Energy decay to Timoshenko’s system with thermoelasticity

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Asymptotic Analysis 86 (2014) 227–247 227DOI 10.3233/ASY-131196IOS Press

Energy decay to Timoshenko’s system withthermoelasticity of type III

L.H. Fatori a,∗, J.E. Muñoz Rivera b and R. Nunes Monteiro a

a Department of Mathematics, Universidade Estadual de Londrina, Campus Universitário, Londrina,CEP 86.051-990, Paraná, BrazilE-mails: [email protected], [email protected] National Laboratory of Scientific Computations, LNCC/MCT, Rua Getúlio Vargas 333, Quitandinha,Petrópolis, CEP 25651-070, RJ, BrazilandInstituto de Matemática – UFRJ, Av. Horácio Macedo, Centro de Tecnologia Cidade Universitária –Ilha do Fundão, 21941-972 Rio de Janeiro, RJ, BrasilE-mail: [email protected]

Abstract. We consider the thermoelastic beam system when the oscillations are defined by the Timoshenko’s model and theheat conduction is given by Green and Naghdi theories. Our main result is that the corresponding semigroup is exponentiallystable if and only if the wave speeds associated to the hyperbolic part of the system are equal. In the case of lack of exponentialstability we show that the solution decays polynomially and we prove that the rate of decay is optimal.

Keywords: Timoshenko system, decay rates, optimality of polynomial stability

1. Introduction

In this paper we study the asymptotic properties of the thermoelastic beam system where the transver-sal oscillations are given by the Timoshenko’s model and the balance of the energy is described by theGreen and Naghdi theory, known as thermoelasticity of type III. The motion equation and the balanceof the energy are given by

ρAϕtt − Sx = 0, (1.1)

ρIψtt −Mx + S = 0, (1.2)

ρ3ut + qx + γψxt = 0. (1.3)

The constant ρ denotes the density, A the cross-sectional area and I the area moment of inertia. ByS we denote the shear force, M is the bending moment and q is the heat flux. The function ϕ is thetransverse displacement, ψ is the rotation angle of a filament of the beam and u is the temperaturedifference. Here, t is the time variable and x is the space coordinate along the beam. The constitutive

*Corresponding author. E-mail: [email protected].

0921-7134/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

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228 L.H. Fatori et al. / Energy decay to Timoshenko’s system with thermoelasticity of type III

laws we use are the following:

S = kAG(ϕx + ψ), M = EIψx + βu,

q = −δαx −Kαxt,

where α is the so-called thermal displacement whose time derivative is the empirical temperature u, i.e.,

αt = u,

E and G are elastic constants, k the shear coefficient for measuring the stiffness of materials (k < 1),δ and K denote the thermal conductivity, β the coefficient of linear thermal expansion and γ a couplingconstant. We refer the reader to [6,10,25] and [7,8] for Timoshenko beam and model of thermoelasticitytype III, respectively.

To simplify the notation let us denote by ρ1 = ρA, ρ2 = ρI ,κ = kAG and b = EI . Under theseconditions the system can be written as

ρ1ϕtt − κ(ϕx + ψ)x = 0 in (0,∞) × (0,L),ρ2ψtt − bψxx + κ(ϕx + ψ) + βux = 0 in (0,∞) × (0,L),ρ3utt − δuxx + γψttx −Kutxx = 0 in (0,∞) × (0,L).

(1.4)

In order to exhibit the dissipative nature of system (1.4), it is convenient to introduce a new variable (see[26])

θ(t,x) =∫ t

0u(s,x) ds+

1δχ(x), (1.5)

where χ ∈ H10 (0,L) solves the following Cauchy problem{

χxx = ρ3u1 −Ku0xx + γψ1x in (0,L),χ(x) = 0, x = 0,L.

(1.6)

Then, using (1.5) and (1.6) the starting system (1.4) is transformed to

ρ1ϕtt − κ(ϕx + ψ)x = 0 in (0,∞) × (0,L),ρ2ψtt − bψxx + κ(ϕx + ψ) + βθtx = 0 in (0,∞) × (0,L),ρ3θtt − δθxx + γψtx −Kθtxx = 0 in (0,∞) × (0,L).

(1.7)

We consider initial conditions

ϕ(0, ·) = ϕ0, ϕt(0, ·) = ϕ1,

ψ(0, ·) = ψ0, ψt(0, ·) = ψ1, (1.8)

θ(0, ·) = θ0, θt(0, ·) = θ1

and boundary conditions of type Dirichlet–Dirichlet–Dirichlet

ϕ(t, 0) = ϕ(t,L) = 0, ψ(t, 0) = ψ(t,L) = 0, θ(t, 0) = θ(t,L) = 0, t > 0 (1.9)

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L.H. Fatori et al. / Energy decay to Timoshenko’s system with thermoelasticity of type III 229

or Dirichlet–Neumann–Dirichlet boundary conditions

ϕ(t, 0) = ϕ(t,L) = 0, ψx(t, 0) = ψx(t,L) = 0, θ(t, 0) = θ(t,L) = 0, t > 0. (1.10)

Note that the initial condition for θ, defines also initial condition for u, in the sense that

u(0, ·) = θ1, ut(0, ·) = δθ0,xx − γψ1,x +Kθ1,xx.

The asymptotic behavior of Timoshenko systems with different damping mechanism, have been inves-tigated extensively in the literature. See for example [1,4,9,12,22,24] and references therein to quoted buta few. One of the first papers concerning thermal dissipation is given in [18] where the authors considerthe following Timoshenko system:

ρ1ϕtt − k(ϕx + ψ)x = 0 in (0,∞) × (0,L),ρ2ψtt − bψxx + k(ϕx + ψ) + δθx = 0 in (0,∞) × (0,L),ρ3θt − θxx + δψxt = 0 in (0,∞) × (0,L).

(1.11)

They proved that the system is exponentially stable if and only if the wave speeds are equal, that is

ρ1

k=

ρ2

b. (1.12)

In the model (1.11) the heat flux is given by Fourier’s law. The problem with this constitutive law isthat any thermal disturbance at a single point has an instantaneous effect everywhere in the medium,which is not realist. This phenomenon is known as the infinite speed of propagation.

To overcome this physical paradox, many theories were merged. Green and Naghdi [7,8] re-examinedthe classical model of thermoelasticity and introduced three types of thermoelastic theories based onan entropy equality instead of the usual entropy inequality. In each of these theories, the heat flux isgiven by a different constitutive assumption. As a result, three theories are obtained and were calledthermoelasticity type I, II and III respectively. This theory is developed to obtain a consistent explanation,which will incorporate thermal pulse transmission in a logical manner and elevates the unphysical infinitespeed of heat propagation induced by the classical theory of heat conduction. We recommend the readerto the survey paper of Chandrasekharaiah [3] for more detail on this theory.

Another option to remove the infinity speed of propagation is to consider the Cattaneo’s law for theheat flux. In this case the Timoshenko system is given by

ρ1ϕtt − k(ϕx + ψ)x = 0 in (0,∞) × (0,L),ρ2ψtt − bψxx + k(ϕx + ψ) + δθx = 0 in (0,∞) × (0,L),ρ3θt + qx + δψxt = 0 in (0,∞) × (0,L),τqt + βq + θx = 0 in (0,∞) × (0,L).

(1.13)

The above model was studied by several authors, see for example [5,15,21,23] to quote a few. Recently,in [23] was introduced a new stability number associated to system (1.13), which is defined by

χ0 =

(τ − ρ1

ρ3κ

)(ρ2 −

bρ1

κ

)− τρ1δ

2

ρ3κ. (1.14)

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In [23] it is established that system (1.13) is exponentially stable if and only if χ0 = 0. Note that if τ = 0then Cattaneo’s law reduces to Fourier Law and condition χ0 = 0 is equivalent to (1.12).

The Timoshenko model with thermoelasticity of type III was studies in [14,16,17,26]. In [14] theauthors studied the same model we consider in this article. They proved the exponential decay of thesolution provided the wave speeds of the system are equal, that is (1.12) holds. Although this resultshows the exponential decay of the solution, this result is interesting only from mathematical point ofview because for really application the wave speeds are always different. To see that, let us suppose that(1.12) holds, then we will have that

kG

E= 1.

Since the relation between elastic modulus and shear modulus in Timoshenko’s models is

G =E

2 + 2r

for r ∈ [0, 1/2] (see [6,10,25]) we get

k

2 + 2r= 1.

But this is not possible because k is always a small number less than one.The main result of this paper is to prove that condition (1.12) is a necessary and sufficient condition

to get the exponential decay of the solution. Moreover, we show that in the general case (different wavespeeds) the solution of system (1.7) decays polynomially to zero as t−1/2, when time goes to infinity andwe will prove that the rate of decay is optimal for any initial data taken in the domain of A:

∥∥T (t)w∥∥ � c

t1/2‖w‖D(A).

Using a standard semigroup procedure we can prove that

∥∥T (t)w∥∥ � c

tk/2‖w‖D(Ak),

that is, to say the more regularity, the faster rate of decay.The remaining part of the paper is organized as follows. In Section 2 we show the well-posedness

of the model by using the semigroup theory for different boundary conditions (Dirichlet–Dirichlet–Dirichlet or Dirichlet–Neuman–Dirichlet). In Section 3 we show that condition (1.12) is necessary andsufficient to get exponential stability of the corresponding semigroup for the case (1.10) and for thecase of Dirichlet–Dirichlet boundary condition (1.12) is sufficient to exponential stability of the cor-responding semigroup. Finally, in Section 4 we prove the polynomial stability and the optimality toDirichlet–Neuman–Dirichlet boundary conditions.

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2. Notations and semigroup setting

In this section, we establish the well-posedness of the system. We consider the Hilbert space

H1 = H10 (0,L) × L2(0,L) ×H1

0 (0,L) × L2(0,L) ×H10 (0,L) × L2(0,L)

or

H2 = H10 (0,L) × L2(0,L) ×H1

∗(0,L) × L2∗(0,L) ×H1

0 (0,L) × L2(0,L),

where

L2∗(0,L) =

{u ∈ L2(0,L):

∫ L

0u(x) dx = 0

}

and

H1∗(0,L) =

{u ∈ H1(0,L):

∫ L

0u(x) dx = 0

}.

In this paper, we will use the following (equivalent) norm for Hj ,

‖U‖2Hj

=∥∥(ϕ,Φ,ψ,Ψ , θ,Θ)′

∥∥2Hi

= γρ1‖Φ‖2L2 + ρ2γ‖Ψ‖2

L2 + γb‖ψx‖2L2 + γk‖ϕx + ψ‖2

L2 + βδ‖θx‖2L2 + βρ3‖Θ‖2

L2 ,

where the prime is used to denote the transpose. The initial-boundary value problem (1.7) can be rewrit-ten as the following abstract initial value problem for a first-order evolution equation

Ut(t) = AjU (t), U (0) = U0,

where U0 = (ϕ0,ϕ1,ψ0,ψ1, θ0, θ1)′ and Aj :D(Aj) ⊂ Hj → Hj is the differential operator

Aj =

⎛⎜⎜⎜⎜⎜⎝

0 Id 0 0 0 0kρ−1

1 ∂2x 0 κρ1

−1∂x 0 0 00 0 0 Id 0 0

−kρ−12 ∂x 0 ρ−1

2 (b∂2x − κId) 0 0 −βρ−1

2 ∂x0 0 0 0 0 Id0 0 0 −γρ−1

3 ∂x δρ−13 ∂2

x Kρ−13 ∂2

x

⎞⎟⎟⎟⎟⎟⎠

with domain

D(A1) ={U ∈ H1 | ϕ,ψ ∈ H2(0,L) ∩H1

0 (0,L), δθ +KΘ ∈ H2(0,L),Φ,Ψ ∈ H10 (0,L)

}and

D(A2) ={U ∈ H2 | ϕ ∈ H2(0,L) ∩H1

0 (0,L), δθ +KΘ ∈ H2(0,L),

Φ,ψx ∈ H10 (0,L),Ψ ∈ H1

∗(0,L)}.

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When the boundary conditions make no difference in the definitions we denote Hj and Aj by H and Arespectively.

Lemma 2.1. The operator A generates a C0-semigroup of contractions T (t) = eAt on H .

Proof. Note that A is dissipative, that is

Re(AU ,U )H = −Kβ

∫ L

0|Θx|2 dx � 0. (2.1)

It is not difficult to see that 0 ∈ ρ(A) and therefore by the Lummer–Phillips Theorem (see [19], Theo-rem 4.3) the operator A generates a C0-semigroup of contractions on H . In fact to show that 0 ∈ ρ(A) itis enough to show that for any F ∈ H , there exists only one solution U ∈ D(A) of AU = F . In termsof the components we have

−Φ = f1, (2.2)

−κ(ϕx + ψ)x = ρ1f2, (2.3)

−Ψ = f3, (2.4)

−bψxx + κ(ϕx + ψ) + βΘx = ρ2f4, (2.5)

−Θ = f5, (2.6)

−δθxx + γΨx −KΘxx = ρ3f6. (2.7)

Therefore using (2.2), (2.4) and (2.6) we conclude that there exists only one θ such that

−δθxx = γf3,x −Kf5,xx + ρ3f6, θ(0) = θ(L) = 0.

Also it is not difficult to see that there exists only one solution ϕ and ψ of the system

−κ(ϕx + ψ)x = ρ1f2 ∈ L2(0,L), (2.8)

−bψxx + κ(ϕx + ψ) = βf5,x + ρ2f4 ∈ L2(0,L), (2.9)

from where we have that

‖U‖ � C‖F‖,

which in particular implies that ‖A−1F‖ � C‖F‖, so we have that 0 ∈ ρ(A). �

From now on we denote by c or C a positive constant that can be different in different places.Thus, we have the following result.

Theorem 2.2. Assume that (ϕ0,ϕ1,ψ0,ψ1, θ0, θ1) ∈ D(A), then there exists a unique solution(ϕ,ϕt,ψ,ψt, θ, θt) to the system (1.7)–(1.8) with boundary conditions (1.9) or (1.10) satisfying

(ϕ,ϕt,ψ,ψt, θ, θt) ∈ C([0,∞);D(A)

)∩ C1

([0,∞);H

).

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3. Exponential stability

In this section we will prove the exponential stability of the semigroup if and only if condition (1.12)holds. To prove this, we use the Gearhart–Herbst–Prüb-Huang result (see [20]) which gives a necessaryand sufficient condition to get exponential stability. That is, we have the following theorem.

Theorem 3.1. Let T (t) = eAt be a C0-semigroup of contractions on Hilbert space. Then T (t) is expo-nentially stable if and only if

ρ(A) ⊇ {iβ: β ∈ R} ≡ iR (3.1)

and

lim|β|→∞

∥∥(iβI −A)−1∥∥ < ∞. (3.2)

Proof. See e.g. [13,20]. �

The main result of this section is the following theorem.

Theorem 3.2. The semigroup T (t) = eAt associated to system (1.7)–(1.8) with boundary condition(1.10) is exponentially stable if and only if

ρ1

ρ2=

κ

b. (3.3)

For the case of Dirichlet–Dirichlet boundary condition the (1.9) the semigroup T (t) = eAt is exponen-tially stable if (3.3) is verified.

To show the above characterization, following lemmas are in order.

Lemma 3.3. Under the above notations we have that iR ⊂ ρ(A).

Proof. Note that A is a closed operator and D(A) has compact embedding over the phase space Hthen σ(A) are given only for eigenvalues. To prove iR ⊂ ρ(A) it is enough to show that there is noimaginary eigenvalues (see Theorem 6.29 in [11]). Let us reasoning by contradiction. Let us supposethat there exists an imaginary eigenvalue iλ with eigenvector U = (ϕ,Φ,ψ,Ψ , θ,Θ)′ = 0 verifyingiλU −AU = 0. From (3.10) with F = 0 we get Θx = 0. Using this in (3.8) we obtain θ = Θ = 0. Sofrom (3.6) and (3.9) it follows that ψx = Ψx = 0. Using (3.7) we get (ϕx + ψ)x = 0. From (3.4) and(3.5) we conclude that ϕ = Φ = 0.

Finally, from (3.6) and (3.7) we get ψ = Ψ = 0. Therefore, U = 0 which is a contradiction. �

To prove show that the resolvent operator is uniformly bounded over the imaginary axes, let usdenote F = (f1, f2, f3, f4, f5, f6)′. The resolvent equation (λI − A)U = F , with λ ∈ iR and

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U = (ϕ,Φ,ψ,Ψ , θ,Θ)′, in term of its components can be written as

λϕ− Φ = f1, (3.4)

λρ1Φ− κ(ϕx + ψ)x = ρ1f2, (3.5)

λψ − Ψ = f3, (3.6)

λρ2Ψ − bψxx + κ(ϕx + ψ) + βΘx = ρ2f4, (3.7)

λθ −Θ = f5, (3.8)

λρ3Θ − δθxx + γΨx −KΘxx = ρ3f6. (3.9)

With the boundary condition of Dirichlet type (1.9), that is to say

U = (ϕ,Φ,ψ,Ψ , θ,Θ)′ ∈ D(A1).

Or with the boundary condition of Neumann type (1.10), that is to say

U = (ϕ,Φ,ψ,Ψ , θ,Θ)′ ∈ D(A2).

From (2.1) it follows that there is a positive constant c, such that

‖Θx‖2L2 � c‖F‖H‖U‖H . (3.10)

The next lemma will be important in that follows.

Lemma 3.4. Under the above notations we have that there is a positive constant c, such that

‖Θ‖2L2 �

c

|λ|[‖Θx‖L2 + ‖Ψ‖L2 + ‖F‖H

]‖U‖1/2

H ‖F‖1/2H

for λ ∈ iR and |λ| � 1.

Proof. Using an interpolation inequality we get

‖Θ‖L2 � ‖Θ‖1/2H−1‖Θ‖1/2

H1 .

From Eqs (3.9) and (3.8) we find that

|λ|‖Θ‖H−1 � c[‖θx‖L2 + ‖Ψ‖L2 + ‖Θx‖L2 + ‖F‖H

]� c

[‖Ψ‖L2 + ‖Θx‖L2 + ‖F‖H

].

Using Poincare’s inequality and (3.10) our conclusion follows. �

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Lemma 3.5. Under the above notations the solution of the resolvent system (3.4)–(3.9) verifies

b‖ψx‖2L2 � c

[‖Ψ‖2

L2 + ‖U‖H‖F‖H +1|λ|‖ϕx + ψ‖L2‖Ψ‖L2

]

for a positive constant c and |λ| � 1.

Proof. Multiplying Eq. (3.7) by ψ and using (3.6), we obtain

b

∫ L

0|ψx|2 dx= ρ2

∫ L

0|Ψ |2 dx+ ρ2

∫ L

0Ψf3 dx+

κ

λ

∫ L

0(ϕx + ψ)Ψ dx

+κ

λ

∫ L

0(ϕx + ψ)f3 dx− β

∫ L

0Θψx dx− ρ2

∫ L

0f4ψ dx.

From the Cauchy–Schwartz, Poincaré, Young inequalities and (3.10) our conclusion follows. �

Lemma 3.6. Under the above notations we have that there is a positive constant c such that

ρ1‖Φ‖2L2 � c

[‖ϕx + ψ‖2

L2 + ‖Ψ‖L2‖U‖H + ‖F‖H‖U‖H]


Proof. Multiplying Eq. (3.5) by ϕ and integrating from 0 to L, it follows that∫ L

0

[λρ1Φ− κ(ϕx + ψ)x

]ϕ dx = ρ1

∫ L

0f2ϕ dx.

Using Eq. (3.4), we get

ρ1

∫ L

0|Φ|2 dx = κ

∫ L

0|ϕx + ψ|2 dx− κ

∫ L

0(ϕx + ψ)ψ dx− ρ1

∫ L

0f1Φ dx+ ρ1

∫ L

0f2ϕ dx.

Recalling the definition of norm in H , we have

ρ1

∫ L

0|Φ|2 dx � κ‖ϕx + ψ‖2

L2 + κ‖ϕx + ψ‖L2‖ψ‖L2︸︷︷︸:=I1

+ c‖F‖H‖U‖H .

Using the Poincaré and Young inequalities in I1, we obtain

ρ1

∫ L

0|Φ|2 dx � c‖ϕx + ψ‖2

L2 + c‖ψx‖2L2 + c‖F‖H‖U‖H .

Applying Lemma 3.5, our conclusions follows. �

For the case Dirichlet–Dirichlet–Dirichlet boundary condition we need the following results.

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Lemma 3.7. Under the above notations we have that there is a positive constant c such that

∣∣ϕx(L)∣∣2 + ∣∣ϕx(0)

∣∣2 � c[‖ϕx + ψ‖2

L2 + ‖Ψ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H]

and ∣∣ψx(L)∣∣2 + ∣∣ψx(0)

∣∣2 � c[‖ψx‖L2‖ϕx + ψ‖L2 + ‖Ψ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

].

Proof. We consider p(x) = x − L2 . Multiplying Eq. (3.5) by (ϕx + ψ)p and integrating over [0,L], we

have

λρ1

∫ L

0Φ(ϕx + ψ)p dx− κ

∫ L

0(ϕx + ψ)x(ϕx + ψ)p dx = ρ1

∫ L

0f2(ϕx + ψ)p dx. (3.11)

Note that, from Eqs. (3.4), (3.6) and integrating by parts, we find

Re

{λρ1

∫ L

0Φ(ϕx + ψ)p dx

}= Re

{−ρ1

∫ L

0Φ(Φx + Ψ )p dx− ρ1

∫ L

0Φ(f1,x + f3)p dx

}� c

[‖Φ‖2

L2 + ‖Ψ‖L2‖U‖H + ‖F‖H‖U‖H].

Using Lemma 3.6, we get

Re

{λρ1

∫ L

0Φ(ϕx + ψ)p dx

}� c

[‖ϕx + ψ‖2

L2 + ‖Ψ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H].

Integrating by parts, we obtain the following inequality

Re

{κ

∫ L

0(ϕx + ψ)x(ϕx + ψ)p dx

}

=κL

4

(∣∣ϕx(t,L)∣∣2 + ∣∣ϕx(t, 0)

∣∣2)− κ

2

∫ L

0|ϕx + ψ|2 dx. (3.12)

Taking the real part in (3.11) and using the above inequality, we obtain the first estimate. Now, multiply-ing Eq. (3.7) by ψxp where p(x) = x− L

2 and integrating over [0,L], we have

∫ L

0

[λρ2Ψ − bψxx + κ(ϕx + ψ) + βΘx

]pψx dx = ρ2

∫ L

0f4ψxw dx. (3.13)

Using (3.6) and integrating by parts, we obtain

Re

{λρ1

∫ L

0Ψψxp dx

}� ρ2

2

∫ L

0|Ψ |2 dx+ c‖F‖H‖U‖H . (3.14)

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On the other hand,

Re

{b

∫ L

0ψxxψxp dx

}=

bL

4

(∣∣ψx(t,L)∣∣2 + ∣∣ψx(t, 0)

∣∣2)− b

2

∫ L

0|ψx|2 dx (3.15)

and from the Cauchy–Schwartz inequality and definition of norm in H , we get

Re

{κ

∫ L

0(ϕx + ψ)ψxp dx

}� c‖ψx‖L2‖ϕx + ψ‖L2 . (3.16)

Taking the real part in (3.13), using (3.14)–(3.16) and Lemma 3.5 we obtain our conclusion. �

To simplify notations we denote

ξ =

∣∣∣∣ρ2 −b

kρ1

∣∣∣∣.Lemma 3.8. Under the above notations we have that there is a positive constant c, such that

κ‖ϕx + ψ‖2L2 � c

[‖Ψ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

]+ c

(|λ|2ξ2 + 1

)‖Ψ‖2

L2


Proof. Multiplying Eq. (3.7) by ϕx + ψ and integrating from 0 to L, we find that

κ

∫ L

0|ϕx + ψ|2 dx =

∫ L

0[−λρ2Ψ + bψxx − βΘx + ρ2f4](ϕx + ψ) dx.

Integrating by parts, using Eqs. (3.4)–(3.6), we have

κ

∫ L

0|ϕx + ψ|2 dx= ρ2

∫ L

0Ψ [Φx + Ψ + f1,x + f3] dx− b

k

∫ L

0ψx[λρ1Φ− ρ1f2] dx

+ bϕxψx|L0 − β

∫ L

0Θx(ϕx + ψ) dx+ ρ2

∫ L

0f4(ϕx + ψ) dx.

Now, using Eq. (3.5) and the definition of norm in H , we have

κ

∫ L

0|ϕx + ψ|2 dx�

∣∣∣∣(ρ2 −

b

kρ1

)∫ L

0ΨΦx dx

∣∣∣∣︸︷︷︸:=I2

+∣∣bϕxψx|L0

∣∣

+ c[‖Ψ‖2

L2 + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H]. (3.17)

Now, we estimate the term I2. For this, we derive for the variable x Eq. (3.4) and add with (3.6), and sowe get the following equality

Φx + Ψ = λ(ϕx + ψ) − (f1,x + f3). (3.18)

AUTHOR COPY


Thus, from (3.18), we obtain that∣∣∣∣ξ∫ L

0ΨΦx dx

∣∣∣∣ � [ξ|λ|‖Ψ‖L2‖ϕx + ψ‖L2︸︷︷︸

:=I3

+ cξ‖F‖H‖U‖H].

Applying the Young inequality in I3, we get∣∣∣∣ξ∫ L

0ΨΦx dx

∣∣∣∣ � c|λ|2ξ2‖Ψ‖2L2 +

κ

2‖ϕx + ψ‖2

L2 + c‖F‖H‖U‖H . (3.19)

Substituting this inequality in (3.17), we have

κ‖ϕx + ψ‖2L2 � c

[‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

]+ c

(|λ|2ξ2 + 1

)‖Ψ‖2

L2 + c∣∣ϕxψx|x=L

x=0

∣∣. (3.20)

Using Young inequality, we conclude that for any ε > 0 there exists a positive constant cε, such that∣∣ϕxψx|x=Lx=0

∣∣ � ε[∣∣ϕx(t,L)

∣∣2 + ∣∣ϕx(t, 0)∣∣2]+ cε

[∣∣ψx(t,L)∣∣2 + ∣∣ψx(t, 0)

∣∣2].From Lemmas 3.7 and 3.5, we get∣∣ϕxψx|x=L

x=0

∣∣ � ε‖ϕx + ψ‖2L2 + cε

[‖Ψ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

].

Inserting this inequality in (3.20), for ε small enough, we have our conclusion. �

Lemma 3.9. Under the above notations we have that for λ ∈ iR and |λ| � 1 there is a positive constantsc such that

‖Ψ‖2L2 � c

[‖Θx‖L2‖U‖H + ‖Θ‖L2‖U‖H + ‖F‖H‖U‖H

]+

c

|λ|1/2

[‖Θx‖1/2

L2 ‖U‖3/2H + ‖F‖1/2

H ‖U‖3/2H + ‖F‖3/2

H ‖U‖1/2H + ‖F‖2

H

](3.21)

if we consider Dirichlet boundary condition (1.9). For the case Neumann boundary condition (1.10) weget

‖Ψ‖2L2 � c

[‖Θx‖L2‖U‖H + ‖Θ‖L2‖U‖H + ‖F‖H‖U‖H

]. (3.22)

Proof. First we consider the boundary conditions given by (1.9). Let us denote by

p(x) =∫ x

0Ψ ds.

Multiplying Eq. (3.9) by p and integrating over [0,L], we get

λρ3

∫ L

0Θp dx− δ

∫ L

0θxxp dx+ γ

∫ L

0Ψxp dx−K

∫ L

0Θxxp dx = ρ3

∫ L

0f6p dx.

AUTHOR COPY


Since p(0) = 0, we can rewrite the above equation as

γ

∫ L

0|Ψ |2 dx= λρ3

∫ L

0Θp dx︸︷︷︸

:=I4

−[δθx(t,L)p(L) +KΘx(t,L)p(L)

]︸︷︷︸:=I5

+ δ

∫ L

0θxΨ dx+K

∫ L

0ΘxΨ dx− ρ3

∫ L

0f6p dx. (3.23)

Let us denote by ω the function such that{−ωxx = Θ in (0,L),ω = 0, x = 0,L.

Then we have that

I4 = λρ3ωx(L)p(L) − λρ3

∫ L

0ωxΨ dx. (3.24)

Note that∣∣λρ3ωx(L)p(L)∣∣ � c|λ|‖Θ‖H−1

∣∣p(L)∣∣. (3.25)

On the other hand, from Eq. (3.7) and Lemma 3.7 we get∣∣p(L)∣∣� c

|λ|[(∣∣ψx(L)

∣∣+ ∣∣ψx(0)∣∣)+ ‖U‖H + ‖F‖H

]� c

|λ|[‖U‖H + ‖F‖H

]. (3.26)

Using (3.26) in (3.25) and using Lemma 3.7, we obtain∣∣λρ3ωx(L)p(L)∣∣ � c‖Θ‖H−1

[‖U‖H + ‖F‖H

]. (3.27)

Now, using Eq. (3.6), it follows that

λ

∫ L

0ωxΨ dx=

1ρ2

∫ L

0ωx

[bψxx − κ(ϕx + ψ) − βΘx + ρ2f4

]dx

=b

ρ2ωxψx|L0 − b

ρ2

∫ L

0Θxψ dx− 1

ρ2

∫ L

0ωx

[κ(ϕx + ψ) + βΘx − ρ2f4

]dx.

Note that, from Lemma 3.7, we have that

b

ρ2

∣∣ωxψx|L0∣∣� c‖Θ‖H−1

[|ψx(L)|+ |ψx(0)|

]� c‖Θ‖H−1

[‖U‖H + ‖F‖H

]

AUTHOR COPY


and thus∣∣∣∣λρ3

∫ L

0ωxΨ dx

∣∣∣∣ � c[‖Θ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖U‖H‖F‖H

]. (3.28)

From the estimates (3.27) and (3.28),

|I4| � c[‖Θ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

].

Finally we will estimate the term I5. Multiplying Eq. (3.9) by q := x(δθx +KΘx) and integrating over[0,L], we get

λρ3

∫ L

0Θq dx︸︷︷︸

I6

−∫ L

0(δθxx +KΘxx)q dx︸︷︷︸

I7

+ γ

∫ L

0Ψxq dx︸︷︷︸I8

= ρ3

∫ L

0f6q dx. (3.29)

From Eq. (3.8), for |λ| � 1, we get

Re{I6} � c|λ|[‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

].

Now, integrating by parts and using (3.8), we have

Re{I7} = c∣∣δθx(L) +KΘx(L)

∣∣2 + c

∫ L

0

∣∣∣∣ δλ (Θx + f5,x) +KΘx

∣∣∣∣2 dx.

Using Eqs. (3.6) and (3.8), for |λ| � 1, we obtain

Re{I8} � c|λ|[‖Θx‖L2‖U‖H + ‖F‖H‖U‖H + ‖F‖2

H

].

Therefore, taking the real part in (3.29) and replacing the estimates of I6, I7 and I8, for |λ| � 1, it followsthat ∣∣δθx(L) +KΘx(L)

∣∣2 � c|λ|[‖Θx‖L2‖U‖H + ‖F‖H‖U‖H + ‖F‖2

H

].

From the previous estimate we get

|I5| � c|λ|1/2[‖Θx‖1/2

L2 ‖U‖1/2H + ‖F‖1/2

H ‖U‖1/2H + ‖F‖H

]∣∣p(L)∣∣.

Using (3.10) and (3.26), we get

|I5| �c

|λ|1/2

[‖Θx‖1/2

L2 ‖U‖3/2H + ‖F‖1/2

H ‖U‖3/2H + ‖F‖3/2

H ‖U‖1/2H + ‖F‖H‖U‖H + ‖F‖2

H

].

Therefore, our conclusion follows for the case boundary conditions (1.9). In the case (1.10) we havethat p(L) = 0 together with (3.23), (3.24) and (3.28) we get (3.22). The proof of the lemma is nowcomplete. �

Now we are in the position to prove the main result of this section.

AUTHOR COPY


Proof of Theorem 3.2. Let us show the sufficient condition. From Lemma 3.3 it follows that the imag-inary axis is contained in the resolvent of the operator. It remains to show that the resolvent of theoperator is uniformly bounded over the imaginary axis. To that end let U = (ϕ,Φ,ψ,Ψ , θ,Θ)′ andF = (f1, f2, f3, f4, f5, f6)′ satisfy (3.4)–(3.9). Multiplying Eqs. (3.5) and (3.7) by γ

λΦ and γλΨ respec-

tively adding the product result, we get for |λ| > 1

γκ‖ϕx + ψ‖2L2 + γb‖ψx‖2

L2 � c

[‖Φ‖2 + ‖Ψ‖2 +

1|λ|‖Θx‖L2‖U‖H +

1|λ|‖F‖H‖U‖H

]� c

[‖Φ‖2 + ‖Ψ‖2 + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

].

Now, summing β‖θx‖2L2 , βρ3‖Θ‖2

L2 , γρ1‖Φ‖2L2 and γρ2‖Ψ‖2

L2 , we obtain

‖U‖2H � c

[‖Φ‖2

L2 + ‖Ψ‖2L2 + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

].

Using Lemma 3.6, we get

‖U‖2H � c

[‖ϕx + ψ‖2

L2 + ‖Ψ‖L2‖U‖H + ‖Ψ‖2L2 + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

].

From Lemma 3.8, we have

‖U‖2H � c

[‖Ψ‖L2‖U‖H + ‖Θx‖L2‖U‖H + ‖F‖H‖U‖H

]+ c

(|λ|2ξ2 + 1

)‖Ψ‖2

L2 .

Applying the Young inequality we have that for any ε > 0, there exists a positive constant cε, such that

‖U‖2H � cε‖Θx‖2

L2 + cε‖F‖2H + cε

(|λ|2ξ2 + 1

)‖Ψ‖2

L2 .

Using (3.10) and applying the Young inequality we get that there exists a positive constant c such that

‖U‖2H � c‖F‖2

H + c(|λ|2ξ2 + 1

)‖Ψ‖2

L2 . (3.30)

From the hypothesis, we have that ξ = 0. From Lemma 3.9 and applying the Young inequality successivetimes, we obtain using Lemma 3.4 and (3.10)

‖U‖H � c‖F‖H

for a positive constant c. Thus, we obtain∥∥(λId −A)−1∥∥ � c.

Therefore from Lemma 3.3, the system is exponentially stable.To show that condition (3.3) is also necessary we use the boundary condition (1.10). We will show

that there exists a sequence of complex numbers (λμ)μ∈N ⊂ iR with limμ→∞ |λμ| = ∞ and a boundedsequence (Fμ)μ∈N ⊂ H , such that ‖(λμI −A)−1‖ → ∞ when λμ → ∞.

For any μ ∈ N, we take a bounded sequence Fμ = (0, ρ−11 sin(μπ

L x), 0, 0, 0, 0)′.

AUTHOR COPY


Let us denote by Uμ = (ϕμ,Φμ,ψμ,Ψμ, θμ,Θμ)′ the solution of the resolvent equation (λμId−A)Uμ =Fμ, that is

λμϕμ − Φμ = 0, (3.31)

λμρ1Φμ − κ(ϕμ,x + ψμ)x = sin

(μπ

Lx

), (3.32)

λμψμ − Ψμ = 0, (3.33)

λμρ2Ψμ − bψμ,xx + κ(ϕμ,x + ψμ) + βΘμ,x = 0, (3.34)

λμθμ −Θμ = 0, (3.35)

λρ3Θμ − δθμ,xx + γΨμ,x −KΘμ,xx = 0. (3.36)

Eliminating Φμ, Ψμ and Θμ, we obtain the following system for ϕμ, ψμ and θμ

ρ1λ2μϕμ − κ(ϕμ,x + ψμ)x = sin

(μπ

Lx

), (3.37)

ρ2λ2μψμ − bψμ,xx + κ(ϕμ,x + ψμ) + βλμθμ,x = 0, (3.38)

ρ3λ2μθμ − δθμ,xx + γλμψμ,x −Kλμθμ,xx = 0. (3.39)

Due to the boundary conditions, we can assume that the solution has the form

ϕμ(x) = Aμ sin

(μπ

Lx

), ψμ(x) = Bμ cos

(μπ

Lx

), θμ(x) = Cμ sin

(μπ

Lx

), (3.40)

where Aμ = Aμ(μ), Bμ = Bμ(μ), Cμ = Cμ(μ) and will be determined. In order to satisfy (3.37)–(3.39)with the solution (3.40), it is necessary and sufficient that the coefficients Aμ, Bμ and Cμ satisfy

Aμp1(λμ) +Bμk

(μπ

L

)= 1, (3.41)

Aμκ

(μπ

L

)+Bμp2(λμ) − Cμβλμ

(μπ

L

)= 0, (3.42)

Cμp3(λμ) −Bμγλμ

(μπ

L

)= 0, (3.43)

where

p1(λμ) := ρ1λ2μ + κ

(μπ

L

)2

,

p2(λμ) := ρ2λ2μ + b

(μπ

L

)2

+ κ,

p3(λμ) := ρ3λ2μ + [δ +Kλμ]

(μπ

L

)2

.

AUTHOR COPY


Note that under this conditions we have

Aμ =p2p3 − γβλ2

μ(μπL )2

p1p2p3 − γβλ2μ(μπ

L )2p1 − κ2(μπL )2p3

.

At this point, we define the sequence (λμ)μ∈N ⊂ iR such that p1 = c, that is

λμ = ±i

√κ

ρ1

(μπ

L

)2

− c

ρ1≈ iμ, (3.44)

where c := κ2[b− ρ2ρ1κ]−1.

Note that, using the hypothesis on the coefficients we can write

p2(λμ) =

[b− ρ2

ρ1κ

](μπ

L

)2

+ cρ2

ρ1+ κ. (3.45)

Now, combining (3.44)–(3.45) we deduce

p1p2p3 − γβλ2μ

(μπ

L

)2

p1 − κ2

(μπ

L

)2

p3 = p3

[p1p2 − κ2

(μπ

L

)2]− γβλ2

μ

(μπ

L

)2

p1

= p3

[κ2

[b− ρ2

ρ1κ

]−1(cρ2

ρ1+ κ

)]− γβλ2

μ

(μπ

L

)2

p1

≈ cμ4 (at least)

and

p2p3 − γβλ2μ

(μπ

L

)2

≈ cμ5 (at least).

Then we obtain that Aμ is (at least) of the order of μ. Finally, recalling the definition of ϕμ, we knowthat

‖Uμ‖2H � ρ1

∫ L

0|Φμ|2 dx = ρ1|λμ|2

∫ L

0

∣∣∣∣Aμ sin

(μπ

Lx

)∣∣∣∣2 dx

� C|λμ|2|Aμ|2 ≈ cμ4 (at least), (3.46)

which implies

limμ→∞

‖Uμ‖H = ∞.

Therefore the system is not exponentially stable. �

AUTHOR COPY


4. Polynomial stability and optimality

In the previous section, we showed that the solution of the system does not decay exponentially whenrelationship ρ1

ρ2= k

b is satisfied. Now, we show that there is polynomial decay for the solution of thissystem and the decay rate is optimal. Our result is based on the following theorem (see [2], Theorem 2.4).

Theorem 4.1. Let T (t) = eAt be a bounded C0-semigroup on a Hilbert space H with generator A suchthat iR ⊂ ρ(A). Then for any α > 0 and x ∈ H we have∥∥R(λ,A)

∥∥ = O(|λ|α

), |λ| → ∞ ⇐⇒

∥∥T (t)A−1x∥∥H

= o(t−1/α

), t → ∞. (4.1)

Our main result is summarized in the next theorem.

Theorem 4.2. Let us suppose that ρ1ρ2

= kb , then the semigroup associated to system (1.7) with boundary

conditions (1.10) decays polynomially to zero as time goes to infinity, that is, there exists a positiveconstant c, such that∥∥T (t)U0

∥∥H

� c

t12

‖U0‖D(A).

Moreover, the rate of decay is optimal for any initial data taken in D(A).

Proof. The main idea to show the polynomial decay is to estimate ‖Ψ‖L2 in terms of ‖Θ‖L2 to getthe uniform estimate of the resolvent operator in terms of λ. To show the optimality we use inequality(3.44) where the choice of λμ, such that p1(λμ) = c was the key point of the calculations. Now ξ =

|ρ2 − bkρ1| = 0. From Lemma 3.3 we know that the imaginary axis is contained into the resolvent of the

operator. From (3.30) we have that

‖U‖2H � c‖F‖2

H + c|λ|2‖Ψ‖2L2

for |λ| � 1. Using (3.22) we have

‖U‖2H � c|λ|2

[‖Θ‖L2‖ϕx + ψ‖H−1 + ‖U‖H‖F‖H

]+ ε‖ϕx + ψ‖2

L2 + c‖F‖2H

which implies

‖U‖2H � c|λ|2

[‖Θ‖L2‖ϕx + ψ‖H−1 + ‖U‖H‖F‖H

]+ c‖F‖2

H .

Applying the Young inequality we have that for any ε1 > 0, there exists a positive constant cε1 , such that

‖U‖2H � c|λ|2

[cε1‖Θ‖2

L2 + ε1‖ϕx + ψ‖2H−1 + ‖U‖H‖F‖H

]+ c‖F‖2

H .

Note that from (3.4) and (3.6) we get

|λ|‖ϕx + ψ‖H−1 � c[‖Φ‖L2 + ‖Ψ‖H−1 + ‖F‖H−1

]� c

[‖U‖H + ‖F‖H

].

AUTHOR COPY


From this

‖U‖2H � c|λ|2

[‖Θ‖2

L2 + ‖U‖H‖F‖H]+ c‖F‖2

H .

Using Lemma 3.4 implies

‖U‖2H � c‖F‖2

H + c|λ|2‖U‖H‖F‖H + c|λ|‖Θx‖L2‖U‖1/2H ‖F‖1/2

H

+ c|λ|‖Ψ‖L2‖U‖1/2H ‖F‖1/2

H + c|λ|‖U‖1/2H ‖F‖3/2

H .

Finally, using (3.10) and (3.22) we obtain

‖U‖2H � c‖F‖2

H + c|λ|2‖U‖H‖F‖H + c|λ|‖U‖H‖F‖H+ c|λ|‖U‖3/2

H ‖F‖1/2H + c|λ|‖U‖1/2

H ‖F‖3/2H .

Applying the Young inequality successive times we get

‖U‖2H � c|λ|4‖F‖2

H

which is equivalent to∥∥R(λ,A)∥∥ � O

(|λ|2

)as |λ| → ∞.

Now, in order to prove the inverse inequality we use contradiction arguments. Suppose that O(|λ|2) isnot optimal, there exists δ > 0 such that∥∥R(λ,A)

∥∥ = O(|λ|2−δ

)as |λ| → ∞, (4.2)

which implies that, for all F ∈ H , there exists C > 0 such that

1|λ|2−δ

‖U‖H � C‖F‖H ∀λ ∈ iR, λ = 0,

where U ∈ H is the solution of the resolvent equation (λI −A)U = F in H .This is a contradiction because, taking advantage of Theorem 3.2 we can construct sequences

(λμ)μ∈N ⊂ iR, (Uμ)μ∈N ⊂ D(A) and (Fμ)μ∈N ⊂ H such that

‖Uμ‖2H � c|λμ|2‖Fμ‖2

H (see inequality (3.46)),

which implies that

1|λμ|2−δ

∥∥R(λμ,A)∥∥ � c0|λμ|δ → ∞ (μ → ∞),

contradicting (4.2).

AUTHOR COPY


Therefore we have∥∥R(λ,A)∥∥ = O

(|λ|2

)as |λ| → ∞.

Using Theorem 4.1 for α = 2 and Lemma 3.3 we have∥∥T (t)A−1x∥∥H

= o(t−1/2

), t → ∞, ∀x ∈ H ,

where T (t)t�0 is the C0-semigroup generated by a on H . Finally, for U0 ∈ D(A) (U0 = 0) choosingx := ‖U0‖−1

D(A)AU0 ∈ H our conclusion follows. �

Acknowledgement

L.H. Fatori was supported by Fundação Araucária/SETI.

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AUTHOR COPY


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Energy decay to Timoshenko’s system with thermoelasticity