this paper, we shalluse C = C(Ω, ω, T, d, β,M) to denote a generic positive constant, which may changefrom line to line.

Denote by Δ =∑d

i,j=1 ∂xi(αij∂xj

) the “Laplacian” associated to the matrix M.We consider the following Kirchhoff plate equation in an inhomogeneous media:

(1.1)

⎧⎪⎪⎨⎪⎪⎩∂2t u + Δ2u− ∂2

t Δu = 0 in Ω × R,

u = Δu = 0 on ∂Ω × R,

u(·, 0) = u0, ∂tu(·, 0) = u1 in Ω.

Let

H �={z ∈ H3(Ω)

∣∣∣ z = Δz = 0 on ∂Ω}×(H2(Ω) ∩H1

0 (Ω)).

Clearly, H is a Hilbert space with the norm

||(u0, u1)||H�= ‖(∇Δu0, u1,Δu1)‖(L2(Ω))d×H1

0 (Ω)×L2(Ω) .

∗Received by the editors March 9, 2007; accepted for publication (in revised form) February 8,2008; published electronically June 6, 2008. This work was supported by the NSF of China undergrants 10525105 and 10771149, grant MTM2005-00714 of the Spanish MEC, and the Chinese Post-doctoral Science Foundation.

http://www.siam.org/journals/siap/68-6/68482.html†Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China (kim dang phung@

yahoo.fr).‡Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese

Academy of Sciences, Beijing 100080, China, and Yangtze Center of Mathematics, Sichuan University,Chengdu 610064, China ([email protected]).

1535

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1536 KIM DANG PHUNG AND XU ZHANG

Here and henceforth, ∇ = (∑d

j=1 β1j∂xj

, . . . ,∑d

j=1 βdj∂xj

). It is easy to rewrite (1.1)as an abstract Cauchy problem in H, with an unbounded operator A : D(A) ⊂ H → Has the generator of the underlying C0-group. Hence, for any initial data (u0, u1) ∈ H,system (1.1) is well-posed in H. From the standard operator semigroup theory, D(Ak)(k ∈ N) are themselves Hilbert spaces with the graph norms.

For any z ∈ C(R;H), we denote by E(z, t) the functional

(1.2) E(z, t)�=

1

2

∫Ω

[|∇Δz(x, t)|2 + |∇∂tz(x, t)|2 + |Δ∂tz(x, t)|2

]dx.

It is clear that H is the finite energy space of system (1.1), and its energy E(·, t) isconservative in the sense that for any u solution of (1.1) and all t ∈ R,

(1.3) E(u, t) =1

2||(u0, u1)||2H.

The main purpose of this paper is to investigate the state-observation problem forsystem (1.1), which is formulated as follows: To determine the initial data (u0, u1) ofa solution u of (1.1) from the single interior measurement u|ω×(0,T ). It is well knownthat the state-observation problem is closely related to the inverse source problem, i.e.,to determine the source term which causes the evolution process from the boundaryand/or interior measurement. Inverse source problems of PDEs have been the objectof numerous studies in recent years. Extensive related references can be found, say,in [18, 24, 25, 26] for the hyperbolic equations, in [22] for the Euler–Bernoulli plateequation, and in other works cited therein. Most of the references on inverse sourceproblems cited above are addressed to global uniqueness and stability; here we givea constructive strategy to recover the initial data from a partial measurement of thesolution. Our strategy for identification of source is inspired by the time reversalmethod and may be more practical than the formal tools of control theory (e.g., [23]).By means of an iterative time reversal technique, we further establish an asymptoticformula to reconstruct the desired initial state (u0, u1) of (1.1) by superposing differentsolutions of some Kirchhoff plates depending only on the measurement u|ω×(0,T ).

More precisely, the knowledge of u on ω× (0, T ) allows us to consider a sequenceof solutions {v(j)}j≥0 given as follows. First, let U (−1) = 1

2u in ω × (0, T ). Next,

define v(j) = v(j)(x, t) (j = 0, 1, 2, . . .) inductively to be the solution of the followingsystem:

⎧⎪⎪⎨⎪⎪⎩∂2t v

(j) + Δ2v(j) − ∂2t Δv(j) + ∂tΔv(j) · 1|ω = −2∂tΔU (j−1)(·, T − t) · 1|ω in Q,

v(j) = Δv(j) = 0 on Σ,

v(j)(·, 0) = ∂tv(j)(·, 0) = 0 in Ω,

(1.4)

where U (j) = U (j)(x, t) is given by

U (j)(x, t) =

{v(0)(x, t) − u(x, T − t), j = 0,

v(j)(x, t) − U (j−1)(x, T − t), j > 0,for (x, t) ∈ ω × (0, T ).

(1.5)

Note that the values of functions U (j) are defined only in ω × (0, T ). Nevertheless,by system (1.4), it suffices to determine the values of v(j) in the whole domain Q

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1537

from ∂tΔU (j−1)∣∣ω×(0,T )

. It is easy to see that the functions v(j) depend only on ∂tΔu

restricted to ω × (0, T ).We say that (Ω, ω, T0) satisfies the classical geometric control condition (GCC),

introduced in [2, 3], if ∂Ω is C∞ with no contact of infinite order with its tangent,and any generalized bicharacteristic ray (x(ρ), t(ρ)) of ∂2

t − Δ starting at ρ = 0 witht(0) = 0 meets ω × (0, T0) (see also [4] for an improvement on the regularity of ∂Ωand of M). Notice that GCC can be rephrased by a geodesic condition (see [17]).

The main results of this paper are stated as follows.Theorem 1.1. Under GCC, for any T ≥ T0 there exists a constant σ > 0 such

that for any initial data (u0, u1) ∈ H and any N > 0, it holds that

(1.6)

∥∥∥∥∥(

N∑k=0

v(2k)(·, T ) − u0,

N∑k=0

∂tv(2k)(·, T ) + u1

)∥∥∥∥∥H

≤ Ce−σN ‖(u0, u1)‖H .

Theorem 1.2. Suppose

J3�= sup

j>0

∥∥∥∥∥(

j∑k=0

v(2k)(·, T ),

j∑k=0

∂tv(2k)(·, T )

)∥∥∥∥∥D(A3)

< +∞

and that Ω is connected. Then, for any nonempty open subset ω of Ω and anyβ ∈ (0, 1), there exists a time T > 0 such that for any initial data (u0, u1) ∈ D(A3)and any N > 0, it holds that(1.7)∥∥∥∥∥(

N∑k=0

v(2k)(·, T ) − u0,

N∑k=0

∂tv(2k)(·, T ) + u1

)∥∥∥∥∥H

≤ C

lnβ(1 + N)[J3+‖(u0, u1)‖D(A3)].

The above results say that (∑N

k=0 v(2k)(·, T ), −

∑Nk=0 ∂tv

(2k)(·, T )) can be em-ployed to serve as an asymptotic formula to recover the initial state (u0, u1) of system(1.1). The key point to do this is the time reversibility of Kirchhoff plate. Fink (see[6, 7]) experimented with the time reversal mirror and succeeded in generating manyapplications (e.g., in biomedical engineering and telecommunication). Next, manymathematicians were also interested in this phenomenon (e.g., [1, 8, 19]). Thanksto the refocusing properties of the time-reversed waves, the time reversal techniquehas been successfully used to solve inverse problems for acoustic waves or electro-magnetic waves (see, e.g., [5, 12]). Nevertheless, the main novelty in Theorems1.1 and 1.2 is, respectively, the explicit exponential and logarithmical convergencerates for (

∑Nk=0 v

(2k)(·, T ), −∑N

k=0 ∂tv(2k)(·, T )) to approximate (u0, u1) in the strong

topology of H. Note also that Theorem 1.2 is for the case without GCC on (Ω, ω, T ),for which one can usually expect a much weaker result than the case with GCC (werefer the reader to [14, 20] for a different yet related topic for the hyperbolic equa-tions).

Technically, the proofs of Theorems 1.1 and 1.2 are reduced to suitable observabil-ity estimates for system (1.1). Under GCC, the desired observability estimate followsfrom the known result in [2, 3] for the wave equation. For the treatment in the casewithout GCC, by the Fourier–Bros–Iagolnitzer transformation given in [14], the ob-taining of the desired observability estimate for the evolution system (1.1) depends onsome quantitative unique continuation property for a fourth order elliptic-like equa-tion with multiple characteristics (see (3.1)), which, in turn, will be established by

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1538 KIM DANG PHUNG AND XU ZHANG

means of global Carleman estimate. Although global Carleman estimates are wellunderstood for many PDEs with single characteristics or without characteristics, itseems that there is no reference for the multiple-characteristic PDEs. The crucialpoint for the possibility of applying the Carleman estimate to the above-mentionedmultiple-characteristic equation is that this equation can be rewritten equivalently astwo coupled elliptic equations of second order, and that, based on a useful pointwiseestimate for second order differential operators with symmetric coefficients (withoutany sign condition), we are successful in using Carleman estimates with a commonweight function for these equations.

To end this section, we remark that, if the first equation in (1.4) is replaced by

∂2t v

(j) +Δ2v(j) − ∂2t Δv(j) − (−Δ)−1

(∂tv

(j) · 1|ω)

= 2(−Δ)−1(∂tU

(j−1)(·, T − t) · 1|ω),

while (u0, u1) is assumed only to belong to D(A), then, based on inequality (5.7) inTheorem 5.2, the estimate (1.7) in Theorem 1.2 becomes∥∥∥∥∥(

N∑k=0

v(2k)(·, T ) − u0,

N∑k=0

∂tv(2k)(·, T ) + u1

)∥∥∥∥∥H

≤ C

lnβ(1 + N)[J1 + ‖(u0, u1)‖D(A)].

The rest of this paper is organized as follows. In section 2, we derive the desiredpointwise estimate for second order differential operators with symmetric coefficients.Section 3 shows an interpolation inequality for the fourth order elliptic-like equationwith multiple characteristics mentioned above. Section 4 is devoted to a quantitativeunique continuation property for system (1.1). In section 5, we establish two observ-ability estimates for solutions of (1.1). The proofs of Theorems 1.1 and 1.2 are givenin section 6.

2. Pointwise estimate for second order differential operators with sym-metric coefficients. In this section, we will establish a pointwise estimate for secondorder differential operators with symmetric coefficients (without any sign condition),which will play a key role in what follows.

Let m ∈ N. For simplicity, for a function u, we will use the notation ui = ∂u∂xi

,where xi is the ith coordinate of a generic point (x1, . . . , xm) in R

m.For any

(2.1) aij = aji ∈ C1(Rm), i, j = 1, 2, . . . ,m,

we recall the following known identity (see [10, Theorem 4.1], and also [9, Theorem 1.1]for a variant version).

Lemma 2.1. Assume u, �,Ψ ∈ C2(Rm). Let θ = e� and v = θu. Then

(2.2)

θ2

∣∣∣∣∣∣m∑

i,j=1

(aijui)j

∣∣∣∣∣∣2

+ 2

m∑j=1

{2

m∑i,i′,j′=1

aijai′j′�i′vivj′ −

m∑i,i′,j′=1

aijai′j′�ivi′vj′

+ Ψ

m∑i=1

aijviv −m∑i=1

aij[(A + Ψ)�i +

Ψi

2

]v2

}j

= 2

m∑i,j=1

cijvivj + Bv2 +

∣∣∣∣∣∣m∑

i,j=1

(aijvi)j −Av

∣∣∣∣∣∣2

+ 4

∣∣∣∣∣∣m∑

i,j=1

aij�ivj

∣∣∣∣∣∣2

,

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1539

where

(2.3)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A�= −

m∑i,j=1

(aij�i�j − aijj �i − aij�ij

)− Ψ,

B�= 2

⎧⎨⎩AΨ −m∑

i,j=1

[(A + Ψ)aij�i

]j

⎫⎬⎭+ Ψ2 −m∑

i,j=1

(aijΨj

)i,

cij�=

m∑i′,j′=1

[2aij

′(ai

′j�i′)j′ − (aijai′j′�i′)j′

]+ Ψaij .

In what follows, for any function ψ ∈ C4(Rm), and any (large) parameters ς > 1and κ > 1, we choose the function � in Lemma 2.1 as follows:

(2.4) � = ςϕ, ϕ = eκψ.

It is easy to check that

(2.5) �i = ςκϕψi, �ij = ςκ2ϕψiψj + ςκϕψij , i, j = 1, 2, . . . ,m.

For n ∈ N, we denote by O(κn) a function of order κn for large κ (which isindependent of ς); by Oκ(ςn) a function of order ςn for fixed κ and for large ς. Thedesired pointwise estimate for the operator “

∑mi,j=1

∂∂xj

(aij ∂

∂xi

)” is stated as follows.

Theorem 2.2. Assume (2.1) holds, and u ∈ C2(Rm). Let

(2.6) θ = e�, v = θu, Ψ = 2

m∑i,j=1

aij�ij .

Then

(2.7)

θ2

∣∣∣∣∣∣m∑

i,j=1

(aijui)j

∣∣∣∣∣∣2

+ 2

m∑i,j=1

{m∑

i′,j′=1

[2aijai

′j′�i′vivj′ − aijai′j′�ivi′vj′

]

+ Ψaijviv − aij[(A + Ψ)�i +

Ψi

2

]v2

}j

≥ 2

m∑i,j=1

cijvivj + Bv2,

where A, B, and cij are given in (2.3). Moreover, for ς and κ large enough, thefollowing estimates hold uniformly in any bounded set of R

m:(2.8)

m∑i,j=1

cijvivj ≥ ςκϕ

⎛⎝ m∑i,j=1

aijψiψj

⎞⎠⎛⎝ m∑i,j=1

aijvivj

⎞⎠+

m∑i,j,i′,j′=1

[2aij

′ai

′jψi′j′

+ aijai′j′ψi′j′ + 2aij

′ai

′jj′ ψi′ − (aijai

′j′)j′ψi′

]vivj

},

B = 2ς3κ4ϕ3

⎛⎝ m∑i,j=1

aijψiψj

⎞⎠2

+ ς3ϕ3O(κ3) + Oκ(ς2).

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1540 KIM DANG PHUNG AND XU ZHANG

Proof. Clearly, (2.7) is a direct consequence of Lemma 2.1. Recalling (2.3) forcij , and noting (2.6) and (2.5), we have

m∑i,j=1

cijvivj

=

m∑i,j,i′,j′=1

[2aij

′ai

′j�i′j′ + aijai′j′�i′j′ + 2aij

′ai

′jj′ �i′ − (aijai

′j′)j′�i′]vivj

= 2ςκ2ϕ

⎛⎝ m∑i,j=1

aijψivj

⎞⎠2

+ ςκ2ϕ

⎛⎝ m∑i,j=1

aijψiψj

⎞⎠⎛⎝ m∑i,j=1

aijvivj

⎞⎠+ ςκϕ

m∑i,j,i′,j′=1

[2aij

′ai

′jψi′j′ + aijai′j′ψi′j′ + 2aij

′ai

′jj′ ψi′ − (aijai

′j′)j′ψi′

]vivj

≥ ςκ2ϕ

⎛⎝ m∑i,j=1

aijψiψj

⎞⎠⎛⎝ m∑i,j=1

aijvivj

⎞⎠+ ςκϕ

m∑i,j,i′,j′=1

[2aij

′ai

′jψi′j′ + aijai′j′ψi′j′ + 2aij

′ai

′jj′ ψi′ − (aijai

′j′)j′ψi′

]vivj ,

which gives the first inequality in (2.8).On the other hand, by (2.5), recalling the definitions of Ψ and A, we see that

Ψ = 2ςκ2ϕ

m∑i,j=1

aijψiψj + ςϕO(κ), A = −ς2κ2ϕ2m∑

i,j=1

aijψiψj + Oκ(ς).

Hence, from the definition of B, we have

B = 2

{− 2ς3κ4ϕ3

⎛⎝ m∑i,j=1

aijψiψj

⎞⎠2

+ ς3ϕ3O(κ3) + Oκ(ς2)

+ ςκm∑

i,j=1

⎡⎣⎛⎝ς2κ2ϕ3m∑

i′,j′=1

ai′j′ψi′ψj′ + Oκ(ς)

⎞⎠ aijψi

⎤⎦j

}+ Oκ(ς2)

= 2

{− 2ς3κ4ϕ3

⎛⎝ m∑i,j=1

aijψiψj

⎞⎠2

+ ς3ϕ3O(κ3) + Oκ(ς2)

+ ςκm∑

i,j=1

⎛⎝3ς2κ3ϕ3m∑

i′,j′=1

ai′j′ψi′ψj′ + ς2ϕ3O(κ2) + Oκ(ς)

⎞⎠ aijψiψj

}

= 2ς3κ4ϕ3

⎛⎝ m∑i,j=1

aijψiψj

⎞⎠2

+ ς3ϕ3O(κ3) + Oκ(ς2),

which yields the second inequality in (2.8).

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1541

3. Interpolation inequality for a fourth order elliptic-like equation withmultiple characteristics. As a crucial preliminary, we derive in this section thefollowing a priori estimate for a fourth order elliptic-like equation with multiple char-acteristics.

Theorem 3.1. Suppose that Ω is connected. Then, for any nonempty opensubset ω of Ω, there exists a constant C0 = C0(Ω, ω, d,M) > 0 such that for anyw = w(x, s) ∈ H2(Ω × (−2, 2)) and f = f(x, s) ∈ L2(Ω × (−2, 2)) with

(3.1)

{−∂2

sw + Δ2w + ∂2sΔw = f in Ω × (−2, 2) ,

w = Δw = 0 on ∂Ω × (−2, 2)

we have

∫ 1

−1

∫Ω

|Δw|2 dxds ≤ C0eC0/ε

[∫ 2

−2

∫ω

(|w|2 + |Δw|2

)dxds +

∫ 2

−2

∫Ω

|f |2 dxds]

+ e−2/ε

∫ 2

−2

∫Ω

(|Δw|2 + |∂sΔw|2

)dxds ∀ ε > 0.

(3.2)

Notice that this interpolation estimate (3.2) or Holder dependence continuous in-equality has already appeared in [13] for second order elliptic operators in the frame-work of null controllability for the heat equation.

Before proving Theorem 3.1, we remark that inequality (3.2) is a kind of quan-titative unique continuation of (3.1) in the following sense: If w ∈ H2(Ω × (−2, 2))solves (3.1) with f = 0 in Ω × (−2, 2), and w = 0 in ω × (−2, 2), then, by Theorem3.1, w = 0 in Ω× (−1, 1). On the other hand, it is easy to verify that any solution wto (3.1) with f = 0 in Ω × (−2, 2) is of the form

w(x, s) =∞∑k=1

(ake

s√

1+λk− 11+λk + bke

−s√

1+λk− 11+λk

)ϕk(x), ak, bk ∈ C,

where {λk}k≥1 are the eigenvalues of −Δ with homogeneous Dirichlet boundary condi-tion and {ϕk}∞k=1 the corresponding eigenvectors (constituting an orthonormal basisof L2(Ω)). Therefore, w(·, s) is analytic with respect to s, which, in turn, impliesw = 0 in Ω × (−2, 2).

Note also that (3.1) is not elliptic in the classical sense. Indeed, the symbol ofits principal operator reads ξ4 + ξ2η2, which vanishes for ξ = 0 and any η ∈ R.As mentioned in the introduction, we use global Carleman estimates to establish(3.2). To do this, a key observation is the possibility of decomposing the operator−∂2

s + Δ2 + ∂2sΔ as follows:

(3.3) −∂2s + Δ2 + ∂2

sΔ = (∂2s + Δ)(−I + Δ) + Δ,

where I is the identity. Consequently, in order to derive the desired inequality (3.2),it is natural to proceed in cascade by applying the global Carleman estimates tothe second order elliptic operators ∂2

s + Δ and Δ. Thanks to Theorem 2.2, this ispossible because only the symmetry of the matrix (aij)1≤i,j≤m is required. Therefore,Theorem 2.2 applies to both the operators ∂2

s + Δ and Δ. We remark that, due tothe necessity of using same weight function for these two different elliptic operators

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1542 KIM DANG PHUNG AND XU ZHANG

of second order, there seems no existing Carleman estimate in the literature for ourpurpose.

Proof of Theorem 3.1. The proof is divided into four steps.Step 1. Choice of the weight function. In order to apply Theorem 2.2 in cascade

to the operators ∂2s + Δ and Δ, it is important to choose a common weight function

θ = θ (x, s) for these two different operators.

It is well known that (see [11] or [21], for example) there is a function ψ ∈ C4(Ω)

such that ψ > 0 in Ω, ψ = 0 on ∂Ω, and

(3.4) 0 <d∑

i=1

∣∣∣∂xi ψ(x)∣∣∣2 ≤ C

∣∣∣∇ψ(x)∣∣∣2 ∀ x ∈ Ω \ω0

where ω0 ⊂ ω is an arbitrary fixed nonempty open subset of Ω such that ω0 ⊂ ω.Therefore,

(3.5) h�=

1

||ψ ||L∞(Ω)

minx∈Ω\ω0

|∇ψ(x)| > 0.

Let us introduce

(3.6) b =

√1 +

1

κln (2 + eκ), b0 =

√b2 − 1

κln

(1 + eκ

),

where κ > ln 2 is the parameter that appeared in Theorem 2.2 and is chosen largeenough. It is easy to see that

1 < b0 < b ≤ 2.

Further, we choose

(3.7) ψ(x, s) =ψ(x)

||ψ ||L∞(Ω)

+ b2 − s2.

By (2.4) and (2.6), this gives the function ϕ (x, s) = eκψ(x,s) and the desired weight

function θ (x, s) = eςeκψ(x,s)

(recall Theorem 2.2 for the parameter ς). It is easy tocheck that

(3.8)

{ϕ (·, s) ≥ 2 + eκ for any s satisfying |s| ≤ 1,

ϕ (·, s) ≤ 1 + eκ for any s satisfying b0 ≤ |s| ≤ b.

Step 2. Reduction of (3.1) to a cascade system. Let

(3.9) z = −w + Δw.

Then, in view of (3.3), system (3.1) can be written equivalently as the following ellipticsystem of second order in cascade:

(3.10)

⎧⎪⎪⎨⎪⎪⎩Δw = z + w in Ω × (−2, 2) ,

∂2sz + Δz = f − z − w in Ω × (−2, 2) ,

w = z = 0 on ∂Ω × (−2, 2) .

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1543

Note, however, that there is no (homogeneous) boundary condition for w (andhence z) at s = ±2. Now, we introduce a cut-off function φ = φ (s) ∈ C∞

0 (−b, b) ⊂C∞

0 (R) such that

(3.11)

{0 ≤ φ(s) ≤ 1, |s| < b,

φ(s) ≡ 1, |s| ≤ b0.

Let

(3.12) w = φw, z = φz.

Then, noticing that φ does not depend on x, it follows by (3.10) that

(3.13)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Δw = z + w in Ω × (−2, 2) ,

∂2s z + Δz = φf + 2∂sφ∂sz + z∂2

sφ− z − w in Ω × (−2, 2) ,

w = z = 0 on ∂Ω × (−2, 2) ,

supp w(x, ·)⋃

supp z(x, ·) ⊂ (−b, b), x ∈ Ω.

Step 3. Carleman estimates. First, we apply Theorem 2.2 with m = d + 1,xd+1 = s, (aij)1≤i,j≤d+1 = (M 0

0 0 ), u replaced by w, and the weight function θ givenabove. In this case, by the definition of cij in (2.3), it is easy to check that

(3.14) cij = 0 whenever one of i and j is equal to d + 1.

Moreover, by (2.8), recalling (3.7) for the definition of ψ and (3.5) for the positiveconstant h, we conclude that there is a constant κ0 > 1 such that for any κ ≥ κ0,one can find a constant ς0 > 1 so that for any ς ≥ ς0, the following estimates holduniformly for (x, s) ∈ Ω \ ω0 × [−b, b]:

(3.15)

d∑i,j=1

cijvivj ≥ ςϕ{κ2|∇ψ|2 + O(κ)

}|∇v|2 ≥ h2

2ςκ2ϕ|∇v|2,

B = 2ς3κ4ϕ3|∇ψ|4 + ς3ϕ3O(κ3) + Oκ(ς2) ≥ h4ς3κ4ϕ3,

where v = θw. Now, integrating inequality (2.7) (with u replaced by w) of Theo-

rem 2.2 in Ω× (−b, b), recalling that φ vanishes near s = ±b, ψ = 0 on ∂Ω, and v = 0on ∂Ω × (−b, b), noting (3.15) and the first equation in (3.13), one arrives at

(3.16)

1

C

{ςκ2

∫ b

−b

∫Ω

ϕ|∇v|2dxds + ς3κ4

∫ b

−b

∫Ω

ϕ3|v|2dxds}

≤∫ b

−b

∫Ω

θ2 |z + w|2 dxds +ςκ

||ψ ||L∞(Ω)

∫ b

−b

∫∂Ω

ϕ∂ψ

∂ν

∣∣∣∣ ∂v

∂νM

∣∣∣∣2 d(∂Ω)ds

+ C

{ςκ2

∫ b

−b

∫ω0

θ2ϕ|∇w|2dxds + ς3κ4

∫ b

−b

∫ω0

θ2ϕ3|w|2dxds},

where ∂ψ∂ν =

∑di=1 ψiν

i, ∂v∂νM

=∑d

i,j=1 αijviν

j , and ν =(ν1, . . . , νd

)= ν(x) is the

unit outward normal vector of Ω at x ∈ ∂Ω. For the boundary term in (3.16), we

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1544 KIM DANG PHUNG AND XU ZHANG

have used that

d+1∑i,j=1

⎧⎨⎩d+1∑

i′,j′=1

[2aijai

′j′�i′vivj′ − aijai′j′�ivi′vj′

]⎫⎬⎭ νj

=ςκϕ

||ψ ||L∞(Ω)

(d∑

i=1

ψiνi

)∣∣∣∣∣∣d∑

i,j=1

aijviνj

∣∣∣∣∣∣2

=ςκϕ

||ψ ||L∞(Ω)

∂ψ

∂ν

∣∣∣∣ ∂v

∂νM

∣∣∣∣2 ,which follows from the fact that on ∂Ω × (−b, b), we have for j = 1, . . . , d,

vj =

(d∑

i=1

viνi

)νj , �j = ςκϕψj =

ςκϕ

||ψ ||L∞(Ω)

ψj =ςκϕ

||ψ ||L∞(Ω)

(d∑

i=1

ψiνi

)νj ,

and νd+1 = 0.Choose a cut-off function g ∈ C∞

0 (ω) with g ≡ 1 in ω0 and 0 ≤ g ≤ 1 in ω.Multiplying the first equation in (3.13) by gθ2ϕw and integrating it in Ω × (−b, b),using integration by parts, one obtains

(3.17)

∫ b

−b

∫ω0

θ2ϕ|∇w|2dxds ≤ C

[ςκ2

∫ b

−b

∫ω

θ2ϕ2|w|2dxds +

∫ b

−b

∫ω

θ2|z|2dxds].

Recalling that v = θw, by (2.5), we get

(3.18)1

Cθ2(|∇w|2 + ς2κ2ϕ2|w|2) ≤ |∇v|2 + ς2κ2ϕ2v2 ≤ Cθ2(|∇w|2 + ς2κ2ϕ2|w|2).

By the choice of ψ, one can check that ∂ψ∂ν < 0 on ∂Ω. Therefore, by (3.16) and

noting (3.17)–(3.18), we end up with

ςκ2

∫ b

−b

∫Ω

θ2ϕ|∇w|2dxds + ς3κ4

∫ b

−b

∫Ω

θ2ϕ3|w|2dxds

≤ C

(∫ b

−b

∫Ω

θ2 |z|2 dxds + ςκ2

∫ b

−b

∫ω

θ2 |z|2 dxds + ς3κ4

∫ b

−b

∫ω

θ2ϕ3|w|2dxds).

(3.19)

Next, we apply Theorem 2.2 with m = d + 1, xd+1 = s, (aij)1≤i,j≤d+1 = (M 00 1 ),

u replaced by z, and the weight function θ as the above. In this case, for any fixedb1 ∈ (0, b), by (2.8), recalling again (3.7) for the definition of ψ and (3.5) for thepositive constant h, we conclude that there is a constant κ1 ≥ κ0 such that for anyκ ≥ κ1, one can find a constant ς1 ≥ ς0 so that, for any ς ≥ ς1, the following estimateshold uniformly for (x, s) ∈ (Ω × (−b, b)) \ (ω0 × (−b1, b1)):

(3.20)

d+1∑i,j=1

cijpipj ≥ ςϕ[κ2(|∇ψ|2 + |∂sψ|2) + O(κ)

](|∇p|2 + |∂sp|2)

≥ h2

2ςκ2ϕ(|∇p|2 + |∂sp|2),

B = 2ς3κ4ϕ3(|∇ψ|2 + |∂sψ|2)2 + ς3ϕ3O(κ3) + Oκ(ς2) ≥ h4ς3κ4ϕ3,

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1545

where p = θz. Using (3.20) and the second equation in (3.13), similar to the proof of(3.16), one obtains(3.21)

1

C

{ςκ2

∫ b

−b

∫Ω

ϕ(|∇p|2 + |∂sp|2)dxds + ς3κ4

∫ b

−b

∫Ω

ϕ3|p|2dxds}

≤∫ b

−b

∫Ω

θ2∣∣φf + 2∂sφ∂sz + z∂2

sφ− z − w∣∣2 dxds

+ςκ

||ψ ||L∞(Ω)

∫ b

−b

∫∂Ω

ϕ∂ψ

∂ν

∣∣∣∣ ∂p

∂νM

∣∣∣∣2 d(∂Ω)ds

+ C

{ςκ2

∫ b1

−b1

∫ω0

θ2ϕ(|∇z|2 + |∂sz|2)dxds + ς3κ4

∫ b1

−b1

∫ω0

θ2ϕ3|z|2dxds}.

By the second equation in (3.13), similar to (3.17), one has(3.22)∫ b1

−b1

∫ω0

θ2ϕ(|∇z|2 + |∂sz|2)dxds

≤ C

[ςκ2

∫ b

−b

∫ω

θ2ϕ2|z|2dxds +

∫ b

−b

∫ω

θ2∣∣φf + 2∂sφ∂sz + z∂2

sφ− w∣∣2 dxds] .

Now, similar to (3.19), from (3.21) and (3.22), it follows that

(3.23)

ςκ2

∫ b

−b

∫Ω

θ2ϕ(|∇z|2 + |∂sz|2)dxds + ς3κ4

∫ b

−b

∫Ω

θ2ϕ3|z|2dxds

≤ C

{ςκ2

∫ b

−b

∫Ω

θ2∣∣φf + 2∂sφ∂sz + z∂2

sφ∣∣2 dxds +

∫ b

−b

∫Ω

θ2 |w|2 dxds

+ ςκ2

∫ b

−b

∫ω

θ2 |w|2 dxds + ς3κ4

∫ b

−b

∫ω

θ2ϕ3|z|2dxds}.

Combining (3.19) and (3.23), we find that for any ς and κ large enough,

(3.24)

∫ b

−b

∫Ω

θ2ϕ(|∇z|2 + |∂sz|2)dxds + ς2κ2

∫ b

−b

∫Ω

θ2ϕ3|z|2dxds

+ ς3κ4

∫ b

−b

∫Ω

θ2ϕ|∇w|2dxds + ς5κ6

∫ b

−b

∫Ω

θ2ϕ3|w|2dxds

≤ C

(∫ b

−b

∫Ω

θ2∣∣φf + 2∂sφ∂sz + z∂2

sφ∣∣2 dxds + ς3κ4

∫ b

−b

∫ω

θ2 |z|2 dxds

+ ς5κ6

∫ b

−b

∫ω

θ2ϕ3|w|2dxds).

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1546 KIM DANG PHUNG AND XU ZHANG

Recall that z = Δw − w. Therefore, (3.24) leads to

(3.25)

ς2κ2

∫ b

−b

∫Ω

θ2ϕ3|Δw|2dxds

≤ C

[∫ b

−b

∫Ω

θ2(|f |2 +∣∣2∂sφ∂sz + z∂2

sφ∣∣2)dxds

+ ς3κ4

∫ b

−b

∫ω

θ2ϕ3|Δw|2dxds + ς5κ6

∫ b

−b

∫ω

θ2ϕ3|w|2dxds].

Step 4. Completion of the proof. Denote c0 = 2 + eκ > 1, and recall (3.6) forb0 ∈ (1, b). Fixing the parameter κ in (3.25), by (3.9), using (3.8) and (3.11), onefinds

(3.26)

ς2e2ςc0

∫ 1

−1

∫Ω

|Δw|2 dxds

≤ CeCς

[∫ b

−b

∫Ω

|f |2dxds +

∫ b

−b

∫ω

(|w|2 + |Δw|2)dxds]

+ Ce2ς(c0−1)

∫(−b,−b0)∪(b0,b)

∫Ω

∣∣2∂sφ∂s(Δw − w) + (Δw − w)∂2sφ∣∣2 dxds.

From (3.26), one concludes that there exists ε0 > 0 such that the desired inequal-ity (3.2) holds for ε ∈ (0, ε0], which, in turn, implies that it holds for any ε > 0. Thiscompletes the proof of Theorem 3.1.

4. Quantitative unique continuation for Kirchhoff plate. This sectionshows the following quantitative unique continuation for solutions of (1.1).

Theorem 4.1. Suppose that Ω is connected. Then, for any nonempty open subsetω of Ω and any β ∈ (0, 1), there exists a time T > 0 such that the solution u of (1.1)satisfies

‖(u0, u1)‖2H2(Ω)×H1(Ω) ≤ e

C

[‖(u0,u1)‖H

‖(u0,u1)‖H2(Ω)×H1(Ω)

]1/β ∫ T

0

∫ω

|u (x, t)|2 dxdt

∀ (u0, u1) ∈ H \ {0}.

(4.1)

Remark 4.1. Estimate (4.1) is equivalent to

(4.2) ‖(u0, u1)‖2H2(Ω)×H1(Ω) ≤

C

ln2β

(1 +

‖(u0,u1)‖2H

‖u‖2L2(ω×(0,T ))

) ‖(u0, u1)‖2H

or, equivalently,(4.3)

‖(u0, u1)‖2H2(Ω)×H1(Ω) ≤ CeC/μ

∫ T

0

∫ω

|u (x, t)|2 dxdt + μ2β ‖(u0, u1)‖2H ∀μ > 0.

This kind of interpolation estimate or logarithmic dependence continuous inequal-ity has already appeared in [14, 20] in the framework of boundary control and stabi-lization for hyperbolic equations.

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1547

In order to prove Theorem 4.1, we need the following known result from [14,p. 473].

Proposition 4.2. For any N ∈ N, the function

(4.4) F (z)�=

1

∫R

eizτe−τ2N

is holomorphic in C and there exist four positive constants A, c0, c1, and c2 such that{|F (z)| ≤ Aec0|Im z|α ,

|Im z| ≤ c2 |Re z| =⇒ |F (z)| ≤ Ae−c1|z|α ,

where α = 2N2N−1 .

Proof of Theorem 4.1. Fix any nonempty open subset ω1 of ω such that ω1 ⊂ ω.We claim that it suffices to show that

(4.5)‖(u0, u1)‖2

H2(Ω)×H1(Ω) ≤ CeC/μ

∫ T

0

∫ω1

[|u (x, t)|2 + |Δu (x, t)|2

]dxdt

+ μ2β ‖(u0, u1)‖2H ∀μ > 0.

To see this, we choose a cut-off function � ∈ C∞0 (ω) such that � = 1 in ω1 and

0 ≤ � ≤ 1 in ω. Then, for any μ > 0,

(4.6)

∫ T

0

∫ω1

[|u (x, t)|2 + |Δu (x, t)|2

]dxdt ≤

∫ T

0

‖�u (·, t)‖2H2(Rd) dt

≤ C

[1

μ2

∫ T

0

‖�u (·, t)‖2L2(Rd) dt + μ

∫ T

0

‖�u (·, t)‖2H3(Rd) dt

]

≤ C

[1

μ2

∫ T

0

∫ω

|u (x, t)|2 dxdt + μ ‖(u0, u1)‖2H

].

Combining (4.5) and (4.6), we arrive at (4.3). By Remark 4.1, this yields the desiredinequality (4.1).

We now prove (4.5) and divide the proof into three steps.Step 1. Reducing the problem to a fourth order elliptic-like equation. Let β ∈

(0, 1), and choose N ∈ N such that 0 < β + 12N < 1. Let γ = 1

α = 1 − 12N > β.

Recall the definition of F (z) in (4.4) and the constant C0 > 0 in Theorem 3.1 (withω replaced by ω1). By Proposition 4.2, for any λ ≥ 1, the holomorphic function

(4.7) Fλ(z)�= λγF (λγz) ≡ 1

∫R

eizτe−( τλγ )

2N

satisfies

(4.8)

{|Im z| ≤ 2 =⇒ |Fλ (z)| ≤ Aλγe2αc0λ,

|Im z| ≤ 2 and 2c2

≤ |Re z| =⇒ |Fλ (z)| ≤ Aλγe−c1λ|Re z|α .

In what follows, we fix a time T satisfying

(4.9) T > 8 max

(2,

2

c2, α

√1 + 2αc0C0

c1

).

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1548 KIM DANG PHUNG AND XU ZHANG

For any Φ = Φ(t) ∈ C∞0 (0, T ) ⊂ C∞

0 (R) and any solution u of (1.1), following [14],we introduce the following Fourier–Bros–Iagolnitzer transformation of u:

(4.10) W�0,λ(x, s) =

∫R

Fλ(�0 + is− �)Φ(�)u(x, �)d�, s, �0 ∈ R.

Clearly, ∂2sFλ(�0 + is− �) = −∂2

�Fλ(�0 + is− �). Hence

(4.11)

∂2s (I − Δ)W�0,λ(x, s)

= −∫

R

∂2�Fλ(�0 + is− �)Φ(�) (I − Δ)u(x, �)d�

= −∫

R

Fλ(�0 + is− �) [Φ′′(�) (I − Δ)u(x, �) + 2Φ′(�)∂t (I − Δ)u(x, �)] d�

−∫

R

Fλ(�0 + is− �)Φ(�)∂2t (I − Δ)u(x, �)d�.

Since u is a solution of (1.1), it follows from (4.11) that W�0,λ satisfies the followingfourth order elliptic-like equation with multiple characteristics:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−∂2sW�0,λ(x, s) + Δ2W�0,λ(x, s) + ∂2

sΔW�0,λ(x, s)

=

∫R

Fλ(�0 + is− �) [Φ′′(�) (I − Δ)u(x, �) + 2Φ′(�)∂t (I − Δ)u(x, �)] d�,

(x, s) ∈ Ω × R,

W�0,λ(x, s) = ΔW�0,λ(x, s) = 0, (x, s) ∈ ∂Ω × R,

W�0,λ(x, 0) = (Fλ ∗ Φu(x, ·)) (�0), x ∈ Ω.

(4.12)

On the other hand, we have

‖ΦΔu (x, ·)‖L2(T2 −1,T2 +1)

≤ ‖ΦΔu(x, ·) − Fλ ∗ ΦΔu(x, ·)‖L2(T2 −1,T2 +1) + ‖Fλ ∗ ΦΔu(x, ·)‖L2(T

2 −1,T2 +1)

≤ ‖ΦΔu(x, ·) − Fλ ∗ ΦΔu(x, ·)‖L2(R) +

(∫ T/2+1

T/2−1

|ΔWt,λ(x, 0)|2 dt)1/2

.

Denote by F (f) the Fourier transform of f . Therefore, using Parseval’s equality

and the fact that F (Fλ) (τ) = e−( τλγ )

2N

, one finds

‖ΦΔu(x, ·) − Fλ ∗ ΦΔu(x, ·)‖L2(R) =1√2π

∥∥∥F(ΦΔu(x, ·) − Fλ ∗ ΦΔu(x, ·))∥∥∥

L2(R)

=1√2π

(∫R

∣∣∣(1 − e−( τλγ )

2N)F(ΦΔu(x, ·)

)(τ)∣∣∣2 dτ)1/2

≤ C

(∫R

∣∣∣ τλγ

F(ΦΔu(x, ·)

)(τ)∣∣∣2 dτ)1/2

≤ C

λγ‖∂t (ΦΔu(x, ·))‖L2(R) .

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1549

Hence,

(4.13)

∫ T/2+1

T/2−1

|Φ(t)Δu(x, t)|2 dt

≤ C

[1

λ2γ‖∂t (ΦΔu(x, ·))‖2

L2(R) +

∫ T/2+1

T/2−1

|ΔWt,λ(x, 0)|2 dt].

For any t ∈(T2 − 1, T

2 + 1)⊂ R, by Cauchy’s integral theorem (in the theory of

complex variable functions) and Holder’s inequality, we deduce that

|ΔWt,λ(x, 0)| ≤ 1

π

∫|�0−t|≤1

∫|s|≤1

|ΔW�0,λ(x, s)| dsd�0

≤ 2

π

(∫|�0−t|≤1

∫|s|≤1

|ΔW�0,λ(x, s)|2 dsd�0

)1/2

.

Hence

∫ T/2+1

T/2−1

|ΔWt,λ(x, 0)|2 dt ≤ 4

π2

∫ T/2+1

T/2−1

dt

∫|�0−t|≤1

∫|s|≤1

|ΔW�0,λ(x, s)|2 dsd�0

≤ 4

π2

∫ T/2+1

T/2−1

dt

∫ T/2+2

T/2−2

d�0

∫|s|≤1

|ΔW�0,λ(x, s)|2 ds

≤ 8

π2

∫ T/2+2

T/2−2

d�0

∫|s|≤1

|ΔW�0,λ(x, s)|2 ds.

(4.14)

Noticing that ∫Ω

‖∂t (ΦΔu(x, ·))‖2L2(R) dx ≤ C ‖(u0, u1)‖2

H ,

combining (4.13) and (4.14), we get

(4.15)

∫ T/2+1

T/2−1

∫Ω

|Φ(t)Δu(x, t)|2 dxdt

≤ C

[1

λ2γ‖(u0, u1)‖2

H +

∫ T/2+2

T/2−2

d�0

∫|s|≤1

∫Ω

|ΔW�0,λ(x, s)|2 dxds].

Step 2. The estimate on∫ T/2+2

T/2−2d�0∫|s|≤1

∫Ω|ΔW�0,λ(x, s)|2 dxds. We now fix

any Φ = Φ(t) ∈ C∞0 (0, T ) satisfying 0 ≤ Φ ≤ 1 in (0, T ) and Φ ≡ 1 on

[T4 ,

3T4

].

Applying Theorem 3.1 (with ω replaced by ω1) to W�0,λ, we obtain that for allε > 0, ∫

|s|≤1

∫Ω

|ΔW�0,λ(x, s)|2 dxds

≤ e−2/ε

∫|s|≤2

∫Ω

[|ΔW�0,λ(x, s)|2 + |∂sΔW�0,λ(x, s)|2

]dxds(4.16)

+ C0eC0/ε

∫|s|≤2

∫ω1

[|W�0,λ(x, s)|2 + |ΔW�0,λ(x, s)|2

]dxds

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1550 KIM DANG PHUNG AND XU ZHANG

+ C0eC0/ε

∫|s|≤2

∫Ω

∣∣∣ ∫R

Fλ(�0 + is− �)[Φ′′(�) (I − Δ)u(x, �)

+ 2Φ′(�)∂t (I − Δ)u(x, �)]d�∣∣∣2dxds.

Using the first conclusion in (4.8), we deduce that

(4.17)

∫ T/2+2

T/2−2

d�0

∫|s|≤2

∫Ω

[|ΔW�0,λ(x, s)|2 + |∂sΔW�0,λ(x, s)|2

]dxds

=

∫ T/2+2

T/2−2

d�0

∫|s|≤2

∫Ω

[∣∣∣∣∫R

Fλ(�0 + is− �)Φ(�)Δu(x, �)d�

∣∣∣∣2

+

∣∣∣∣∂s ∫R

Fλ(�0 + is− �)Φ(�)Δu(x, �)d�

∣∣∣∣2]dxds

≤∫ T/2+2

T/2−2

d�0

∫|s|≤2

∫Ω

⎡⎣∣∣∣∣∣∫ T

0

(Aλγe2αc0λ

)|Δu(x, �)| d�

∣∣∣∣∣2

+

∣∣∣∣∣∫ T

0

(Aλγe2αc0λ

)|Δ∂�u(x, �) + Φ′(�)Δu(x, �)| d�

∣∣∣∣∣2⎤⎦ dxds

≤ Cλ2γe2α+1c0λ ‖(u0, u1)‖2H .

Similarly,

(4.18)

∫ T/2+2

T/2−2

d�0

∫|s|≤2

∫ω1

[|W�0,λ(x, s)|2 + |ΔW�0,λ(x, s)|2

]dxds

≤ Cλ2γe2α+1c0λ

∫ T

0

∫ω1

[|u(x, t)|2 + |Δu(x, t)|2

]dxdt.

Further, by the choice of Φ, it is obvious that

supp(∂2t Φ)⊂ supp (∂tΦ) ⊂ K

�=

[0,

T

4

]⋃[3T

4, T

].

Let K0 =[3T8 , 5T

8

]. Then dist (K,K0) = T

8 . The choice of T in (4.9) implies T > 16

and T > 16/c2. Hence,(T2 − 2, T

2 + 2)⊂ K0, and

|�0 − �| ≥ T

8≥ 2

c2∀ (�0, �) ∈ K0 ×K.

Therefore, using the second conclusion in (4.8), we deduce that∫ T/2+2

T/2−2

d�0

∫|s|≤2

∫Ω

∣∣∣∣∣∫

R

Fλ(�0 + is− �)[Φ′′(�) (I − Δ)u(x, �)(4.19)

+2Φ′(�)∂t (I − Δ)u(x, �)]d�

∣∣∣∣∣2

dxds

≤ C

∫K0

d�0

∫|s|≤2

∫Ω

∣∣∣∣∣∫K

(Aλγe−c1λ|�0−�|α

)(|(I − Δ)u(x, �)|

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1551

+ |∂t (I − Δ)u(x, �)|)d�

∣∣∣∣∣2

dxds

≤ C∣∣∣Aλγe−c1λ(dist(K,K0))

α∣∣∣2∫

Ω

∣∣∣∣∫K

[|(I − Δ)u(x, �)| + |∂t (I − Δ)u(x, �)|] d�∣∣∣∣2dx

≤ Cλ2γe−2c1λ(T8 )

α

‖(u0, u1)‖2H .

Combining (4.17)–(4.19), we arrive at

(4.20)

∫ T/2+2

T/2−2

d�0

∫|s|≤1

∫Ω

|ΔW�0,λ(x, s)|2 dxds

≤ Cλ2γ

{e2α+1c0λe−2/ε ‖(u0, u1)‖2

H + e−2c1λ(T8 )

α

eC0/ε ‖(u0, u1)‖2H

+ e2α+1c0λeC0/ε

∫ T

0

∫ω1

[|u(x, t)|2 + |Δu(x, t)|2

]dxdt

}.

Step 3. Choice of ε and completion of the proof. We deduce from (4.15) and(4.20) that

(4.21)

∫ T/2+1

T/2−1

∫Ω

|Φ(t)Δu(x, t)|2 dxdt

≤ C

{[1

λ2γ+ λ2γe2α+1c0λe−2/ε + λ2γe−2c1λ(T

8 )α

eC0/ε

]‖(u0, u1)‖2

H

+ λ2γe2α+1c0λeC0/ε

∫ T

0

∫ω1

[|u(x, t)|2 + |Δu(x, t)|2

]dxdt

}.

We now choose

(4.22) ε =1

2α+1c0λ.

Recall the choice of T in (4.9), which gives −c1(T8

)α+ 2αc0C0 ≤ −1. Hence

(4.23) e−2c1λ(T8 )

α

eC0/ε = exp

{2

[−c1

(T

8

+ 2αc0C0

}≤ e−2λ.

Combining (4.21)–(4.23), we deduce that, for all λ ≥ 1,

(4.24)

∫ T/2+1

T/2−1

∫Ω

|Δu(x, t)|2 dxdt =

∫ T/2+1

T/2−1

∫Ω

|Φ(t)Δu(x, t)|2 dxdt

≤ C

λ2γ‖(u0, u1)‖2

H + CeCλ

∫ T

0

∫ω1

[|u(x, t)|2 + |Δu(x, t)|2

]dxdt.

Finally, by means of the energy method, it is easy to show that

(4.25) ‖(u0, u1)‖2H2(Ω)×H1(Ω) ≤ C

∫ T/2+1

T/2−1

∫Ω

|Δu(x, t)|2 dxdt.

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1552 KIM DANG PHUNG AND XU ZHANG

Combining (4.24) and (4.25), it is easy to conclude that, for any μ ∈ (0, 1),

(4.26)

‖(u0, u1)‖2H2(Ω)×H1(Ω)

≤ CeC/μ

∫ T

0

∫ω1

[|u(x, t)|2 + |Δu(x, t)|2

]dxdt + Cμ2β ‖(u0, u1)‖2

H .

Note that this inequality is trivially true for any μ ≥ 1 and for some C > 1. Con-sequently, inequality (4.5) holds true for all μ > 0. This completes the proof ofTheorem 4.1.

5. Observability estimates for Kirchhoff plate. The purpose of this sectionis to establish two observability estimates which quantify the unique continuationproperty of (1.1) saying that if its solution u satisfies ∂tΔu = 0 in ω × (0, T ), then(u0, u1) = 0. Due to the finite speed of propagation, the time T > 0 has to be chosenlarge enough.

First, when GCC is assumed, we have the following estimate.Theorem 5.1. Under GCC, for any T ≥ T0, the solution u of (1.1) satisfies

(5.1) E(u, 0) ≤ C

∫ T

0

∫ω

|∂tΔu(x, t)|2 dxdt ∀ (u0, u1) ∈ H.

Proof. Under GCC, it follows from [2, 3] that for any g ∈ L2 (Ω × (0, T0)) andany (χ0, χ1) ∈ H1

0 (Ω) × L2 (Ω), the solution χ of the wave equation

(5.2)

⎧⎪⎪⎨⎪⎪⎩∂2t χ− Δχ = g in Ω × (0, T0) ,

χ = 0 on ∂Ω × (0, T0) ,

χ (·, 0) = χ0, ∂tχ (·, 0) = χ1 in Ω

satisfies the following observability estimate:(5.3)

‖(χ0, χ1)‖2H1

0 (Ω)×L2(Ω) ≤ C

[∫ T0

0

∫ω

|∂tχ (x, t)|2 dxdt +

∫ T0

0

∫Ω

|g (x, t)|2 dxdt].

Note that the solution u of (1.1) solves

⎧⎪⎪⎨⎪⎪⎩∂2t (I − Δ)u− Δ (I − Δ)u = −Δu in Ω × R,

(I − Δ)u = 0 on ∂Ω × R,

(I − Δ)u (·, 0) = (I − Δ)u0, ∂t (I − Δ)u (·, 0) = (I − Δ)u1 in Ω.

(5.4)

Applying estimate (5.3) to system (5.4) (with χ = (I − Δ)u and g = −Δu), weconclude that

(5.5)

E (u, 0) ≤ C

[∫ T0

0

∫ω

|∂t(I − Δ)u (x, t)|2 dxdt +

∫ T0

0

∫Ω

|Δu (x, t)|2 dxdt]

≤ C

[∫ T0

0

∫ω

|∂tΔu (x, t)|2 dxdt + ‖(u0, u1)‖2H2(Ω)×H1(Ω)

].

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1553

However, by (5.5) and using the classical uniqueness-compactness argument (e.g.,[15]), it follows that

(5.6) ‖(u0, u1)‖2H2(Ω)×H1(Ω) ≤ C

∫ T0

0

∫ω

|∂tΔu (x, t)|2 dxdt.

Therefore, combining (5.5) and (5.6), we deduce the desired estimate (5.1). Thiscompletes the proof of Theorem 5.1.

Next, when GCC is not assumed, we have the following weaker estimate.Theorem 5.2. Suppose that Ω is connected. Then, for any nonempty open subset

ω of Ω, any β ∈ (0, 1), and the time T > 0 given in Theorem 4.1, the solution u of(1.1) satisfies

(5.7)E(u, 0) ≤ CeC/μ

∫ T

0

∫ω

|∂tu (x, t)|2 dxdt + μ2β ‖(u0, u1)‖2D(A)

∀ (u0, u1) ∈ D(A) \ {0}, ∀ μ > 0,

and

(5.8)E(u, 0) ≤ CeC/μ

∫ T

0

∫ω

|∂tΔu (x, t)|2 dxdt + μ2β ‖(u0, u1)‖2D(A3)

∀ (u0, u1) ∈ D(A3) \ {0}, ∀ μ > 0.

Proof. By (4.3) in Remark 4.1, one sees that the solution u of system (1.1) satisfies(5.9)∫ T

0

∫Ω

|Δu (x, t)|2 dxdt ≤ CeC/μ

∫ T

0

∫ω

|u (x, t)|2 dxdt + μ2β ‖(u0, u1)‖2H ∀μ > 0,

and

(5.10)

∫ T

0

∫Ω

|u (x, t)|2 dxdt ≤ CeC/μ

∫ T

0

∫ω

|u (x, t)|2 dxdt+μ2β ‖(u0, u1)‖2H ∀μ > 0.

It remains to apply (5.9) (resp., (5.10)) with u replaced by ∂tu (resp., ∂tΔu) to getthe desired estimate (5.7) (resp., (5.8)) by using the following inequality:

E (u, 0) ≤ C

∫ T

0

∫Ω

|∂tΔu (x, t)|2 dxdt,

which, in turn, follows from the usual energy method.

6. Proof of Theorems 1.1 and 1.2. We begin with the proof of Theorem 1.2.Recall that the functions v(j) and U (j), defined, respectively, in (1.4) and (1.5), dependonly on ∂tΔu (x, T − t) · 1|ω . Let

(6.1) w(j)(x, t) =

{v(0)(x, t) − u(x, T − t), j = 0,

v(j)(x, t) − w(j−1)(x, T − t), j > 0,for (x, t) ∈ Q.

Clearly, w(j) = U (j) in ω × (0, T ). Also, it is easy to check that w(j) solves

(6.2)

{∂2tw

(j) + Δ2w(j) − ∂2t Δw(j) + ∂tΔw(j) · 1|ω = 0 in Q,

w(j) = Δw(j) = 0 on Σ.

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1554 KIM DANG PHUNG AND XU ZHANG

By (6.1) and the third equation in system (1.4), noticing the conservative law(1.3), we deduce that

(6.3)

{E(w(j), 0) = E(w(j−1), T ), j > 0,

E(w(0), 0) = E(u, T ) = E(u, 0).

First, applying the standard energy method to system (6.2), it follows that

(6.4) E(w(j), T ) − E(w(j), 0) +

∫ T

0

∫ω

∣∣∣∂tΔw(j) (x, t)∣∣∣2 dxdt = 0 ∀ T > 0.

By (5.8) in Theorem 5.2 and using a well-known perturbation argument (e.g., [16,section 5]), we conclude that the solution w(j) of (6.2) satisfies, for any μ > 0,(6.5)

E(w(j), 0) ≤ CeC/μ

∫ T

0

∫ω

∣∣∣∂tΔw(j)(x, t)∣∣∣2 dxdt + μ2β

∥∥∥(w(j)(·, 0), ∂tw(j)(·, 0)

)∥∥∥2

D(A3).

Combining (6.4)–(6.5) and the first line in (6.3), it follows that, for any μ > 0,(6.6)

E(w(j), 0) ≤ CeC/μ[E(w(j), 0) − E(w(j+1), 0)

]+μ2β

∥∥∥(w(j)(·, 0), ∂tw(j)(·, 0)

)∥∥∥2

D(A3).

In view of the dissipation law for system (6.2), noticing again (6.1) and the thirdequation in system (1.4), one has(6.7)∥∥∥(w(2j+1)(·, T ), ∂tw

(2j+1)(·, T ))∥∥∥

D(A3)≤ C

∥∥∥(w(2j)(·, T ), ∂tw(2j)(·, T )

)∥∥∥D(A3)

.

Therefore, by (6.6)–(6.7) and denoting M�= supj>0

∥∥(w(2j)(·, T ), ∂tw(2j)(·, T )

)∥∥2

D(A3),

we conclude that

(6.8) E(w(j), 0) ≤ CeC/μ[E(w(j), 0) − E(w(j+1), 0)

]+μ2βM ∀ μ > 0.

Now, by (6.8) and similar to Remark 4.1, one deduces that the solution w(j) of (6.2)satisfies

(6.9)E(w(j), 0

)M

≤ C ln−2β

(1 +

M

E(w(j), 0

)− E

(w(j+1), 0

)) .

Let

αn =E(w(n), 0

)M

.

Then,

(6.10) αn+1 =E(w(n+1), 0

)M

=E(w(n), T

)M

≤ αn.

Combining (6.9) and (6.10), we obtain

(6.11) αn+1 ≤ C ln−2β

(1 +

1

αn − αn+1

)∀ n ∈ N.

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TIME REVERSAL FOCUSING FOR KIRCHHOFF PLATE 1555

Similar to [14, 27], starting from (6.11), one deduces that

(6.12) αn+1 ≤ C ln−2β (1 + n) ∀ n ∈ N,

which gives

(6.13)E(w(2N), T

)M

≤ C ln−2β (1 + 2N) .

Now it remains to compute w(2N) (·, T ). By induction, it is easy to verify that, forany N ≥ 1,

w(2N) (·, t) =

N∑k=1

[v(2k) (·, t) − v(2k−1) (·, T − t)

]+ v(0) (·, t) − u (·, T − t) .

Therefore,

(6.14) w(2N) (·, T ) =

N∑k=0

v(2k) (·, T ) − u0, ∂tw(2N) (·, T ) =

N∑k=0

∂tv(2k) (·, T ) + u1.

Finally, by (6.13)–(6.14) and J3 < +∞, one arrives at the desired estimate (1.7).Similarly, the proof of Theorem 1.1 follows from (6.3), (6.14), and Theorem

5.1.

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1556 KIM DANG PHUNG AND XU ZHANG

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