Uniqueness and stability of an inverse kernel problem for type III thermoelasticity

J. Math. Anal. Appl. 402 (2013) 242–254

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Uniqueness and stability of an inverse kernel problem for type III thermoelasticity Bin Wu a,∗ , Jun Yu b , Zewen Wang c a

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

b

Department of Mathematics and Statistics, The University of Vermont, Burlington, VT 05401, United States

c

Department of Mathematics, School of Science, East China Institute of Technology, Nanchang 330013, China

article

abstract

info

Article history: Received 19 November 2012 Available online 24 January 2013 Submitted by Hyeonbae Kang

In this paper, we establish a global Carleman estimate for a thermoelastic system of type III. Based on this estimate, we furthermore study an inverse problem of determining a spatially varying thermal kernel function. We derive Hölder stability for the coefficient inverse problem only by displacement measurements over a given subdomain ω along a sufficiently large time interval and temperature measurements at time t0 over whole domain Ω . The uniqueness for such an inverse problem is yielded as a direct result. © 2013 Elsevier Inc. All rights reserved.

Keywords: Inverse problem Type III thermoelasticity Carleman estimate Integro-differential system Hölder stability

1. Introduction In this paper, we discuss the inverse problem of determining an unknown spatially varying function in the thermal kernel in thermoelasticity of type III. This system is used to describe the elastic and thermal behavior of elastic, heat conductive media in some environment with low temperature. This case allows heat transmission at finite speed, which is much different from the classical thermoelasticity based on Fourier law [12,30]. We set Ω ⊂ R3 a bounded domain with smooth boundary ∂ Ω := Γ and QT = Ω × (0, T ), ωT = ω × (0, T ), ΣT = Γ × (0, T ) with a given T > 0. Then the thermoelastic system of type III is as follows [27]: utt − µ∆u − (µ + ρ)∇ divu + β∇v = 0,

vt − ν ∆v −

(x, t ) ∈ QT ,

(1.1)

t



g (x, t − s)∆v(x, s)ds + β divut = 0,

(x, t ) ∈ QT

(1.2)

0

with initial and boundary conditions:

¯ 0 (x), u(x, 0) = u

¯ 1 (x), ut (x, 0) = u

u(x, t ) = v(x, t ) = 0,

v(x, 0) = v¯ 0 (x),

(x, t ) ∈ ΣT .

x ∈ Ω,

(1.3) (1.4)

Here u = (u1 , u2 , u3 )T , and v are the displacement of the solid elastic material and the temperature respectively, the subscript T represents the transpose of matrices. Here, for simplicity, we assume that the Lamé coefficients µ, ρ , the thermal coefficient ν and the coupling coefficient β are positive constants.

∗

Corresponding author. E-mail address: [email protected] (B. Wu).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2013.01.023

B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

243

We assume that the kernel function g is given by g (x, t ) = k(t )f (x),

(x, t ) ∈ QT ,

(1.5)

where function k is known, which is usually assumed to be strongly positive-definite in the sense that k′ (t ) < 0, k′′ (t ) > 0 for any t > 0 and k decays exponentially to zero as time goes to infinity [13]. In this paper, we take k the same form as the one in [30]: k(t ) = γ e−α t ,

t >0

(1.6)

with two positive constants γ and α . Moreover in order to guarantee the kernel function g > 0 we assume that f (x) ≥ c0 ,

¯ x∈Ω

(1.7)

with some c0 > 0. Furthermore, let x0 be a fixed point such that x0 ∈ R \ Ω , c1 = αν + γ c0 , and then assume that the condition 3

3 2

|∇ log (αν + γ f (x))| |x − x0 | ≤ 1 −

r0

(1.8)

c1

holds for some r0 ∈ (0, c1 ). Remark 1.1. Condition (1.8) is a technical condition imposed by the method we use to prove the stability of our inverse problem below by means of the Carleman estimate. More precisely, (1.8) will be used for the Carleman estimate of a hyperbolic operator ∂tt − (αν + γ f (x))∆; see Isakov and Kim [17,18] or Bellassoued and Yamamoto [2]. Remark 1.2. Under assumption (1.6), we easily see that k′ (t ) + α k(t ) = 0,

t > 0.

(1.9)

Furthermore, we set α = 0 and f ≡ c with a positive constant c. Then system (1.1)–(1.4) becomes the following initial and boundary value problem:

 u − µ∆u − (µ + ρ)∇ divu + β∇v = 0,   tt vtt − ν ∆vt − c γ ∆v + β divutt = 0, u(x, 0) = u¯ 0 (x), ut (x, 0) = u¯ 1 (x), v(x, 0) = v¯ 0 (x), vt (x, 0) = v¯ 1 (x),  u(x, t ) = v(x, t ) = 0,

(x, t ) ∈ QT , (x, t ) ∈ QT , x ∈ Ω, (x, t ) ∈ ΣT

(1.10)

with

v¯ 1 (x) = µ∆v¯ 0 (x) − β divu¯ 1 (x). In [37], Zhang and Zuazua showed the well-posedness of system (1.10). Moreover, in one dimension case, Rivera and Qin [27] proved the global existence, uniqueness and exponential stability for system (1.1)–(1.4). For mathematical studies on the direct problem related to elastic theory, we refer reader to [13,24,25]. In this paper we deal with an inverse problem of recovering a thermal kernel in type III thermoelastic system (1.1)–(1.4). More precisely, we have the following. Inverse problem. Our task is to determine function f (x) in type III thermoelastic system (1.1)–(1.4) by observation data u|ω×(0,T ) ,

and

v(x, t0 ), x ∈ Ω ,

where ω satisfies ∂ω ⊃ ∂ Ω . Remark 1.3. It is noted that we do not need the measurements of v in ω × (0, T ) and u(·, t0 ) over Ω . Remark 1.4. In the proof of our main result, we need to find the Carleman estimate of divergence of Eq. (1.1). The condition ∂ω ⊃ ∂ Ω is used to deal with the lack of divu boundary condition. Inverse problems related to the thermoelastic system have recently been studied by several authors. Without taking memory effect into consideration, i.e. g (x, t ) ≡ 0, Bellassoued and Yamamoto [3] proved a Hölder stability for an inverse heat source problem. Meanwhile, Wu and Liu [32] studied an inverse problem of determining two spatially varying coefficients in the same classical thermoelastic model and gave a Lipschitz stability for this inverse problem under some a priori information. When ν = 0, the system (1.1)–(1.4) becomes a hyperbolic thermoelastic system, i.e. the so-called type II thermoelasticity. In this case, in [33] Hölder stability for an inverse source problem has been obtained by authors. In comparison with these papers, the type of system discussed in this paper is different from the ones in [3,32,33]. Due to this reason, we have to deal with a Carleman estimate for a strongly damped wave equation. Moreover, our inverse problem of determining kernel function is more complicated than inverse source and coefficient problems in [3,32,33]. To our knowledge, there is little work discussing determination of spatially varying kernel function by means of the Carleman estimate. The main work related to such an inverse problem is [29,10]. But in those two papers, the authors discussed a hyperbolic equation with memory term rather than a parabolic equation with memory term. This yields that the method to derive the Carleman estimate used in [29,10] could not work in our case; see Remark 2.1.

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The Carleman estimate is a key tool in the proof of stability and uniqueness results of our inverse problem. The Carleman estimate is a class of weighted energy estimates which is in connection with the differential operator. Bukhgeim and Klibanov [7] proposed a remarkable method based on a Carleman estimate and established the uniqueness for an inverse problem for scalar partial differential equations. Since then it was a useful methodology for studying the stability of coefficient inverse problems for various types of partial differential equations; see [6,19,22,14,15,34,35,16,4,5,28,11,36,9]. The Carleman estimate was also applied to the integro-differential equation to prove the stability of inverse problems [26]. See also Cavaterra, Lorenzi and Yamamoto [8], Romanov and Yamamoto [29], de Buhan and Osses [10]. Moreover see Klibanov [20] and Bellassoued, Cristofol and Soccorsi [1] for Bukhgeim–Klibanov method applications to inverse problems for the Maxwell system. For a recent survey of the Bukhgeim–Klibanov method, we refer the reader to [21]. Throughout this paper, we use t and x = (x1 , x2 , x3 ) to denote the time variable and the spatial variable, respectively. For vector function v = (v1 , v2 , v3 )T , we set

 ∇v =

∂vi ∂ xj



,

∇x,t v = (∇ v, vt )

1≤i,j≤3

and

 |v| =

3 

 12

 ,

|vi |2

|∇ v| =

 1 3    ∂vi 2 2   .  ∂x 

i ,j = 1

i=1

j

In addition we use C to denote generic positive constants which may be different from line to line. To establish our main results, we need the following two assumptions. (A1) t0 and T satisfy min{t0 , T − t0 } ≥ max



1

1

√ ,√ r0



µ

sup |x − x0 |,

(1.11)

x∈Ω

where r0 is the same one as in (1.8). (A2) u ∈ H 5 (QT ), v ∈ H 4 (QT ), and there exists a positive constant M0 such that

∥u∥H 5 (QT ) + ∥v∥H 4 (QT ) ≤ M0 .

(1.12)

Next, for positive constants M1 and ϵ0 , we define the admissible set of unknown coefficient f as W = f ∈ C (Ω ) satisfying (1.7) and (1.8); ∥f ∥C (Ω¯ ) ≤ M1 , |∆v[f ](x, t0 )| ≥ ϵ0 .





(1.13)

We then state our main result of this paper as follows. Theorem 1.1. Let assumptions (A1) and (A2) be held and let f1 , f2 ∈ W . We further assume k has the form (1.6). Then there exist two constants C = C (Ω , T , x0 , M0 , M1 , ϵ0 ) > 0 and κ = κ(Ω , T , x0 , M0 , M1 , ϵ0 ) ∈ (0, 1) such that

 κ ∥f1 − f2 ∥L2 (Ω ) ≤ C ∥u1 − u2 ∥H 5 (ω×(0,T )) + ∥v1 (·, t0 ) − v2 (·, t0 )∥H 2 (Ω ) ,

(1.14)

where (u1 , v1 ) and (u2 , v2 ) are two solutions of system (1.1)–(1.4) corresponding to f1 and f2 respectively. The following uniqueness is a direct result from Theorem 1.1. Corollary 1.1. Under the same assumptions as in Theorem 1.1 and if u1 (x, t ) = u2 (x, t ),

(x, t ) ∈ ω × (0, T ) and v1 (x, t0 ) = v2 (x, t0 ),

x ∈ Ω,

then f1 = f2 in Ω . The remainder of the paper is organized as follows. In the next section, we prove a Carleman estimate for a type III thermoelastic system. In last section, based on the Carleman estimate of Section 2, we give the proof of stability result of our inverse problem. 2. A Carleman estimate This section is devoted to the derivation of a global Carleman estimate for the following strong coupling system Utt − µ∆U − (µ + ρ)∇ divU + β∇ V = F, Vt − ν ∆V −

(x, t ) ∈ QT ,

(2.1)

t



k(t − s)f1 (x)∆V (x, s)ds + β divUt = G,

(x, t ) ∈ QT

(2.2)

0

U(x, t ) = V (x, t ) = 0,

( x , t ) ∈ ΣT .

(2.3)

B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

245

¯ , t0 ∈ (0, T ) and η such that 0 < η < min{µ, r0 }, we introduce a function Φ in the following To do this, for fixed x0 ∈ R3 \ Ω way: Φ (x, t ) = |x − x0 |2 − η(t − t0 )2 + M ,

for all (x, t ) ∈ Ω × (0, T ),

(2.4)

where constant M satisfies M ≥ sup(x,t )∈QT |x − x0 |2 − η(t − t0 )2 . Furthermore, we define the weight function in the Carleman estimate as:



ϕ(x, t ) = eλΦ (x,t ) ,



for all (x, t ) ∈ Ω × (0, T )

with a positive parameter λ. By condition (A1), we have





inf ϕ(x, t0 ) > max sup ϕ(x, 0), sup ϕ(x, T ) .

x∈Ω

x∈Ω

x∈Ω

Therefore, for a fixed sufficiently small positive constant δ , there exists ϵ depending on δ such that [t0 − ϵ, t0 + ϵ] ⊂ [2ϵ, T − 2ϵ] and for all (x, t ) ∈ Ω × [t0 − ϵ, t0 + ϵ]

ϕ(x, t ) ≥ d − δ,

(2.5)

and for all (x, t ) ∈ Ω × [0, 2ϵ] ∪ [T − 2ϵ, T ]

ϕ(x, t ) ≤ d − 2δ,

(2.6)

with

 

d := exp λ inf |x − x0 |2 + M x∈Ω



.

(2.7)

Remark 2.1. It is noted that the form of ϕ here is different from the one in [29,10]. The reason to choose such ϕ is that the principal part in Eq. (2.2) is parabolic type rather than hyperbolic type in [29,10]. In those two papers, the weight function in the Carleman estimate is chosen as

ϕ( ˜ x, t ) = |x − x0 |2 − ηt 2 ,

(x, t ) ∈ QT ,

which leads to t





2  C |k(t − s)f1 (x)∆v(x, s)|ds e2sϕ dxdt ≤ s

0

QT

|∆v|2 e2sϕ dxdt QT

by Lemma 1 in [29] or Lemma 1 in [10]. But this equality is not correct in our case. So the method used in [29,10] cannot be applied to derive our Carleman estimate. Now we state the main result in this section. Theorem 2.1. Let (U, V ) ∈ H 4 (QT ) × H 3 (QT ) satisfy (2.1)–(2.3), f1 ∈ W and let (A1) be held. Then there exist positive constants λ = λ0 (Ω , T , x0 , M1 ) and s = s0 (Ω , T , x0 , M1 , λ) and C = C (Ω , T , x0 , M1 , λ) such that for all λ ≥ λ0 and s ≥ s0 , the following estimate holds





s3 |V |2 + s|∇ V |2 + s4 |Vt |2 + s2 |∇ Vt |2 + s2 |∆V |2 + |Vtt |2 + |∆Vt |2 e2sϕ dxdt



QT



s|∇x,t Ut |2 + s|∇x,t divUt | + s3 |Ut |2 + s3 |divUt |2 e2sϕ dxdt



+



QT



 2

≤C

s

   |F|2 + |Ft |2 + |divF|2 + |divFt |2 + s |G|2 + |Gt |2 e2sϕ dxdt

QT

   + Cs3 ∥U∥2H 4 (ω ) + Cs4 e2s(d−2δ) ∥U∥2H 3 (Ω ) + ∥V ∥2H 3 (Ω ) + C T

s|∆V (x, t0 )|2 e2sϕ dxdt .

(2.8)

QT

2.1. Carleman estimate for a strongly damped wave equation Now we consider a Carleman estimate for the strongly damped wave operator ∂tt − b∆ − α ∆t . In this section, we prove a Carleman estimate for the strongly damped wave equation with two large parameters. A similar estimate was discussed in [31], but with a factor not containing λ. On the contrary, as our inverse problem includes a parabolic equation, we need a large λ to eliminate the coupling term in Eq. (2.2); see (2.45). Moreover, we apply a new method to estimate  sufficiently 2 2sϕ |∇ y | e dxdt and Q |y|2 e2sϕ dxdt, because ∇ y(x, t0 ) and y(x, t0 ) are not zeros in our case. In Section 2.3 we need this Q T

estimate for

 QT

T

|∇ y|2 e2sϕ dxdt to prove Theorem 2.1. In the remainder of this section, we will prove the following Theorem.

Theorem 2.2. Let b ∈ C (Ω ) satisfy b > 0 and 32 |∇(log b)| |x − x0 | < 1 for x ∈ Ω and let α be a positive constant. Then there exist positive constants λ1 = λ1 (Ω , T , x0 ) and s1 = s1 (Ω , T , x0 , λ1 ) and C = C (Ω , T , x0 ) such that for all λ ≥ λ1 and s ≥ s1 ,

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B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

the following estimate holds



(s2 λ3 ϕ 2 |y|2 + λ|∇ y|2 + s3 λ4 ϕ 3 |yt |2 + sλ2 ϕ|∇ yt |2 + sϕ|∆y|2 + s−1 ϕ −1 |ytt |2 + s−1 ϕ −1 |∆yt |2 )e2sϕ dxdt QT



|ytt − b∆y − α ∆yt |2 e2sϕ dxdt + C

≤C



sϕ|∆y(x, t0 )|2 e2sϕ dxdt

(2.9)

QT

QT

for all y ∈ H 1 (0, T ; H 2 (Ω )) ∩ H 2 (0, T ; L2 (Ω )) satisfying



ytt − b∆y − α ∆yt ∈ L2 (QT ), Supp(y(·, t )) ⊂ Ω , y(x, 0) = y(x, T ) = yt (x, 0) = yt (x, T ) = 0,

t ∈ (0, T ), x ∈ Ω.

(2.10)

First, we give two Lemmas which will be used in the proof of Theorem 2.2. The first lemma is an existing Carleman estimate for the hyperbolic operator with a second large parameter, which was proved by Isakov and Kim [17,18]. With the aid of integration by parts, Bellassoued and Yamamoto [2] gave a more simple proof of this estimate.

¯ . There exist positive constants λ2 = λ2 (x0 , Ω , T ) and Lemma 2.1. Let a > 0 satisfy 23 |∇(log a)||x − x0 | < 1 for x ∈ Ω s2 = s2 (x0 , Ω , T , λ2 ) and C = C (x0 , Ω , T ) such that for all λ ≥ λ2 and s ≥ s2 , the following estimate holds 



sλϕ|∇ z |2 + sλϕ|zt |2 + s3 λ3 ϕ 3 |z |2 e2sϕ dxdt ≤ C



QT



|ztt − a∆y|2 e2sϕ dxdt

(2.11)

QT

for all z ∈ H 2 (0, T ; L2 (Ω )) ∩ L2 (0, T ; H 2 (Ω )) satisfying ztt − a∆y ∈ L2 (QT ), Supp(z (·, t )) ⊂ Ω , z (x, 0) = z (x, T ) = zt (x, 0) = zt (x, T ) = 0,



t ∈ (0, T ), x ∈ Ω.

(2.12)

Next lemma is a technical result, whose proof is similar to Lemma 3.1.1 in [23] and Lemma 1 in [8]. Moreover, in [31] we gave a detailed proof for t0 = 0. So, we omit the proof here. Lemma 2.2. There exists C > 0 such that



ϕe

2sϕ

2  t     w(x, τ )dτ  dxdt ≤ Cs−1   t0

QT

e2sϕ |w(x, t )|2 dxdt

(2.13)

QT

for all w ∈ L2 (QT ), where C is not dependent on λ and s. Now we give the proof of Theorem 2.2. Proof of Theorem 2.2. Applying a Carleman estimate by Yuan and Yamamoto (Theorem 2.1 in [36] or Theorem 9.1 in [34]) ¯ = λ( ¯ x0 , Ω , T ), s¯ = s¯(x0 , Ω , T , λ) ¯ and C = C (x0 , Ω , T ) such that to yt , we obtain that there exist λ



s3 λ4 ϕ 3 |yt |2 + sλ2 ϕ|∇ yt |2 + s−1 ϕ −1 |ytt |2 + s−1 ϕ −1 |∆yt |2 e2sϕ dxdt ≤ C





QT



|ytt − α ∆yt |2 e2sϕ dxdt

(2.14)

QT

¯ and s ≥ s¯. for all λ ≥ λ Obviously, we have 

|ytt − α ∆yt |2 e2sϕ dxdt ≤ QT



|ytt − b∆y − α ∆yt |2 e2sϕ dxdt + C QT



|∆y|2 e2sϕ dxdt .

(2.15)

QT

We next set L [y] := ytt − b∆y − α ∆yt

and c2 :=

b

α

,

c3 :=

1

α

.

Then we see that

∆yt + c2 ∆y = −c3 (L [y] − ytt ) ,

t ∈ (0, T ),

(2.16)

from which, we obtain

∆y(x, t ) = e−c2 (t −t0 )



t

ec2 (τ −t0 ) c3 (ytt − L [y]) (x, τ )dτ + e−c2 (t −t0 ) ∆y(x, t0 )

t0

−c2 (t −t0 )

= c3 yt (x, t ) − c3 e



yt (x, t0 ) +



t

e t0

c2 (τ −t0 )

(c2 yt + L [y]) (x, τ )dτ



+ e−c2 (t −t0 ) ∆y(x, t0 ).

(2.17)

B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

247

So that, by Lemma 2.2, we have



sϕ|∆y|2 e2sϕ dxdt ≤ C



sϕ |yt (x, t0 )|2 + |∆y(x, t0 )|2 e2sϕ dxdt





QT

QT

2  t   c1 (τ −t0 )  e sϕ e  sϕ|yt | e dxdt + C +C (c1 yt + L [y]) (x, τ ) dxdt t0 QT Q   T  c (t −t )   2  2sϕ 2 2 e 1 0 c1 yt + L [y]2 e2sϕ dxdt sϕ |yt | + |yt (x, t0 )| + |∆y(x, t0 )| e dxdt + C ≤C 



2 2sϕ

2sϕ

QT

QT





≤C

 sϕ|yt |2 + sϕ|yt (x, t0 )|2 + sϕ|∆y(x, t0 )|2 + |L [y]|2 e2sϕ dxdt .

(2.18)

QT

Adding up (2.14) and (2.18), and then substituting (2.15) and taking s large enough to absorb the term ∆y on the right-hand side in (2.15), we derive





s3 λ4 ϕ 3 |yt |2 + sλ2 ϕ|∇ yt |2 + sϕ|∆y|2 + s−1 ϕ −1 |ytt |2 + s−1 ϕ −1 |∆yt |2 e2sϕ dxdt



QT



2 2sϕ

|L [y]| e

≤C

 dxdt + C

QT

sϕ |yt (x, t0 )|2 + |∆y(x, t0 )|3 e2sϕ dxdt .





(2.19)

QT

Next we estimate Q ϕ|yt (x, t0 )|2 e2sϕ dxdt. Choose a function σ1 ∈ C 1 [0, T ] such that σ1 (0) = 0 and σ1 (t0 ) = 1. We T recall that ϕ(x, t0 ) is the maximum of ϕ(x, t ) on [0, T ]. Then, together with Young’s inequality, we have





sϕ|yt (x, t0 )|2 e2sϕ dxdt QT

 ≤ Cs Ω

t0



d



σ1 (t )ϕ(x, t )|yt (x, t )|2 e2sϕ(x,t ) dxdt   ′  |σ1 | + sλϕ|σ1 ∥Φt | + λ|σ1 ∥Φt | ϕ|yt |2 e2sϕ dxdt + Cs ϕ|σ1 ||yt ||ytt |e2sϕ dxdt

ϕ(x, t0 )|yt (x, t0 )|2 e2sϕ(x,t0 ) dx = s

dt

0

 ≤ Cs

Ω

QT

≤ C (ε)

QT





s2 λϕ + s3 ϕ

 3

|yt |2 e2sϕ dxdt + ε

QT



s−1 ϕ −1 |ytt |2 e2sϕ dxdt

(2.20)

QT

for any sufficiently small ε > 0. Thus, taking λ ≥ C (ε) sufficiently large, and substituting (2.20) into (2.19) leads to





s3 λ4 ϕ 3 |yt |2 + sλ2 ϕ|∇ yt |2 + sϕ|∆y|2 + s−1 ϕ −1 |ytt |2 + s−1 ϕ −1 |∆yt |2 e2sϕ dxdt



QT



|L [y]|2 e2sϕ dxdt + C

≤C



QT

sϕ|∆y(x, t0 )|2 e2sϕ dxdt .

(2.21)

QT

We easily see that ytt − b∆y = L [y] + α ∆yt .

(2.22)

Set y˜ = ϕ −1/2 y. Then a direct calculation leads to y˜ tt − b∆y˜ = ϕ −1/2 (L [y] + α ∆yt ) + q,

(2.23)

where

 q=

1 4

   1   λ2 b|∇ Φ |2 − |Φt |2 + λ (b∆Φ − Φtt ) y˜ + λ b∇ Φ · ∇ y˜ − Φt y˜ t . 2

Applying the Carleman estimate (2.11) for a hyperbolic operator, we obtain





sλϕ|˜yt |2 + sλϕ|∇ y˜ |2 + s3 λ3 ϕ 3 |˜y|2 e2sϕ dxdt ≤ C



QT

QT

for sufficiently large s and λ, where we have used that

|q| ≤ C (λ2 |˜y|2 + λ|∇ y˜ |2 + λ|˜yt |2 ). Recall that

|∇ y˜ |2 ≥ ϕ −1 |∇ y|2 − C λϕ −1 |y|2 ,



|˜y|2 ≥ ϕ −1 |y|2 .

  ϕ −1 |L [y]|2 + |∆yt |2 e2sϕ dxdt

(2.24)

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B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

So that from (2.24) we deduce that







 λ|∇ y|2 + s2 λ3 ϕ 2 |y|2 e2sϕ dxdt ≤ C

QT

s−1 ϕ −1 |L [y]|2 + |∆yt |2 e2sϕ dxdt .





(2.25)

QT

Finally, multiplying (2.21) by C + 1, then adding up (2.21) and (2.25), we can obtain the desired estimate (2.9). This completes the proof.  2.2. A preliminary estimate on the measurement domain In order to eliminate the measurements V on ωT′ , we need a preliminary estimate in which the measurements of V on ωT′ are controlled by some suitable norm of U on a larger domain ωT , where ω′ ⊂ ω satisfies ∂ω′ ⊃ ∂ Ω . The proof follows the ideas used in [3]. First we denote the following quantity by Ms,ϕ [V ]:   Ms,ϕ [V ] = s3 |V |2 + s|∇ V |2 + s4 |Vt |2 + s2 |∇ Vt |2 + s2 |∆V |2 + |Vtt |2 + |∆Vt |2 e2sϕ . We next recall a technique lemma proved by Bellassoued and Yamamoto [3], where this result is obtained from a direct computation based on the divergence theorem. Lemma 2.3. There exist positive constants λ3 = λ3 (Ω , T , x0 ) and s3 = s3 (Ω , T , x0 , λ3 ) and C = C (Ω , T , x0 , λ) such that for all λ ≥ λ3 and s ≥ s3 , the following estimate holds

 ωT



s4 λ4 ϕ 4 |w|2 e2sϕ dxdt ≤ C

s2 λ2 ϕ 2 |∇w|2 e2sϕ dxdt

ωT

(2.26)

for all w ∈ H 1 (ωT ) such that w(x, t ) ≡ 0 on ∂ω × (0, T ). Now we are in a position to prove the following. Lemma 2.4. Let (U, V ) satisfy (2.1)–(2.3). Then exist positive constants λ4 = λ4 (x0 , Ω , T ) and s4 = s4 (x0 , Ω , T , λ4 ) and C = C (Ω , T , x0 , λ) > 0 such that for all λ ≥ λ4 and s ≥ s4 , the following estimate holds

 ωT′

Ms,ϕ [V ]dxdt ≤ C





s|V |2 + s2 |Vt |2 + |G|2 + |Gt |2 e2sϕ dxdt



QT



s2 |F|2 + |Ft |2 + |divF|2 + |divFt |2 e2sϕ dxdt + Cs2 eCs ∥U∥2H 4 (ω ) . T



+C QT



(2.27)

Proof. Let χ1 ∈ C ∞ (R3 ) such that 0 ≤ χ1 (x) ≤ 1,

x ∈ Ω,

χ1 (x) ≡ 1,

x ∈ ω′

χ1 (x) ≡ 0,

and

x ∈ Ω \ ω.

(2.28)

Together with (2.3), we then have χ1 (x)V (x, t ) ≡ 0 on ∂ω × (0, T ). Furthermore, recalling that the regular weight function ϕ has positive upper and lower bounds and using Lemma 2.3, we get    s3 |V |2 e2sϕ dxdt ≤ s3 |χ1 V |2 e2sϕ dxdt ≤ C (λ) s|∇(χ1 V )|2 e2sϕ dxdt ωT′

ωT

ωT

≤ C (λ)



s|∇ V |2 e2sϕ dxdt + C (λ)

ωT



s|V |2 e2sϕ dxdt .

(2.29)

QT

Similarly,

 ωT′

s4 |Vt |2 e2sϕ dxdt ≤ C (λ)



s2 |∇ Vt |2 e2sϕ dxdt + C (λ)

ωT



s2 |Vt |2 e2sϕ dxdt .

(2.30)

QT

So that, we obtain

 ωT′

Ms,ϕ [V ]dxdt ≤ C (λ)



s|∇ V |2 + s2 |∇ Vt |2 + s2 |∆V |2 + |∆Vt |2 + |Vtt |2 e2sϕ dxdt

 ωT

+ C (λ)





s|V |2 + s2 |Vt |2 e2sϕ dxdt .





(2.31)

QT

By Eq. (2.1), we have



 ωT′

s|∇ V |2 + s2 |∇ Vt |2 e2sϕ dxdt ≤ C



 QT

s|F|2 + s2 |Ft |2 e2sϕ dxdt + Cs2 eC (λ)s ∥U∥2H 3 (ω ) T





(2.32)

B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

249

and



s2 |∆V |2 + |∆Vt |2 e2sϕ dxdt ≤ C

 ωT′





s|divF|2 + |divFt |2 e2sϕ dxdt + Cs2 eC (λ)s ∥U∥2H 4 (ω ) .





(2.33)

T

QT

Moreover, from Eqs. (1.6) and (2.2), it follows that Vtt − ν ∆Vt − (αν + γ f1 (x))∆V + α Vt + αβ divUt + β divUtt = α G + Gt ,

(2.34)

which implies



2 2sϕ

ωT′

|Vtt | e



(|∆Vt | + |∆V | )e 2

dxdt ≤ C ωT′

2

2sϕ

 dxdt + C QT

(|G|2 + |Gt |2 + |Vt |2 )e2sϕ dxdt + CeC (λ)s ∥U∥2H 3 (ω ) . (2.35) T

By (2.31)–(2.35), we can get (2.27). This completes the proof of Lemma 2.4.



2.3. Proof of Theorem 2.1 Under condition (1.6), from (2.2) and (2.3), we have



Vtt − ν ∆Vt − (αν + γ f1 (x))∆V + α Vt + αβ divUt + β divUtt = α G + Gt , V (x, t ) = 0,

(x, t ) ∈ QT , (x, t ) ∈ ΣT .

In order to apply Theorem 2.2, we introduce two cut-off functions χ2 ∈ C 2 (R3 ) such that 0 ≤ χ2 (x) ≤ 1,

x ∈ Ω,

χ2 (x) ≡ 1,

x ∈ Ω \ ω′

and

Supp(χ2 ) ⊂ Ω

(2.36)

and σ2 ∈ C 2 (R) such that 0 ≤ σ2 (t ) ≤ 1,

t ∈ (0, T ),

σ2 (t ) ≡ 1,

t ∈ (2ϵ, T − 2ϵ),

and

Supp(σ2 ) ⊂ (ϵ, T − ϵ),

(2.37)

where ϵ > 0 is the same as that in (2.5) and (2.6). Set Vˆ (x, t ) = σ2 (t )χ2 (x)V (x, t ). Then we have

 Vˆ tt − ν ∆Vˆ t − (αν + γ f (x))∆Vˆ = H1 , Vˆ (x, 0) = Vˆ (x, T ) = Vˆ t (x, 0) = Vˆ t (x, T ) = 0,  Supp(Vˆ (·, t )) ⊂ Ω ,

(x, t ) ∈ QT , x ∈ Ω, t ∈ (0, T ),

(2.38)

where the function H1 is given by H1 = σ2 χ2 (α G + Gt − αβ divUt − β divUtt − α Vt ) − σ2 ν(∆χ2 Vt + 2∇χ2 · ∇ Vt )

− σ2 (αν + γ f1 (x))(∆χ2 V + 2∇χ2 · ∇ V ) − νσ2 (2∇χ2 · ∇ Vt + ∆χ2 Vt ) + 2σ2′ χ2 Vt + σ2′′ χ2 V − νσ2′ (χ2 ∆V + 2∇χ2 · ∇ V + ∆χ2 V ) such that

|H1 | ≤ C (|G| + |Gt | + |divUt | + |divUtt | + |∆V | + |∇ Vt | + |Vt | + |∇ V | + |V |) . Noting that χ2 ≡ 1 in Ω \ ω′ and σ2 ≡ 1 in (2ϵ, T − 2ϵ), and applying Theorem 2.2 to Vˆ , we obtain



T −2ϵ



2ϵ

Ω \ω′

(s2 λ3 ϕ 2 |V |2 + λ|∇ V |2 + s3 λ4 ϕ 3 |Vt |2 + sλ2 ϕ|∇ Vt |2

+ sϕ|∆V |2 + s−1 ϕ −1 |Vtt |2 + s−1 ϕ −1 |∆Vt |2 )e2sϕ dxdt  ≤C (|G|2 + |Gt |2 + |divUt |2 + |divUtt |2 + |∆V |2 + |∇ Vt |2 + |Vt |2 QT 2sϕ

+ |∇ V | + |V | )e 2

2



sϕ|∆V (x, t0 )|2 e2sϕ dxdt

dxdt + C

(2.39)

QT

for all λ ≥ λ1 and s ≥ s1 , which yields



T −2 ϵ 2ϵ

 Ω

(s3 λ3 ϕ 2 |V |2 + sλ|∇ V |2 + s4 λ4 ϕ 3 |Vt |2 + s2 λ2 ϕ|∇ Vt |2

+ s2 ϕ|∆V |2 + ϕ −1 |Vtt |2 + ϕ −1 |∆Vt |2 )e2sϕ dxdt  ≤C s(|G|2 + |Gt |2 + |divUt |2 + |divUtt |2 + |∆V |2 + |Vt |2 + |∇ Vt |2 QT   2 2 2sϕ C (λ) + |∇ V | + |V | )e dxdt + C (λ)e Ms,ϕ [V ](x, t )dxdt + C sϕ|∆V (x, t0 )|2 e2sϕ dxdt . ωT′

QT

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B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

Then, together with (2.6), and choosing λ sufficiently large, we have



(s3 λ3 ϕ 2 |V |2 + sλ|∇ V |2 + s4 λ4 ϕ 3 |Vt |2 + s2 λ2 ϕ|∇ Vt |2 + s2 ϕ|∆V |2 + ϕ −1 |Vtt |2 + ϕ −1 |∆Vt |2 )e2sϕ dxdt    ≤C s |G|2 + |Gt |2 + |divUt |2 + |divUtt |2 e2sϕ dxdt QT  + C (λ)eC (λ) Ms,ϕ [V ](x, t )dxdt + C (λ)s4 e2s(d−2δ) ∥V ∥2H 3 (Q )

QT

T

ωT′



sϕ|∆V (x, t0 )|2 e2sϕ dxdt .

+C

(2.40)

QT

On the other hand, we set

ˆ (x, t ) = σ2 (t )χ2 (x)ϕ −1/2 (x, t )divUt (x, t ). Π

ˆ (x, t ) = σ2 (t )χ2 (x)Ut (x, t ), U Then by a simple calculation we have

 Uˆ tt − µ∆Uˆ − (µ + ρ)∇ divUˆ = σ2 χ2 (Ft − β∇ Vt ) + H2 , ˆ (x, 0) = Uˆ (x, T ) = Uˆ t (x, 0) = Uˆ t (x, T ) = 0, U  ˆ (·, t )) ⊂ Ω , Supp(U

(x, t ) ∈ QT , x ∈ Ω, t ∈ (0, T )

(2.41)

 ˆ tt − (2µ + ρ)∆Π ˆ = σ2 χ2 ϕ −1/2 (divFt − β ∆Vt ) + H3 , Π ˆ (x, 0) = Π ˆ ( x, T ) = Π ˆ t (x, 0) = Π ˆ t (x, T ) = 0, Π  ˆ (·, t )) ⊂ Ω , Supp(Π

(x, t ) ∈ QT , x ∈ Ω, t ∈ (0, T )

(2.42)

and

with H2 = χ2 2σ2′ Utt + σ2′′ Ut − µσ2 (2∇χ2 · ∇ Ut + ∆χ2 Ut ) − (µ + ρ)σ2 ∇(∇χ2 · Ut ),





H3 = χ2 2(σ2 ϕ −1/2 )′t divUtt + (σ2 ϕ −1/2 )′′t divUt − (2µ + ρ)σ2 2∇(χ2 ϕ −1/2 ) · ∇ divUt + ∆(χ2 ϕ −1/2 )divUt .









So that, by applying Lemma 2.1 to (2.41) and (2.42), together with (2.36) and (2.37), we have T −2 ϵ





2ϵ



Ω \ω′





 



 

2 2 sλϕ |∇x,t Ut |2 + ∇x,t ϕ −1/2 divUt  + s3 λ3 ϕ 3 |Ut |2 + ϕ −1/2 divUt 







e2sϕ dxdt

 2  |Ft | + ϕ −1 |divFt |2 + |∇ Vt |2 + ϕ −1 |∆Vt |2 + |H2 |2 + |H3 |2 e2sϕ dxdt

≤C QT

for all λ ≥ λ2 and s ≥ s2 . Furthermore, from

   ∇x,t ϕ −1/2 divUt  ≥ ϕ −1/2 |∇x,t divUt | − C λϕ −1/2 |divUt |, we deduce that T −2 ϵ





2ϵ

 Ω \ω′





≤C



sλϕ|∇x,t Ut |2 + sλ ∇x,t divUt  + s3 λ3 ϕ 3 |Ut |2 + s3 λ3 ϕ 2 |divUt |2 e2sϕ dxdt



2

 |Ft |2 + ϕ −1 |divFt |2 + |∇ Vt |2 + ϕ −1 |∆Vt |2 + |H2 |2 + |H3 |2 e2sϕ dxdt ,

QT

if we choose s such that s3 λ3 ϕ 2 > 2C λϕ −1/2 . Next similar to arguments from (2.39) to (2.40), we can get



sλϕ|∇x,t Ut |2 + sλ|∇x,t divUt |2 + s3 λ3 ϕ 3 |Ut |2 + s3 λ3 ϕ 2 |divUt |2 e2sϕ dxdt





QT





≤C

 |Ft |2 + ϕ −1 |divFt |2 + |∇ Vt |2 + ϕ −1 |∆Vt |2 + |H2 |2 + |H3 |2 e2sϕ dxdt

QT

+ C (λ)s3 eC (λ)s ∥U∥2H 3 (ω′ ) + C (λ)s3 e2s(d−2δ) ∥U∥2H 3 (Q ) . T

T

Then noting that

  |H2 | + |H3 | ≤ C (λ) |Ut | + |∇x,t Ut | + |divUt | + |∇x,t divUt | ,

(2.43)

B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

251

and then choosing s sufficiently large to absorb the terms in H2 and H3 , we obtain



sλϕ|∇x,t Ut |2 + sλ|∇x,t divUt |2 + s3 λ3 ϕ 3 |Ut |2 + s3 λ3 ϕ 2 |divUt |2 e2sϕ dxdt



 QT



  |F +t |2 + ϕ −1 |divFt |2 + |∇ Vt |2 + ϕ −1 |∆Vt |2 e2sϕ dxdt

≤C QT

+ C (λ)s3 eC (λ)s ∥U∥2H 3 (ω′ ) + C (λ)s3 e2s(d−2δ) ∥U∥2H 3 (Q ) .

(2.44)

T

T

Multiplying (2.40) by C + 1 and adding up (2.44), we have





sλϕ|∇x,t Ut |2 + sλ|∇x,t divUt |2 + s3 λ3 ϕ 3 |Ut |2 + s3 λ3 ϕ 2 |divUt |2 e2sϕ dxdt



QT



s3 λ3 ϕ 2 |V |2 + sλ|∇ V |2 + s4 λ4 ϕ 3 |Vt |2 + s2 λ2 ϕ|∇ Vt |2 + s2 ϕ|∆V |2



+ QT

 + ϕ −1 |Vtt |2 + (C + 1)ϕ −1 |∆Vt |2 e2sϕ dxdt    ≤ C (C + 1) s |G|2 + |Gt |2 + |divUt |2 + |divUtt |2 e2sϕ dxdt QT  C (λ) + (C + 1)C (λ)e Ms,ϕ [V ](x, t )dxdt + (C + 1)C (λ)s4 e2s(d−2δ) ∥V ∥2H 3 (Q ωT′





+C

T)

 |Ft |2 + ϕ −1 |divFt |2 + |∇ Vt |2 + ϕ −1 |∆Vt |2 e2sϕ dxdt

QT

+ C (λ)s3 eC (λ)s ∥U∥2H 3 (ω′ ) + C (λ)s3 e2s(d−2δ) ∥U∥2H 3 (Q ) T T  + C (C + 1) sϕ|∆V (x, t0 )|2 e2sϕ dxdt .

(2.45)

QT

Obviously, by taking λ such that λ ≥ C (C + 1), we could absorb the term of divUtt and the terms of divUt , ∇ Vt , ∆Vt on the right-hand side in (2.45). Finally, inserting (2.27) into (2.45), we then obtain the desired estimate (2.8). This completes the proof of Theorem 2.1.  3. Proof of Theorem 1.1 We are now in a position to prove the stability result, Theorem 1.1, for our inverse problem. Henceforth, we fix λ sufficiently large to satisfy the condition in Theorem 2.1, and use C to denote a generic positive constant depending on x0 , Ω , T , ϵ0 and λ, but not depending on s. For simplicity, we set

˜ (x, t ) = u1 (x, t ) − u2 (x, t ), U

V˜ (x, t ) = v1 (x, t ) − v2 (x, t )

and h(x) = f1 (x) − f2 (x), where (u1 , v1 ) and (u2 , v2 ) are two solutions of system (1.1)–(1.4) corresponding to f1 and f2 respectively. Then from (1.1) to (1.4) we easily see that

 ˜ tt − µ∆U˜ − (µ + ρ)∇ divU˜ + β∇ V˜ = 0, U    t   ˜ ˜ t = G˜ , Vt − ν ∆V˜ − k(t − s)f1 (x)∆V˜ (x, s)ds + β divU 0   ˜ (x, 0) = U˜ t (x, 0) = V˜ (x, 0) = 0,  U  ˜ U(x, t ) = V˜ (x, t ) = 0,

(x, t ) ∈ QT , (x, t ) ∈ QT ,

(3.1)

x ∈ Ω, (x, t ) ∈ ΣT ,

˜ is given by where the function G ˜ (x, t ) = G

t



k(t − s)h(x)∆v2 (x, s)ds.

(3.2)

0

Next we give a Lemma which will be used in the proof of Theorem 1.1, which is a modified form of Lemma 3.4 in [3] or Lemma 3.2 in [1].

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B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

Lemma 3.1. Let



2



 

2 

 

2

2



2



2



˜ ](x, t ) = V˜ tt (x, t ) + ∆V˜ t (x, t ) + V˜ t (x, t ) + divU˜ tt (x, t ) + divU˜ t (x, t ) , N1 [V˜ , U 2 

2 

 

2 

 

2 

 

˜ ](x, t ) = V˜ ttt (x, t ) + ∆V˜ tt (x, t ) + V˜ tt (x, t ) + divU˜ ttt (x, t ) + divU˜ tt (x, t ) . N2 [V˜ , U Then we have



˜ ](x, t0 )e2sϕ(x,t0 ) dx ≤ Cs N1 [V˜ , U

Ω



˜ ]e2sϕ dxdt + C ε s−1 N1 [V˜ , U



˜ ]e2sϕ dxdt N2 [V˜ , U

(3.3)

QT

QT

for any ε > 0, where C is independent of s. Proof. Noting that σ2 (t0 ) = 1 and σ2 (0) = 0 by (2.37), we have



    ∂  2  2sϕ(x,t ) ˜ ˜ dx dt σ2 (t )N1 [V , U](x, t ) e ∂t Ω 0 Ω     t0     ˜ ] + σ22  ∂ N1 [V˜ , U˜ ] + 2sσ22 N1 [V˜ , U˜ ]|ϕt | e2sϕ dxdt ≤ 2|σ2 ||σ2′ |N1 [V˜ , U   ∂t 0 Ω     ∂   N1 [V˜ , U˜ ] e2sϕ dxdt . ˜ ]e2sϕ dxdt + C N1 [V˜ , U ≤ Cs  ∂t  QT QT ˜ ](x, t0 )e2sϕ(x,t0 ) dx = N1 [V˜ , U

t0



(3.4)

On the other hand, by Young’s inequality, we see that

  ∂   N1 [V˜ , U˜ ] ≤ C (ε)sN1 [V˜ , U˜ ] + C ε s−1 N2 [V˜ , U˜ ].  ∂t 

(3.5)

Inserting (3.5) into (3.4) yields (3.3). Then the proof is completed.



Now we give the proof of Theorem 1.1.

˜ t (x, t ), V (x, t ) = V˜ t (x, t ). Then, we obtain Proof of Theorem 1.1. Let U(x, t ) = U  U − µ∆U − µ + λ∇ divU + β∇ V = 0,   t  tt Vt − ν ∆V − k(t − s)f1 (x)∆V (x, s)ds + β divUt = G,  0  U(x, t ) = V (x, t ) = 0,

(x, t ) ∈ QT , (x, t ) ∈ QT ,

(3.6)

(x, t ) ∈ ΣT ,

with G(x, t ) =

t



k(t − s)h(x)∆(v2 )t (x, t )ds + k(t )h(x)∆v2 (x, 0).

(3.7)

0

Here in the second equation we have used the following equality

∂ ∂t

t



k(t − s)f1 (x)∆V˜ (x, s)ds =

t



0

k(s)f1 (x)∆V˜ t (x, t − s)ds + k(t )f1 (x)∆V˜ (x, 0) 0 t



k(t − s)f1 (x)∆V˜ t (x, s)ds.

= 0

Applying Theorem 2.1 to (3.6), together with (A2), we then obtain







s3 |V˜ t |2 + s4 |V˜ tt |2 + s2 |∆V˜ t |2 + |V˜ ttt |2 + |∆V˜ tt |2 e2sϕ dxdt +

QT







˜ ttt |2 + s3 |divU˜ tt |2 e2sϕ dxdt s|divU

QT

 ≤C QT

  ˜ t ∥2 4 s |G|2 + |Gt |2 e2sϕ dxdt + Cs3 eCs ∥U + Cs4 e2s(d−2δ) M0 + C H (ω ) T



s|∆V˜ t (x, t0 )|2 e2sϕ dxdt

(3.8)

QT

for all large s. Similarly, applying Theorem 2.1 to (3.1), we obtain



˜ t |2 e2sϕ dxdt ≤ C s |divU 3

QT





QT



˜ |2 + |G˜ t |2 e2sϕ dxdt + Cs3 eCs ∥U˜ ∥2 4 s |G H (ωT )

+ Cs4 e2s(d−2δ) M0 + C

 QT

s|∆V˜ (x, t0 )|2 e2sϕ dxdt .

(3.9)

B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254

253

By (3.2) and (3.7), we easily see that

|G˜ (x, t )| + |G˜ t (x, t )| + |G(x, t )| + |Gt (x, t )| ≤ C |h(x)|,

for all t ∈ (0, T ).

(3.10)

From (3.8) to (3.10), we derive that







˜ ] + N2 [V˜ , U˜ ] + s3 N3 [V˜ , U˜ ] + N4 [V˜ , U˜ ] e2sϕ dxdt s2 N1 [V˜ , U

QT

 ≤C QT

˜ ∥2 5 + Cs4 e2s(d−2δ) M0 s|h|2 e2sϕ dxdt + Cs3 eCs ∥U H (ωT )







s |∆V˜ t (x, t0 )|2 + |∆V˜ (x, t0 )|2 e2sϕ dxdt ,

+C

(3.11)

QT

where



2



 

2 

 

2

2



2



     ˜   ˜  N3 (x, t ) = V˜ t (x, t ) + V˜ tt (x, t ) + divU t (x, t ) + divUtt (x, t ) , 2 

2 

 

2 

 

˜ tt (x, t ) + divU˜ ttt (x, t ) . N3 (x, t ) = V˜ tt (x, t ) + V˜ ttt (x, t ) + divU Furthermore, by (1.6) and the second equation in (3.1), we easily see that

γ h(x)∆v2 (x, t0 ) = V˜ tt (x, t0 ) − ν ∆V˜ t (x, t0 ) − (αν + γ f1 (x)) ∆V˜ (x, t0 ) + α V˜ t (x, t0 ) + β divU˜ tt (x, t0 ) + αβ divU˜ t (x, t0 ).

(3.12)

From (3.12), it follows for all (x, t ) ∈ QT that



   |∆V˜ t (x, t0 )|2 + |∆V˜ (x, t0 )|2 e2sϕ(x,t ) ≤ C |h(x)|2 + N3 [V˜ , U˜ ](x, t0 ) e2sϕ(x,t0 ) .

(3.13)

Similar to Lemma 3.1, we have

 Ω

˜ ](x, t0 )e2sϕ(x,t0 ) dx ≤ Cs N3 [V˜ , U



˜ ]e2sϕ dxdt + C ε s−1 N3 [V˜ , U QT



˜ ]e2sϕ dxdt N4 [V˜ , U

(3.14)

QT

for any ε > 0. Choosing ε sufficiently small, and then substituting (3.13) and (3.14) into (3.11) leads to







˜ ] + N2 [V˜ , U˜ ] e2sϕ dxdt ≤ C s2 N1 [V˜ , U



QT

˜ ∥2 5 s|h|2 e2sϕ dxdt + Cs3 eCs ∥U + Cs4 e2s(d−2δ) M0 . H (ωT )

QT

(3.15)

By (1.13), (3.12), (3.15) and Lemma 3.1, we obtain

 s

 ˜ ](x, t0 )e2sϕ(x,t0 ) dx |h|2 e2sϕ(x,t0 ) dx ≤ CseCs ∥V˜ (·, t0 )∥2H 2 (Ω ) + Cs N1 [V˜ , U Ω Ω    ˜ ] + N2 [V˜ , U˜ ] e2sϕ dxdt ≤ CseCs ∥V˜ (·, t0 )∥2H 2 (Ω ) + C s2 N1 [V˜ , U QT

≤ CseCs ∥V˜ (·, t0 )∥2H 2 (Ω ) + Cs

 QT

|h|2 e2sϕ dxdt + Cs3 eCs ∥U˜ ∥2H 5 (ω ) + Cs4 e2s(d−2δ) M0 . T

(3.16)

Noting that

 QT

  |h|2 e2sϕ dxdt ≤ sup  x∈Ω

T 0

 

e2s(ϕ(x,t )−ϕ(x,t0 )) dt 

Ω

|h|2 e2sϕ(x,t0 ) dx ≤ ε



|h|2 e2sϕ(x,t0 ) dx Ω

for any sufficiently small ε > 0 by the Lebesgue Theorem, we deduce from (3.16) that

 Ω

  |h|2 e2sϕ(x,t0 ) dx ≤ Cs2 eCs ∥V˜ (·, t0 )∥2H 2 (Ω ) + ∥U˜ ∥2H 5 (ω ) + Cs3 e2s(d−2δ) M0 , T

(3.17)

if we choose ε sufficiently small. We recall that ϕ(x, t0 ) ≥ d for all x ∈ Ω . So that, from (3.17) we obtain

  ∥h∥2L2 (Ω ) ≤ Cs2 e(C −2d)s ∥V˜ (·, t0 )∥2H 2 (Ω ) + ∥U˜ ∥2H 5 (ω ) + Cs3 e−4sδ M0 T   ≤ CeCs ∥V˜ (·, t0 )∥2H 2 (Ω ) + ∥U˜ ∥2H 5 (ω ) + Ce−2sδ M0 T

(3.18)

254

B. Wu et al. / J. Math. Anal. Appl. 402 (2013) 242–254 ⋆

for all s ≥ s⋆ with some sufficiently large s⋆ . Setting s := s + s⋆ and replacing C by CeCs , we obtain (3.18) for all s > 0. Finally, minimizing the right-hand side of (3.18) with respect to s, we get

2κ  ∥h∥2L2 (Ω ) ≤ CM01−2κ ∥V˜ (·, t0 )∥2H 2 (Ω ) + ∥U˜ ∥2H 5 (ω ) T

(3.19)

with κ = (C +δ2δ) ∈ (0, 1) and s=

1 C + 2δ

log

2δ M0 .  2 ˜ ∥2 5 C ∥V˜ (·, t0 )∥H 2 (Ω ) + ∥U H (ω ) T

This completes the proof.



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