Exponential stability and analyticity of abstract linear thermoelastic systems

Z. angew. Math. Phys. 48 (1997) 885–9040044-2275/97/060885-20 $ 1.50+0.20/0c© 1997 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Exponential stability and analyticity of abstract linearthermoelastic systems

Kangsheng Liu∗ and Zhuangyi Liu

Abstract. This paper develops an abstract framework for analysis of linear thermoelastic sys-tems. Linear semigroup theory is used to establish the well-posedness. Sufficient conditions forthe exponential stability and analyticity of the associated semigroups for the thermoelastic sys-tems are obtained via the frequency domain method and a contradiction argument. The resultsare applied to linear thermoelastic rods, beams and plates of homogeneous and nonhomogeneousmaterial with various boundary conditions.

Mathematics Subject Classification (1991). 35B40, 73H10, 73B30.

Keywords. Linear thermoelastic system, exponential stability, analyticity, semigroup.

1. Introduction

A deformation of a thermoelastic body which varies in time leads to a change ofthe temperature distribution in the body, and conversely. The internal energy ofthe body depends on both the deformation and the temperature. A thermoelasticsystem describes the above coupled processes. In general, it consists of an elasticequation and a heat equation which are coupled in a fashion such that the transferbetween the mechanical energy and the heat energy is taken into account (see[N]). The asymptotic behavior of the solution and the smoothness of the solutionhave been extensively studied for both the linear and the nonlinear thermoelasticsystems. We refer the readers to a recent paper by R. Racke [Ra] for a generalsurvey on those topics. In this paper, we confine our interests to the linear systemsonly. Our goals are the following:

1. Develop an abstract framework for the linear thermoelastic systems, and studythe well-posedness by the linear semigroup theory.

2. Find sufficient conditions for the exponential stability of the semigroup associ-

∗ Supported partially by the National Natural Science Foundation of China.

886 K. Liu and Z. Liu ZAMP

ated with the abstract linear thermoelastic system.3. Find sufficient conditions for the analyticity of the semigroup associated with

the abstract linear thermoelastic system.

The equations describing a linear thermoelastic rod of homogeneous material[D] are

utt(x, t) − cu′′(x, t) + cαθ′(x, t) = 0,θt(x, t) + αut

′(x, t) − θ′′(x, t) = 0(1.1)

for (x, t) ∈ [0, L]×R+. Here u is the displacement, and θ represents the tempera-ture resultant of the relative temperature. The coefficients α, c > 0 are constants.We have used the prime to denote the spatial derivative.

The equations describing a linear thermoelastic plate of homogeneous material[L] are

wtt − γ∆wtt + ∆2w + α∆θ = 0, in Ω× R+,

βθt − η∆θ + σθ − α∆wt = 0, in Ω× R+,(1.2)

where Ω is a bounded region in R2 with a smooth boundary ∂Ω. w represents thevertical deflection, and θ represents the relative temperature about the stress freestate θ = 0. α, β, η, σ, γ > 0 are constants.

In the past a few years, it has been proved that the heat dissipation alone isstrong enough to induce the exponential decay of energy of the above two systemswith many types of boundary conditions. Generally, boundary conditions of athermoelastic system consist of structural boundary conditions and temperatureboundary conditions. Those having been considered for system (1.1) are

(i) structural boundary conditions

u(i, t) = 0, i = 0, L; (fixed ends) (1.3)

u′(i, t)− αθ(i, t) = 0, i = 0, L. (free ends) (1.4)

(ii) temperature boundary conditions

θ(i, t) = 0, i = 0, L; (zero temperature), (1.5)

θ′(i, t) = 0, i = 0, L. (zero flux), (1.6)

Hansen [Ha] considered the cases of fixed ends–zero flux and free ends–zero tem-perature; Kim [K], Liu and Zheng [LZ1] considered the case of fixed ends–zerotemperature; Burns, Liu and Zheng [BLZ] considered the other combinations ofabove boundary conditions at each end of the rod.

The boundary conditions which have been investigated for system (1.2) are

(i) structural boundary conditions

w =∂w

∂ν= 0; (clamped edge) (1.7)

Vol. 48 (1997) Abstract linear thermoelastic systems 887

w = ∆w + (1− µ)B1w + αθ = 0; (simply supported edge) (1.8) ∆w + (1− µ)B1w + αθ = 0,∂∆w∂ν + (1− µ)∂B2w

∂τ − γ∂w′′

∂ν + α ∂θ∂ν = 0.(free edge) (1.9)

where ν = (ν1, ν2) is the unit outward normal vector, and τ = (−ν2, ν1) is a unittangent vector to ∂Ω; µ is the Poisson ratio; B1 and B2 are boundary operatorsdefined by

B1w = 2ν1ν2wxy − ν21wyy − ν2

2wxx,

B2w = (ν21 − ν2

2)wxy + ν1ν2(wxx − wyy).

(ii) temperature boundary conditions

λ1∂θ

∂ν+ λ2θ = 0 (1.12)

with λ1, λ2 ≥ 0, λ1 + λ2 6= 0. λ1 = 0 corresponds to zero temperature; λ2 = 0corresponds to zero flux; λ1, λ2 6= 0 corresponds to the Newton’s cooling law.

When the rotatory inertia is neglected in the plate equation (1.2), i.e., γ = 0,Kim [K] considered the case of clamped edge–zero temperature; Racke and Rivera[RR] considered the boundary conditions of w = ∆w = 0 and zero temperature;Liu and Zheng [LZ2] considered the case of clamped on Γ0, simply supported onΓ1 (with Γ0 ∪ Γ1 = ∂Ω, Γ1 ∩ Γ1 = ∅), and the Newton’s cooling law temperatureboundary conditions . When γ 6= 0, Avalos and Lasiecka [AL] considered the caseof clamped on Γ0, free on Γ1 with the temperature boundary condition same asin [LZ2].

The properties of the thermoelastic plate equation system (1.2) when γ = 0 israther special, as Liu and Renardy [LR] proved that for two groups of boundaryconditions, the equations are associated with an analytic semigroup.

In 1991, Gibson, Rosen and Tao [GRT] developed an abstract framework forlinear thermoelastic systems. They considered the following evolution equation

zt =

utvtθt

=

0 I 0−A0 0 −L∗0

0 L0 −A1

uvθ

= Az (1.13)

on a Hilbert space H. In the case of

A0 = αL∗0L0, (1.14)

they obtained the exponential stability of the semigroup eAt under certain condi-tions. Their result can be applied to systems (1.1) and (1.2) with some groups ofboundary conditions. However, assumption (1.14) is too restrictive. It excludes


the cases where boundary conditions involve both displacement and temperature,such as (1.8) and (1.9). It also excludes the case of nonhomogeneous material.

D.Russell [Ru] proposed an abstract framework for coupled dissipative systems.Under some assumption concerning the commutativity of the operators, he provedthe analyticity of the associated semigroup. When his result is applied to thethermoelastic plate equation system (1.2) with γ = 0, it only covers the case ofthe boundary conditions w = ∆w = θ = 0.

We will develop an abstract setting for linear thermoelastic systems in section2. Main results on exponential stability and analyticity of the semigroup associ-ated with the abstract linear thermoelastic system are presented in section 3 andsection 4. Our general results cover most of known results mentioned above, moreprecisely, all except the two cases studied in [AL], [K] and [LZ1]. Moreover, theyalso include some new cases which have not been studied. We apply our theoremson exponential stability to a cantilever Rayleigh beam with zero temperature atthe fixed end and Newton’s cooling law temperature condition at the free end,a partially clamped, partially simply supported Kirchhoff plate with Newton’scooling law temperature boundary condition, a simply supported Kirchhoff platewith zero temperature boundary condition. The theorem on analyticity is appliedto a nonhomogeneous clamped Euler-Bernoulli beam with Newton’s cooling lawtemperature boundary condition, and a partially clamped, partially simply sup-ported Kirchhoff plate (γ = 0) with Newton’s cooling law temperature boundarycondition.

2. Abstract setting and semigroup solution

We consider the following abstract variational evolution equation

c1(w(t), w) + c2(θ(t), θ) + a1(w(t), w) + a2(θ(t), θ)

+ c2(θ(t), Bw)− c2(Bw(t), θ) = 0, ∀w ∈ V1, θ ∈ V2, t > 0. (2.1)

Throughout this paper, we assume that

(A1) Vj and Hj are Hilbert spaces with inner products aj(·, ·) and cj(·, ·), respec-tively, for j = 1, 2.

(A2) Vj → Hj are continuous and dense embedding for j = 1, 2.(A3) B is a linear operator from D(B) = V1 ⊂ H1 to H2 and B ∈ L(V1,H2).

Assumptions (A1) and (A2) imply that there exist self-adjoint, positive definiteoperators Aj in Hj such that

D(A12j ) = Vj , aj(u, v) = cj(A

12j u,A

12j v), ∀ u, v ∈ Vj , j = 1, 2. (2.2)

Moreover, assumption (A3) now implies

BA− 1

21 ∈ L(H1,H2). (2.3)


Let H = V1 ×H1 ×H2 with the inducing inner product. Define in H that

D(A) = z = (u, v, θ) ∈ H | v ∈ V1, θ ∈ D(A2), A121 u+ (BA

− 12

1 )∗θ ∈ V1, (2.4)

and

Az =

v

−A121 [A

121 u+ (BA

− 12

1 )∗θ]Bv −A2θ

, (2.5)

where (BA− 1

21 )∗ ∈ L(H2,H1).

Theorem 2.1. A generates a C0-semigroup, eAt, of contractions on H, and 0 ∈ρ(A), the resolvent set of A.

Proof. For any (f, g, h) ∈ H, the equation Az = (f, g, h) has an unique solutionz = (u, v, θ):

v = f,

θ = A−12 (Bf − h),

u = −A−11 g −A−

12

1 (BA− 1

21 )∗A−1

2 (Bf − h).

(2.6)

It is easy to verify directly that z ∈ D(A) and

‖z‖H ≤M‖(f, g, h)‖H (2.7)

for some M > 0, independent of (f, g, h). Therefore, A is closed and 0 ∈ ρ(A).The dissipativeness of A can be seen in the following calculation:

Re〈Az, z〉H = Re[a1(v, u)− a1(u, v)− c1((BA− 1

21 )∗θ,A

121 v)− a2(θ) + c2(Bv, θ)]

= −a2(θ) + Re[c2(Bv, θ)− c2(θ,Bv)]= −a2(θ) ≤ 0. (2.8)

By Pazy [Pa, Th 4.6], D(A) is dense in H. The conclusion immediately followsfrom the Lumer-Phillips theorem [Pa].

Corollary 2.1. Let (w(t), v(t), θ(t)) = eAtz0 for z0 = (w0, v0, θ0) ∈ H. Thenw(·) ∈ C1([0,∞);H1) ∩C([0,∞);V1),θ(·) ∈ C([0,∞);H2),v(·) = w(·)

(2.9)

satisfy ddtc1(w(t), w) + a1(w(t), w) + c2(θ(t), Bw) = 0,d

dtc2(θ(t)−Bw(t), θ)− c2(θ(t), A2θ) = 0

∀w ∈ V1, θ ∈ D(A2), t > 0.

(2.10)


Moreover, if z0 ∈ D(A), thenw ∈ C2([0,∞);H1) ∩ C1([0,∞);V1),θ ∈ C1([0,∞);H2) ∩ C([0,∞);D(A2))

(2.11)

satisfy the variational equation (2.1).

Proof. When z0 ∈ D(A), the results follows readily from the semigroup theory.When z0 ∈ H, the proof can be done by a standard density (i.e., D(A) = H)argument. We omit the details here.

Remark 2.1. In equation (2.5), we do not distribute A121 into the parentheses

since the sum of two terms may be smoother than each term.

3. Exponential stability

Theorem 3.1. In addition to the conditions (A1)-(A3), assume that

B(D(A1)) ⊂ D(A122 ), (3.1)

and‖Bu‖H2 ≥ δ‖u‖V1 , ∀ u ∈ V1, (3.2)

where δ is a positive constant. Then eAt is exponentially stable.

Proof. We will use the frequency domain condition ([G],[Hu],[Pr]) for the expo-nential stability of C0-semigroup of contractions on a Hilbert space:

(FDC)iR ⊂ ρ(A),sup

‖(iω −A)−1‖ | ω ∈ R

< +∞.

Since 0 ∈ ρ(A), it suffices to prove that

‖(iω −A)z‖H ≥ c > 0 (3.3)

for z in D(A) with unit norm and for |ω| > ε0 ≡ 12‖A−1‖−1. If (3.3) is false, then

there exist a sequence of real number ωn (without losing generality) with ωn > ε0,and a sequence of vector zn ∈ D(A) with unit norm such that

‖(iωn −A)zn‖H → 0, (3.4)


i.e.,

iωnun − vn → 0 in V1, (3.5)

iωnvn +A121 yn → 0 in H1, (3.6)

iωnθn −Bvn +A2θn → 0 in H2, (3.7)

where yn ≡ A121 un + (BA

− 12

1 )∗θn ∈ D(A121 ). It follows from (3.4) that

Re〈(iωn −A)zn, zn〉 = ‖A122 θn‖

2H2→ 0. (3.8)

Therefore, θn converges to zero in H2. We add the inner product of (3.5) with vnin H1 to the inner product of (3.6) with un in H1 to get

‖A121 un‖

2H1− ‖vn‖2H1

→ 0. (3.9)

This, combined with the fact that ‖zn‖ = 1 and ‖θn‖H2 → 0, leads to

limn→∞

‖A121 un‖

2H1

= limn→∞

‖vn‖2H1=

12. (3.10)

Furthermore, by (3.6) we also have

limn→∞

∥∥∥∥∥∥A121 ynωn

∥∥∥∥∥∥2

H1

=12. (3.11)

Dividing (3.7) by ωn, from (3.8) we obtain

−Bvnωn

+A2θnωn

→ 0 in H2. (3.12)

Since B ∈ L(V1,H2) and (BA− 1

21 )∗ ∈ L(H2,H1), we have

BA− 1

21 (BA

− 12

1 )∗ ∈ L(H2). (3.13)

It follows from substituting (3.5) and inserting BA− 1

21 (BA

− 12

1 )∗θn into (3.12) that

−iBA−12

1 [A121 un + (BA

− 12

1 )∗θn] +A2θnωn

=A2θnωn

− iBA−12

1 yn → 0 in H2. (3.14)

Since the condition (3.1) implies that A122 BA

−11 ∈ L(H1,H2), from (3.8) and (3.11)

we have

〈A2θnωn

, BA− 1

21 yn〉H2 = 〈A

122 θn, A

122 BA

−11A

121 yn

ωn〉H2 → 0. (3.15)


Equations (3.14)-(3.15) yield

BA− 1

21 yn → 0 in H2, (3.16)

which leads toBun → 0 in H2. (3.17)

Finally, by condition (3.2), un must converge to zero in V1 which contradicts (3.10).

Example 3.1 Consider a thermoelastic Rayleigh beam equation with cantileverboundary condition,

utt − γu′′

tt + u′′′′ + αθ′′ = 0, (x, t) ∈ (0, L)×R+

βθt − αu′′

t − ηθ′′ + σθ = 0, (x, t) ∈ (0, L)×R+

u(0, t) = u′(L, t) = 0(u′′ + αθ)|x=L = (u′′ + αθ − γutt)′|x=L = 0,θ|x=L = (λθ − ηθ′)|x=0 = 0,

(3.18)

plus initial conditions, where λ, σ ≥ 0, α, β, γ, η > 0, λ+ σ > 0.Let

V1 =u ∈ H2(0, L) | u(0) = u′(0) = 0

,

a1(u, v) =∫ L

0u′′v′′dx, ∀ u, v ∈ V1;

H1 =u ∈ H1(0, L) | u(0) = 0

,

c1(u, v) =∫ L

0(γu′v′ + uv)dx, ∀ u, v ∈ H1;

V2 =θ ∈ H1(0, L) | θ(L) = 0

,

a2(u, v) =∫ L

0(ηu′v′ + σuv)dx + λu(0)v(0), ∀ u, v ∈ V2;

H2 = L2β(0, L),

c2(u, v) =∫ L

0βuvdx, ∀ u, v ∈ H2.

Define operator B : H1 → H2,

Bu =α

βu′′, ∀ u ∈ V1 = D(B). (3.19)


It is easy to see that conditions (A1)-(A3) and (3.2) hold. By the definition of A1in (2.2), for u ∈ D(A1), y = A1u, we have∫ L

0u′′v′′dx = c1(A1u, v) = c1(y, v)

=∫ L

0(γy′v′ + yv)dx

=∫ L

0

(γy′ +

∫ L

x

y(s)ds

)v′dx, ∀v ∈ V1. (3.20)

Take v =∫ x

0ξ(s)ds ∈ V1 for any ξ ∈ C∞0 (0, L) in (3.20) to get

∫ L

0u′′ξ′dx =

∫ L

0

(γy′ +

∫ L

x

y(s)ds

)ξ(x)dx. (3.21)

Hence u′′ ∈ H1(0, L) which implies

D(A1) ⊂u ∈ H3(0, L) | u′′(L) = 0

, (3.22)

andBu =

α

βu′′ ∈ V2 = D(A

122 ), ∀u ∈ D(A1). (3.22)

This verifies condition (3.1).

Example 3.2. Consider the following thermoelastic Kirchhoff plate equation [L]wtt − γ∆wtt + ∆2w + α∆θ = 0, in Ω× R+,

βθt − η∆θ + σθ − α∆wt = 0, in Ω× R+,(3.24)

with the boundary conditions

w =∂w

∂ν= 0 on Γ0 × R+, (3.25)

w = ∆w + (1− µ)B1w + αθ = 0 on Γ1 × R+, (3.26)

η∂θ

∂ν+ λθ = 0 on ∂Ω× R+, (3.27)

where Ω is an open set in R2 with sufficient smooth boundary ∂Ω = Γ0 ∪ Γ1,α, β, γ, η > 0, σ, λ ≥ 0, σ + λ > 0, 0 < µ < 1

2 ,

B1w = 2ν1ν2wxy − ν21wyy − ν2

2wxx (3.28)


and ν = (ν1, ν2) is the unit outward normal to ∂Ω. We assume that

Γ0 ∩ Γ1 = ∅. (3.29)

Let

V1 =u ∈ H2(Ω) | u = 0 on ∂Ω,

∂u

∂ν= 0 on Γ0

,

a1(u, v) =∫

Ω[uxxvxx + uyyvyy + µ(uxxvyy + uyyvxx)

+ 2(1− µ)uxyvxy]dxdy, ∀ u, v ∈ V1;

H1 = H10 (Ω),

c1(u, v) =∫

Ω(γ∇u∇v + uv)dxdy, ∀ u, v ∈ H1;

V2 = H1(Ω),

a2(u, v) =∫

Ω(η∇u∇v + σuv)dxdy +

∫∂ΩλuvdΓ, ∀ u, v ∈ V2;

H2 = L2β(Ω),

c2(u, v) =∫

Ωβuvdxdy, ∀ u, v ∈ H2.


Bu =α

β∆u, ∀u ∈ D(B) = V1. (3.30)

Under the above setting, the conditions (A1)-(A3) and (3.2) hold clearly. Thecondition (3.1) can be verified by showing that

D(A1) ⊂ H3(Ω). (3.31)

Let u ∈ D(A1), z = A1u. Then we have

a1(u, v) = c1(z, v) =∫

Ω(γ∇z∇v + zv)dxdy =

∫Ωz(1− γ∆)vdxdy, ∀v ∈ V1.

(3.32)Therefore, u is the solution of the following elliptic boundary value problem:

∆2u = (1− γ∆)z ∈ H−1

u = ∂u∂ν = 0 on Γ0

u = ∆u+ (1− µ)B1u = 0 on Γ1

(3.33)


Since Γ0 ∩ Γ1 = ∅, by the standard elliptic theory [LM] we obtain u ∈ H3(Ω).

Theorem 3.2. Let the conditions (A1)-(A3) hold. Assume that the adjoint op-erator B∗ : D(B∗) ⊂ H2 → H1 satisfies

D(A122 ) ⊂ D(B∗), B∗D(A

322 ) ⊂ V1, (3.34)

and assume ∥∥∥∥A− 12

2 Bv

∥∥∥∥H2

≥ δ‖v‖H1 , ∀v ∈ V1. (3.35)

Then eAt is exponentially stable.

Remark 3.1. The condition D(A2) ⊂ D(B∗) implies that for (u, v, θ) ∈ D(A),(recall (2.4))

(BA− 1

21 )∗θ = A

− 12

1 B∗θ ∈ V1, u ∈ D(A1). (3.36)

Proof of Theorem 3.2. We will verify (3.3) again. First, repeat (3.4)-(3.10). Equa-tion (3.6), combined with (3.36), implies

iωnvn +A1un → 0 in H1. (3.37)

The condition (3.34) implies that

B∗A− 1

22 ∈ L(H2,H1), A

− 12

2 B = (B∗A− 1

22 )∗ ∈ L(H1,H2), (3.38)

andA

121 B∗A− 3

22 ∈ L(H2,H1), (3.39)

It follows from equation (3.7), (3.8) and (3.38) that

iωn‖θn‖2H2→ 0, (3.40)

iωnA− 1

22 θn − A

− 12

2 Bvn → 0 in H2. (3.41)

We take the inner product of (3.37) with B∗A−22 Bvn in H1 to get

iωn‖A−12 Bvn‖2H2

= −c1(A1un, B∗A−2

2 Bvn) + o(1)

= −c1(A121 un, (A

121 B∗A− 3

22 )A

− 12

2 Bvn) + o(1)= O(1). (3.42)


Therefore, from (3.35), (3.40)-(3.42) we obtain

δ2‖vn‖2H1≤ ‖A−

12

2 Bvn‖2H2

= iωnc2(A− 1

22 θn, A

− 12

2 Bvn) + o(1)

= ic2(√ωnθn,

√ωnA

−12 Bvn) + o(1)

= o(1). (3.43)

This contradicts (3.10).

Example 3.3. In Example 3.2, we take Γ0 = ∅ and replace the boundary condition(3.27) by

θ = 0 on ∂Ω× R+. (3.44)

The setting in Example 3.2 is still valid except for V2. Reset

V2 = H10 (Ω).

Then, the conditions (A1)-(A3) hold obviously. Define in H2

A0 = −∆ with D(A0) = H2(Ω) ∩H10 (Ω). (3.45)

We haveV1 = D(A2) = D(A0), H1 = V2 = D(A

120 ) = H1

0 (Ω), (3.46)

B = −αβA0, A2 =

η

βA0 +

σ

βI. (3.47)

A direct calculation yields

c1(u,α

γ[I − (I + γA0)−1]θ) = c2(Bu, θ), ∀ u ∈ V1, θ ∈ V2.

This impliesV2 ⊂ D(B∗), B∗|V2 =

α

γ[I − (I + γA0)−1].

Therefore, B∗D(A2) ⊂ D(A0) = V1, the condition (3.34) holds.On the other hand, we have

‖A−12

2 Bv‖H2 = ‖αβ

(σ

βI +

η

βA0)−

12A0v‖H2 ≥ δ‖A

120 v‖H2

for some δ > 0 and all v ∈ V1. Since ‖A120 · ‖H2 is an equivalent norm in H1, the

condition (3.35) is also satisfied.

Remark 3.2. We are unable to modify the conditions in Theorem 3.1 and The-orem 3.2 to cover two special cases studied in [AL], [K] and [LZ1].


4. Analyticity

Theorem 4.1. Let the conditions (A1)-(A3) hold. Assume that

D(A2) ⊂ D(B∗), (4.1)

and

〈B∗A−12 Bv, v〉H1 ≥ δ〈A

121 v, v〉H1 , ∀ v ∈ D(A

121 ) (4.2)

where δ > 0 is a constant. Then eAt is an exponentially stable, analytic semigroupon H.

Proof. Since A is dissipative and 0 ∈ ρ(A), by the theory of analytic semigroups[Pa], it suffices to show

‖ 1ω

(iω −A)z‖H ≥ c > 0 (4.3)

for z ∈ D(A) with unit norm and for |ω| > ε0 ≡ 12‖A−1‖−1. We will use a

contradiction argument introduced by Liu and Yong in [LY]. If (4.3) is false, thenthere exist a sequence of real number ωn (without losing generality) with ωn > ε0,and a sequence of vector zn ∈ D(A) with unit norm such that

‖ 1ωn

(iωn −A)zn‖H → 0, (4.4)

i.e.,

iun −1ωnvn → 0 in V1, (4.5)

ivn +1ωn

A1un +1ωn

B∗θn → 0 in H1, (4.6)

iθn −1ωn

Bvn +1ωn

A2θn → 0 in H2, (4.7)

It follows from (4.4) that

Re〈 1ωn

(iωn −A)zn, zn〉H =1ωn‖A

122 θn‖

2H2→ 0. (4.8)


Taking the inner product of (4.7) with θn in H2, we obtain

i‖θn‖2H2− 1ωn〈Bvn, θn〉H2 → 0. (4.9)

The condition (4.1) implies B∗A−12 ∈ L(H2,H1). Thus we can act −B∗A−1

2 on(4.7), and add the result to (4.6) to get

ivn − iB∗A−12 θn +

1ωn

A1un +1ωn

B∗A−12 Bvn → 0 in H1, (4.10)

which further leads to

i‖vn‖2H1− i〈B∗A−1

2 θn, vn〉H1 +〈A121 un,

1ωnA

121 vn〉H1 +

1ωn〈B∗A−1

2 Bvn, vn〉H1 → 0.

(4.11)From (4.1), (4.5) and ‖zn‖ = 1, the first three terms in (4.11) are bounded. So

does the last term in (4.11), i.e., the sequence 1√ωnA− 1

22 Bvn is bounded in H2.

Therefore, from (4.8) we know that

1ωn〈Bvn, θn〉H2 = 〈 1√

ωnA− 1

22 Bvn,

1√ωn

A122 θn〉H2 → 0. (4.12)

This, together with (4.9), implies

‖θn‖H2 → 0. (4/13)

Since zn is of unit norm, then

‖un‖2V1+ ‖vn‖2H1

→ 1. (4.14)

On the other hand, by (4.13) and (4.1), we have B∗A−12 θn → 0 in H1. Therefore,

equation (4.10) yields

ivn +1ωnA1un +

1ωn

B∗A−12 Bvn → 0 in H1. (4.15)

We add the inner product of (4.5) with un in V1 to the complex conjugate ofthe inner product of (4.15) with vn in H1. We then see that the imaginary partof the sum

‖un‖2V1− ‖vn‖2H1

→ 0, (4.16)


and that the real part of the sum

1ωn〈B∗A−1

2 Bvn, vn〉H1 → 0. (4.17)

It follows from (4.14) and (4.16) that

limn→∞

‖un‖2V1= limn→∞

‖vn‖2H1=

12. (4.18)

In what follows, we will prove that ‖vn‖H1 converges to zero, thus a contradiction.Equations (4.15) and (4.17) imply

i‖vn‖2H1+

1ωn〈A1un, vn〉H1 → 0. (4.19)

Applying the condition (4.2) to (4.17), we obtain

1ωn〈A

121 vn, vn〉H1 = ‖ 1√

ωnA

141 vn‖

2H1→ 0. (4.20)

From (4.1) and (4.5), it is easy to see that

1ωn

B∗A−12 Bvn = (B∗A−1

2 )(BA− 1

21 )A

121vnωn

(4.21)

is bounded in H1.Then, by (4.15) the sequence 1ωnA1un is also bounded in H1.

Therefore,

1ωn‖A

341 un‖

2H1

=1ωn〈A1un, A

121 un〉H1 (4.22)

must be bounded. This and (4.20) immediately lead to

1ωn〈A1un, vn〉H1 = 〈 1√

ωnA

341 un,

1√ωn

A141 vn〉H1 → 0. (4.23)

We finally conclude from (4.23) and (4.19) that

‖vn‖H1 → 0. (4.24)


Example 4.1. Consider a nonhomogeneous thermoelastic beam equationρ(x)utt(x, t) + [EI(x)u′′(x, t) + α(x)θ(x, t)]′′ = 0, (x, t) ∈ (0, L)×R+,

β(x)θt(x, t) − α(x)u′′t (x, t) − (k(x)θ′(x, t))′ + σ(x)θ(x, t) = 0,

u|x=0,L = u′|x=0,L = 0, t > 0,(λ(x)θ − k(x)θ′)|x=0 = (λ(x)θ + k(x)θ′)|x=L = 0, t > 0

(4.25)where all the coefficient functions are positive and belong to C2[0, L]. We assumethat the function α(x) satisfies∫ L

0(α2(x)|v′|2 − α′′(x)

2|v|2)dx ≥ δ0

∫ L

0|v′|2dx (4.26)

for some δ0 > 0 and all v ∈ H10 (0, L). The assumption (4.26) holds when α′′(x) ≤

0. Let

V1 = H20 (0, L),

a1(u, v) =∫ L

0EI(x)u′′v′′dx, ∀ u, v ∈ V1;

H1 = H2 = L2(0, L),

c1(u, v) =∫ L

0ρ(x)uvdx, ∀ u, v ∈ H1;

V2 = H1(0, L),

a2(u, v) =∫ L

0(k(x)u′v′ + σ(x)uv)dx+ (λuv)(0) + (λuv)(L), ∀ u, v ∈ V2;

c2(u, v) =∫ L

0β(x)uvdx, ∀ u, v ∈ H2.

Define B : H1 → H2,

Bu =α(x)β(x)

u′′, ∀ u ∈ D(B) = V1. (4.27)

Then the conditions (A1)-(A3) hold obviously. Moreover, it is easy to see that

D(B∗) = H2(0, L), B∗u =(α(x)u)′′

ρ(x), (4.28)

D(A2) = u ∈ H2(0, L) | (4.25)4 holds. (4.29)


Thus the condition (4.1) is satisfied.Now we verify (4.2), i.e.,

‖A−12

2 Bv‖2H2≥ δ‖A

141 v‖

2H1

(4.30)

for some δ > 0 and all v ∈ V1. Define in H2

A0u =u′′

β(x), D(A0) = u ∈ H2(0, L) | u′(0) = u′(L) = 0. (4.31)

Then A0 is a self-adjoint positive definite operator in H2 and D(A120 ) = H1(0, L) =

D(A122 ). We also have D(A

141 ) = H1

0 (0, L) since D(A121 ) = V1 = H2

0 (0, L). There-fore, we only need to verify

‖A−12

0 Bv‖2H2≥ δ

∫ L

0|v′|2dx, ∀ v ∈ V1 (4.32)

for some δ > 0. Let

y = A−10 Bv, v ∈ V1. (4.33)

We then have

y′′ = α(x)v′′,

y′ =∫ x

0a(τ)v′′(τ)dτ

= α(x)v′(x)− α′(x)v(x) +∫ x

0α′′(τ)v(τ)dτ.

Furthermore, by the assumption (4.26) we obtain

‖A−12

0 Bv‖2H2=∫ L

0yα(x)v′′dx

=∫ L

0(α(x)y′ + α′(x)y)v′dx

=∫ L

0(α2(x)|v′|2 − a′′(x)

2|v|2)dx+ Re

∫ L

0α′(x)yv′dx

≥ δ0∫ L

0|v′|2dx−

∣∣∣∣∣Re∫ L

0α′(x)yv′dx

∣∣∣∣∣ , ∀ v ∈ V1.


Therefore, the desired estimate (4.32) follows from the fact∣∣∣∣∣Re∫ L

0α′(x)yv′dx

∣∣∣∣∣ ≤ C(

1ε‖A−

12

0 Bv‖2H2+ ε

∫ L

0|v′|2dx

)(4.34)

for some C > 0 and any ε > 0.

Example 4.2. Consider the thermoelastic plate equation in Example 3.2 withoutrotatory inertia, i.e., γ = 0. Replace the temperature boundary condition (3.27)by

θ = 0 on ∂Ω× R+. (4.35)

Let

V1 =u ∈ H2(Ω) | u = 0 on ∂Ω,

∂u

∂ν= 0 on Γ0

,

a1(u, v) =∫

Ω[uxxvxx + uyyvyy + µ(uxxvyy + uyyvxx)

+ 2(1− µ)uxyvxy]dxdy, ∀ u, v ∈ V1;

H1 = L2(Ω),

c1(u, v) =∫

Ωuvdxdy, ∀ u, v ∈ H1;

V2 = H10 (Ω),

a2(u, v) =∫

Ω(η∇u∇v + σuv)dxdy ∀ u, v ∈ V2;

H2 = L2β(Ω),

c2(u, v) =∫

Ωβuvdxdy, ∀ u, v ∈ H2.


Bu =α

β∆u, ∀u ∈ D(B) = V1. (4.36)

It is easy to see that conditions (A1)-(A3) hold. Moreover, we have

A2 =1β

(σI − η∆), D(A2) = H2(Ω) ∩H10 (Ω), (4.37)


and

c2(Bu, θ) = c1(u, α∆θ) (4.38)

for any u ∈ D(B) = V1, θ ∈ D0 ≡ u ∈ H2(Ω) | u = 0 on Γ1. This implies

D0 ⊂ D(B∗), B∗|D0 = α∆. (4.39)

Thus, the condition (4.1) holds. Define in H1 = L2(Ω)

A0 = −∆, D(A0) = H2(Ω) ∩H10 (Ω). (4.40)

Then A0 is a self-adjoint and positive definite operator in H1, and

〈B∗A−12 Bv, v〉H1 =

(α

β

)2c1(A0(σI + ηA0)−1A0v, v)

=(α

β

)2c1(A0(σI + ηA0)−1A

120 v,A

120 v)

≥(α

β

)2 λ0σ + ηλ0

∫Ω|∇v|2dx (4.41)

for v ∈ V1, where λ0 > 0 is the least eigenvalue of A0. Observing D(A141 ) = H1

0 (Ω)we know that the condition (4.2) holds.

Remark 4.1. In Example 4.2, the geometric condition (3.29) is not needed.

References

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Kangsheng LiuDepartment of Applied MathematicsZhejiang UniversityHangzhou, 310027, China

Zhuangyi LiuDepartment of Mathematics and StatisticsUniversity of MinnesotaDuluth, MN 55812-2496

(Received: December 21, 1996)

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Exponential stability and analyticity of abstract linear thermoelastic systems

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