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Volume 26, N. 2, pp. 215–234, 2007
Copyright © 2007 SBMAC
ISSN 0101-8205
www.scielo.br/cam

A transmission problem for the Timoshenko system

C.A. RAPOSO1, W.D. BASTOS2 and M.L. SANTOS3

1Department of Mathematics, UFSJ, Praça Frei Orlando, 170

36307-352 São João del-Rei, MG
2Department of Mathematics, UNESP, Rua Cristovão Colombo, 2265

15054-000 São José do Rio Preto, SP
3Department of Mathematics, UFPA, Rua do Una, 156 – 66113-200 Pará, PA

E-mails: raposo@ufsj.edu.br / waldemar@ibilce.unesp.br /mlsantos@ufpa.br

Abstract. In this work we study a transmission problem for the model of beams developed

by S.P. Timoshenko [10]. We consider the case of mixed material, that is, a part of the beam

has friction and the other is purely elastic. We show that for this type of material, the dissipation

produced by the frictional part is strong enough to produce exponential decay of the solution,

no matter how small is its size. We use the method of energy to prove exponential decay for

the solution.

Mathematical subject classification:35J55, 35J77, 93C20.

Key words: transmission, Timoshenko, beams, exponential decay, frictionaldamping.

1 Introduction

The transverse vibration of a beam is mathematically described by a system of

two coupled differential equations given by

ρutt − (K (ux + ψ))x = 0, in (0, L)× (0,∞),

Iρψt t − (E Iψx)x + K (ux + ψ) = 0, in (0, L)× (0,∞).
(1.1)

Here, L is the length of the beam in its equilibrium position,t is the time

variable andx is the space coordinate along the beam. The functionu = u(x, t)

#668/06. Received: 23/I/07. Accepted: 09/II/07.



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216 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

is the transverse displacement of the beam andψ = ψ(x, t) is the rotation angle

of a filament of the beam. The coefficientsρ, Iρ, E, I andK are the mass per

unit length, the polar moment of inertia of a cross section, Young’s modulus

of elasticity, the moment of inertia of a cross section and the shear modulus

respectively. We denoteρ1 = ρ, ρ2 = Iρ, b = E I, k = K and we obtain

directly from (1.1) the following system

ρ1utt − k(ux + ψ)x = 0, in (0, L)× (0,∞),

ρ2ψt t − b2ψxx + k(ux + ψ) = 0, in (0, L)× (0,∞).
(1.2)

The mathematical model describing the vibrations of beam with fixed extrem-

ities is formed by the system (1.2), boundary conditions

u(0, t) = u(L , t) = ψ(0, t) = ψ(L , t) = 0, t > 0,

and initial data

u(∙, 0) = φ0, ut(∙, 0) = φ1, ψ(∙, 0) = ψ0, ψt(∙, 0) = ψ1, in (0, L).

If friction is taken into account, the system(1.2) becomes

ρ1utt − k(ux + ψ)x + αut = 0, in (0, L)× (0,∞),

ρ2ψt t − b2ψxx + k(ux + ψ)+ βψt = 0, in (0, L)× (0,∞),
(1.3)

whereα andβ are positive constants (we assumeα = β = 1). The termsαut

andβψt represent the attrition acting in the vertical vibrations and in the angle

of rotation of the filaments of the beam, respectively.

Dissipative properties associated to the system(1.3) have been studied by sev-

eral authors by considering dissipative mechanism of frictional or viscoelastic

type. The frictional dissipation, obtained by introduction of a frictional mech-

anism acting on the entire domain or on the boundary, was studied in [7, 8, 9].

The viscoelastic dissipation, given by a memory effect, was considered in [2]

and [6].

An interesting problem comes out when the dissipation acts only on a part of

the domain. In the present paper we consider a frictional mechanism acting only

on the part of the domain given by 0≤ x ≤ L0 with 0 < L0 < L . We prove

that for everyL0 the energy of the system decays exponentially to zero as time

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C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 217

goes to infinity. In other words, our result states that dissipative properties of the

system are transferred to the whole beam and stabilizes the system. The main

result of this paper is Theorem 2 and its corollary, both in section 5.

The mathematical model which deals with this situation is called a transmission

problem. From the mathematical point of view a transmission problem consist

of an initial and boundary value problem for a hyperbolic equation for which the

corresponding elliptic operator has discontinuous coefficients. Hence, we can

not expect to have regular solutions in the role domain. In the next section we

establish the transmission problem and define appropriately the notion of solu-

tion considered. We useHm andL p to denote the usual Sobolev and Lebesgue

spaces [1].

2 The transmission problem

In this section we describe precisely the transmission problem treated in the paper

and establish existence and regularity of solution. We begin by introducing the

notation

ρ j (x) =

{
ρ1

j , if x ∈ (0, L0)

ρ2
j , if x ∈ (L0, L)

}

,

k(x) =

{
k1, if x ∈ (0, L0)

k2, if x ∈ (L0, L)

}

,

b(x) =

{
b1, if x ∈ (0, L0)

b2, if x ∈ (L0, L)

}

,

α =

{
1, if x ∈ (0, L0)

0, if x ∈ (L0, L)

}

,

β =

{
1, if x ∈ (0, L0)]

0, if x ∈ (L0, L)

}

,

u(x, t) =

{
u(x, t), if x ∈ (0, L0)× (0,∞)

v(x, t), if x ∈ (L0, L)× (0,∞)

}

,

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218 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

ψ(x, t) =

{
ψ(x, t), if x ∈ (0, L0)× (0,∞)

φ(x, t), if x ∈ (L0, L)× (0,∞)

}

.

Using the notation above, model (1.3) can be written in the following form:

ρ1
1utt − k1(ux + ψ)x + ut = 0, in (0, L0)× (0,∞), (2.1)

ρ1
2ψt t − b1ψxx + k1(ux + ψ)+ ψt = 0, in (0, L0)× (0,∞), (2.2)

ρ2
1vt t − k2(vx + φ)x = 0, in (L0, L)× (0,∞), (2.3)

ρ2
2φt t − b2φxx + k2(vx + φ) = 0, in (L0, L)× (0,∞), (2.4)

L0 L−L0

Dissipative part

u(x), ψ(x) v(x), φ(x)

Elastic part

with boundary conditions,

u(0, t) = v(L , t) = ψ(0, t) = φ(L , t) = 0, t > 0,

transmission conditions,

k1u(L0, t) = k2v(L0, t), ρ1
1ut(L0, t) = ρ2

1vt(L0, t),

k1ux(L0, t) = k2vx(L0, t), k1ψ(L0, t) = k2φ(L0, t),

ρ1
2ψt(L0, t) = ρ2

2φt(L0, t), b1ψx(L0, t) = b2φx(L0, t),

(2.5)

and initial data

u(∙, 0) = u0, ut (∙, 0) = u1, ψ(∙, 0) = ψ0, ψt (∙, 0) = ψ1, in (0, L0),

v(∙, 0) = v0, vt (∙, 0) = v1, φ(∙, 0) = φ0, φt (∙, 0) = φ1, in (L0, L).
(2.6)

We define the notion of weak solution to the system (2.1)-(2.6) as follows:

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C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 219

Definition 1. LetV ,Hm andL2 be the spaces defined by

V =
{
(w̄, w) ∈ H1(0, L0)× H1(L0, L); w̄(0) = w(L) = 0, w̄(L0) = w(L0)

}

Hm = Hm(0, L0)× Hm(L0, L) and L2 = L2(0, L0)× L2(L0, L).

We say that(u, v, ψ, φ) is a weak solution to the problem(2.1)-(2.6) if for

every(w̄, w) ∈ H1
0 (0, T;H 2 ∩V ) we have:

ρ1
1

∫ L0

0
ut(x, T)w̄(x, T)dx − ρ1

1

∫ L0

0
ut(x, 0)w̄(x, 0)dx

+ ρ2
1

∫ L

L0

vt(x, T)w(x, T)dx − ρ2
1

∫ L

L0

vt(x, 0)w(x, 0)dx

− ρ1
1

∫ T

0

∫ L0

0
ut(x, t)w̄t(x, t)dxdt− ρ2

1

∫ T

0

∫ L

L0

vt(x, t)wt(x, t)dxdt

+
∫ T

0

∫ L

L0

vt(x, t)wx(x, t)dxdt+
∫ T

0

∫ L0

0
k1(ux + ψ)(x, t)w̄x(x, t)dxdt

+
∫ T

0

∫ L

L0

k2(vx + φ)(x, t)wx(x, t)dxdt = 0

and

ρ2
1

∫ L0

0
ψt(x, T)w̄(x, T)dx − ρ2

1

∫ L0

0
ψt(x, 0)w̄(x, 0)dx

+ ρ2
2

∫ L

L0

φt(x, T)w(x, T)dx − ρ2
2

∫ L

L0

φt(x, 0)w(x, 0)dx

− ρ1
2

∫ T

0

∫ L0

0
ψt(x, t)w̄t(x, t)dxdt− ρ2

2

∫ T

0

∫ L

L0

φt(x, t)wt(x, t)dxdt

+
∫ T

0

∫ L0

0
b1ψ(x, t)xw̄(x, t)dxdt+

∫ T

0

∫ L0

0
ψt(x, t)w̄x(x, t)dxdt

+
∫ T

0

∫ L0

0
k1(ux + ψ)w̄(x, t)dxdt+ b2

∫ T

0

∫ L

L0

φx(x, t)wx(x, t)dxdt

+
∫ T

0

∫ L

L0

k2(vx + φ)(x, t)w(x, t)dxdt = 0.

The transmission problem for a single hyperbolic equation was studied by

Dautray and Lions [3], who proved the existence and regularity of solutions for

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220 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

the linear problem. The existence and regularity of solutions to the transmission

problem for the Timoshenko system is given in the following theorem:

Theorem 1. If (u0, v0), (ψ0, φ0) ∈ V and(u1, v1), (ψ1, φ1) ∈ L2, then there

exists a unique weak solution(u, v, ψ, φ) to the system(2.1)-(2.6) satisfying:

(u, v), (ψ, φ) ∈ C(0,∞;V ) ∩ C1(0,∞;L2).

Moreover, if(u0, v0), (ψ0, φ0) ∈ H 2 ∩V and(u1, v1), (ψ1, φ1) ∈ V , then the

weak solution is a strong solution and satisfies

(u, v), (ψ, φ) ∈ C(0,∞;H 2 ∩V ) ∩ C1(0,∞;V ) ∩ C2(0,∞;L2).

Proof. For the proof we proceed in a quite similar manner as in [3]. �

The total energy associated to the system is defined by

E(t) =
1

2

∫ L0

0

{
ρ1

1|ut |
2 + ρ1

2|ψt |
2 + b1|ψx|

2 + k1|ux + ψ |2
}

dx

+
1

2

∫ L

L0

{
ρ2

1|vt |
2 + ρ2

2|φt |
2 + b2|φx|

2 + k2|v
2
x + φ|2

}
dx.

Next we prove that the total energy associated to the system is decreasing for

everyt > 0.

Lemma 1. Let (u, v, ψ, φ) be the strong solution to the system(2.1)-(2.6),

then
d

dt
E(t) = −

∫ L0

0
|ut |

2dx −
∫ L0

0
|ψt |

2dx.

Proof. Multiplying (2.1) byut and integrating by parts over the interval(0, L0),

we get
d

dt

ρ1
1

2

∫ L0

0
|ut |

2dx = k1 (ux(L0)+ ψ(L0)) ut(L0)

− k1

∫ L0

0
(ux + ψ) utxdx −

∫ L0

0
|ut |

2dx.

(2.7)

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C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 221

Now, multiplying (2.2) byψt and integrating by parts over(0, L0) we obtain

d

dt

{
ρ1

2

2

∫ L0

0
|ψt |

2dx +
b1

2

∫ L0

0
|ψx|

2dx

}
= b1ψx(L0)ψt(L0)

− k1

∫ L0

0
(ux + ψ)ψtdx −

∫ L0

0
|ψt |

2dx.

(2.8)

Multiplying (2.3) byvt and integrating by parts on(L0, L), we get

ρ2
1

2

d

dt

∫ L

L0

|vt |
2dx = − k2(vx + φ)(L0)vt(L0)− k2

∫ L

L0

(vx + φ) vt xdx. (2.9)

Multiplying (2.4) byφt and integrating by parts on(L0, L) leads to

d

dt

{
ρ2

2

2

∫ L

L0

|φt |
2dx +

b2

2

∫ L

L0

|φx|
2dx

}
= −b2φx(L0)φt(L0)

− k2

∫ L

L0

(vx + φ) φtdx.

(2.10)

Now observe that

k1

2

d

dt

∫ L0

0

∣
∣φ1

x + ψ1
∣
∣2

dx = k1

∫ L0

0

(
φ1

x + ψ1
) (
φ1

xt + ψ1
t

)
dx, (2.11)

and

k2

2

d

dt

∫ L

L0

∣
∣φ2

x + ψ2
∣
∣2

dx = k2

∫ L

L0

(
φ2

x + ψ2
) (
φ2

xt + ψ2
t

)
dx. (2.12)

Summing up (2.7), (2.8), (2.9) and (2.10), and using (2.11) and (2.12) together

with the hypothesis of transmission we obtain

d

dt
E(t) = −

∫ L0

0
|ut |

2dx −
∫ L

L0

|ψt |
2dx.

3 Technical lemmas

Now we develop a series of technical results in order to facilitate the proof of the

main result of the paper. We begin by constructing a functionalE(t), equivalent

to the energy functional, which satisfiesE(t) ≤ CE(0), C < 1. In order to do

so, we use some multiplier techniques (usually associated to control problems)

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222 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

and the following restrictions on the boundary conditions for the elastic part of

the beam:

|vx(L)|
2 ≤

1

2L

∫ L

L0

|vx|
2 dx. (3.1)

|φx(L)|
2 ≤

1

2L

∫ L

L0

|φx|
2 dx. (3.2)

Lemma 2. Let us define

E1(t) = N1 E(t)+
∫ L0

0
x (ρ1

1uxut +ρ
1
2ψxψt) dx+

∫ L

L0

x (ρ2
1vxvt +ρ

2
2φxφt) dx.

Then

d

dt
E1(t) ≤ −

1

4

∫ L0

0
[ k1|ux|

2 + b1|ψx|
2 ] dx −

1

4

∫ L

L0

[ k2|vx|
2 + b2|φx|

2 ] dx.

Proof. Multiply (2.1) by x ux and integrate by parts over(0, L0) to get

d

dt

∫ L0

0
ρ1

1x ut ux dx =
L0ρ

1
1

2
|ut(L0)|

2 −
ρ1

1

2

∫ L0

0
|ut |

2 dx

+
L0k1

2
|ux(L0)|

2 −
k1

2

∫ L0

0
|ux|

2 dx

+ k1

∫ L0

0
xψx ux dx −

∫ L0

0
x ut ux dx.

(3.3)

Multiply (2.2) by xψx and integrate by parts over(0, L0) to obtain

d

dt

∫ L0

0
ρ1

2xψt ψx dx =
L0ρ

1
2

2
|ψt(L0)|

2 −
ρ1

2

2

∫ L0

0
|ψt |

2 dx

+
L0b1

2
|ψx(L0)|

2 −
b1

2

∫ L0

0
|ψx|

2 dx

+
L0k1

2
|ψ(L0)|

2 −
k1

2

∫ L0

0
|ψ |2 dx

− k1

∫ L0

0
xψx ux dx −

∫ L0

0
xψt ψx dx.

(3.4)

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C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 223

Multiplying (2.3) byx vx and integrating by parts over(L0, L) leads to

d

dt

∫ L

L0

ρ2
1x vt vx dx = −

L0ρ
2
1

2
|vt(L0)|

2 −
ρ2

1

2

∫ L

L0

|vt |
2 dx

+
Lk2

2
|vx(L)|

2 −
L0k2

2
|vx(L0)|

2

−
k2

2

∫ L

L0

|vx|
2 dx + k2

∫ L

L0

x φx vx dx.

(3.5)

Now multiply (2.4) byx φx and integrate over(0, L0) to obtain

d

dt

∫ L

L0

ρ2
2x φt φx dx = −

L0ρ
2
2

2
|φt(L0)|

2 −
ρ2

2

2

∫ L

L0

|φt |
2 dx

+
Lb2

2
|φx(L)|

2 −
L0b2

2
|φx(L0)|

2

−
b2

2

∫ L

L0

|φx|
2 dx −

L0k2

2
|φ(L0)|

2

−
k2

2

∫ L

L0

|φx|
2 dx − k2

∫ L

L0

x φx vx dx.

(3.6)

Summing up (3.3), (3.4), (3.5), (3.6), and making use of the hypothesis on the
transmission, the punctual terms are canceled and we get

d

dt

∫ L0

0
x (ρ1

1uxut + ρ1
2ψxψt ) dx +

d

dt

∫ L

L0

x (ρ2
1vxvt + ρ2

2φxφt ) dx

=
Lk2

2
|vx(L)|

2 +
Lb2

2
|φx(L)|

2 −
k1

2

∫ L0

0
|ux|

2 dx −
∫ L0

0
xutuxdx

−
b1

2

∫ L0

0
|ψx|

2 dx +
∫ L0

0
xψtψxdx −

k2

2

∫ L

L0

|vx|
2 dx −

b2

2

∫ L

L0

|φx|
2 dx.

Now using (3.1), (3.2) and the Young’s inequality [4] we obtain

d

dt

∫ L0

0
x (ρ1

1uxut + ρ1
2ψxψt ) dx +

d

dt

∫ L

L0

x (ρ2
1vxvt + ρ2

2φxφt ) dx

≤
k2

4

∫ L

L0

|vx|
2 dx −

k2

2

∫ L

L0

|vx|
2 dx +

b2

4

∫ L

L0

|φx|
2 dx −

b2

2

∫ L

L0

|φx|
2 dx

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224 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

−
k1

2

∫ L0

0
|ux|

2 dx +
k1

4

∫ L0

0
|ux|

2 dx −
b1

2

∫ L0

0
|ψx|

2 dx +
b1

4

∫ L0

0
|ψx|

2 dx

+ C̄

[ ∫ L0

0
|ψt |

2 dx +
∫ L0

0
|ut |

2 dx

]
.

Now, if we define

E1(t) = N1 E(t)+
∫ L0

0
x

(
ρ1

1uxut +ρ
1
2ψxψt

)
dx+

∫ L

L0

x
(
ρ2

1vxvt +ρ
2
2φxφt

)
dx

and chooseN1 > C̄ we conclude

d

dt
E1(t) ≤ −

1

4

∫ L0

0

[
k1|ux|

2 + b1|ψx|
2
]

dx−
1

4

∫ L

L0

[
k2|vx|

2 + b2|φx|
2
]

dx.

It is worth noticing that the estimate above is important in two aspects. First,

it recovers a part of energy with minus sign. Second, it will play a fundamental

role in the next two lemmas controlling punctual terms which will come up in

the search for other negative terms of the energy.

Lemma 3. Define

E2(t) = N2 E(t)+
∫ L0

0
ρ1

1x ut(ux + ψ) dx.

Then

d

dt
E2(t) ≤

ρ1
1 L0

2
|ut(L0)|

2 +
k1L0

2
|ux(L0)+ ψ(L0)|

2

− C
∫ L0

0

[
|ut |

2 + |ψt |
2 + |ux + ψ |2

]
dx.

Proof. Multiply (2.1) byx (ux+ψ) and integrate by parts over(0, L0) to obtain

d

dt

∫ L0

0
x ρ1

1(ux + ψ)ut =
ρ1

1 L0

2
|ut (L0)|

2 −
ρ1

1 L0

2

∫ L0

0
|ut |

2 dx

+
k1L0

2
|ux(L0)+ ψ(L0)|

2 −
k1L0

2

∫ L0

0
|ux + ψ |2 dx

+ ρ1
1

∫ L0

0
x ut ψt dx −

∫ L0

0
x ut (ux + ψ) dx.

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C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 225

Using Young’s inequality, we get

d

dt

∫ L0

0
x ρ1

1(ux + ψ)ut ≤
ρ1

1 L0

2
|ut(L0)|

2 +
k1L0

2
|ux(L0)+ ψ(L0)|

2

−
k1L0

4

∫ L0

0
|ux + ψ |2 dx + C̃

∫ L0

0
|ut |

2 dx + C̃
∫ L0

0
|ψt |

2 dx.

Now, defining

E2(t) = N2 E(t)+
∫ L0

0
ρ1

1x ut(ux + ψ) dx

we obtain

d

dt
E2(t) ≤

ρ1
1 L0

2
|ut(L0)|

2 +
k1L0

2
|ux(L0)+ ψ(L0)|

2

−
k1L0

4

∫ L0

0
|ux + ψ |2 dx + (C̃ − N2)

∫ L0

0
|ut |

2 dx + (C̃ − N2)

∫ L0

0
|ψt |

2 dx.

If we chooseN2 > C̃ we conclude that there exists̆C > 0 such that

d

dt
E2(t) ≤

ρ1
1 L0

2
|ut(L0)|

2 +
k1L0

2
|ux(L0)+ ψ(L0)|

2

−
k1L0

4

∫ L0

0
|ux + ψ |2 dx − C̆

∫ L0

0
|ut |

2 dx − C̆
∫ L0

0
|ψt |

2 dx,

and then

d

dt
E2(t) ≤

ρ1
1 L0

2
|ut(L0)|

2 +
k1L0

2
|ux(L0)+ ψ(L0)|

2

− C
∫ L0

0

[
|ut |

2 + |ψt |
2 + |ux + ψ |2

]
dx.

Lemma 4. LetE3 be defined as

E3(t) =
ρ2

1 2 L2

ρ2
2

∫ L

L0

x (ρ1
2vxvt +ρ

1
2φxφt) dx+

∫ L

L0

(x−L)
[
ρ2

1vt(vx + φ)
]

dx.

Then

d

dt
E3(t) ≤

ρ2
1(L − L0)

2
|vt(L0)|

2 +
k2(L − L0)

2
|vx(L0)+ φ(L0)|

2

− C
∫ L0

0

[
|vt |

2 + |φt |
2 + |vx + φ|2

]
dx.

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226 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

Proof. Multiply (2.3) by (x − L)(vx + φ) and integrate by parts over(L0, L)
to get

d

dt

∫ L

L0

(x − L)
[
ρ2

1vt (vx + φ)
]

dx =
ρ2

1(L − L0)

2
|vt (L0)|

2 −
ρ2

1

2

∫ L0

0
|vt |

2 dx

+
k2(L − L0)

2
|vx(L0)+ φ(L0)|

2

−
k2

2

∫ L

L0

|vx + φ|2 dx

+ ρ2
1

∫ L

L0

(x − L)vt φt dx.

Using Young’s inequality we obtain

d

dt

∫ L

L0

(x − L)
[
ρ2

1vt (vx + φ)
]

dx ≤
ρ2

1(L − L0)

2
|vt (L0)|

2

+
k2(L − L0)

2
|vx(L0)+ φ(L0)|

2

−
ρ2

1

2

∫ L0

0
|vt |

2 dx +
ρ2

1

2

∫ L0

0
|vt |

2 dx

−
k2

2

∫ L

L0

|vx + φ|2 dx +
L2ρ2

1

2

∫ L

L0

|φt |
2 dx.

Now we set

E3(t) =
ρ2

1 2 L2

ρ2
2

∫ L

L0

x (ρ1
2vxvt +ρ

1
2φxφt) dx+

∫ L

L0

(x − L)
[
ρ2

1vt(vx + φ)
]

dx

and verify that

d

dt
E3(t) ≤

ρ2
1(L − L0)

2
|vt(L0)|

2 +
k2(L − L0)

2
|vx(L0)+ φ(L0)|

2

−
k2

2

∫ L

L0

|vx + φ|2 dx −
L2ρ2

1ρ
2
1

ρ2
2

∫ L

L0

|vt |
2 dx

+
(

L2ρ2
1

2
− L2ρ2

1

) ∫ L

L0

|φt |
2 dx.

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C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 227

It follows then

d

dt
E3(t) ≤

ρ2
1(L − L0)

2
|vt(L0)|

2 +
k2(L − L0)

2
|vx(L0)+ φ(L0)|

2

− C
∫ L0

0
[ |vt |

2 + |φt |
2 + |vx + φ|2 ] dx.

Observe that in the attempt to recover the total energy of the system with

negative sign we introduced the lemmas 3 and 4. Now we need to control them.

It will be achieved with the aid of the next section.

4 Compactness

This section is dedicated to discuss the argument of compactness employed in the

proof of the main result of the paper. First we introduce a notation; the symbol

⇀ is used to denote convergence in the norm of the Sobolev spaceL∞ as in [5].

For sake of completeness, we state the following result due to J.U. Kim.

Lemma 5. Let (uk) be a sequence of functions satisfying

uk ⇀ u in L∞(0, T, Hβ(0, L)),

uk
t ⇀ ut in L2(0, T, Hα(0, L)),

ask → ∞, with α < β. Then

uk → u in C([0, T], Hr (0, L)),

for somer < β.

Proof. See [5].

Lemma 6 (Lemma of compactness). If we define

B(L0, t) =
ρ1

1 L0

2
|ut(L0)|

2 +
k1L0

2
|ux(L0)+ ψ(L0)|

2

+
ρ2

1(L − L0)

2
|vt(L0)|

2 +
k2(L − L0)

2
|vx(L0)+ φ(L0)|

2

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“main” — 2007/6/27 — 17:40 — page 228 — #14

228 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

then, for everyη > 0 there exists a constantCη > 0 independent of the initial

data, such that
∫ T

0
B(L0, t)dt ≤ η

∫ T

0
E(t)dt + Cη

{ ∫ T

0

∫ L0

0

[
|ux|

2 + |ψx|
2
]

dxdt

+
∫ T

0

∫ L

L0

[
|vx|

2 + |φx|
2
]

dxdt

}

for every strong solution(u, v, ψ, φ) to the system(2.1)-(2.6) and sufficiently

large T .

Proof. We use a contradiction argument. Define

Bn(L0, t) =
ρ1

1 L0

2
|un

t (L0)|
2 +

k1L0

2
|un

x(L0)+ ψn(L0)|
2

+
ρ2

1(L − L0)

2
|vn

t (L0)|
2 +

k2(L − L0)

2
|vn

x(L0)+ φn(L0)|
2.

Suppose that there exists a sequence of initial data

(un
0, ψ

n
0 ) ∈ H 2 ∩V , (vn

0, φ
n
0) ∈ H 2 ∩V , (un

1, ψ
n
1 ) ∈ V , (vn

1, φ
n
1) ∈ V ,

and a positive constantη0 > 0 such that the corresponding solution(un, ψn),

(vn, φn) of the problem

ρ1
1un

tt − k1(u
n
x + ψn)x + un

t = 0 in (0, L0)× (0,∞),

ρ1
2ψ

n
tt − b1ψ

n
xx + k1(u

n
x + ψn)+ ψn

t = 0 in (0, L0)× (0,∞),

ρ2
1v

n
tt − k2(v

n
x + φn)x = 0 in (L0, L)× (0,∞),

ρ2
2φ

n
tt − b2φ

n
xx + k2(v

n
x + φn) = 0 in (L0, L)× (0,∞),

with boundary conditions,

un(0, t) = vn(L , t) = ψn(0, t) = φn(L , t) = 0, t > 0,

transmission conditions,

k1u
n(L0, t) = k2v

n(L0, t), ρ1
1un

t (L0, t) = ρ2
1v

n
t (L0, t),

k1u
n
x(L0, t) = k2v

n
x(L0, t), k1ψ

n(L0, t) = k2φ
n(L0, t),

ρ1
2ψ

n
t (L0, t) = ρ2

2φ
n
t (L0, t), b1ψ

n
x (L0, t) = b2φ

n
x(L0, t),

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“main” — 2007/6/27 — 17:40 — page 229 — #15

C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 229

and initial data

un(∙, 0) = un
0, ut (∙, 0) = un

1, ψn(∙, 0) = ψn
0 , ψn

t (∙, 0) = ψn
1 in (0, L0),

vn(∙, 0) = vn
0, vn

t (∙, 0) = vn
1, φn(∙, 0) = φn

0, φn
t (∙, 0) = φn

1 in (L0, L),

satisfies ∫ T

0
Bn(L0, t)dt = 1, n ∈ N, (4.1)

and the following inequality

1 > η0

∫ T

0
En(t)dt + n

{ ∫ T

0

∫ L0

0

[
|un

x|
2 + |ψn

x |2
]

dx dt

+
∫ T

0

∫ L

L0

[
|vn

x |
2 + |φn

x |
2
]

dx dt

}
.

Then the integral
∫ T

0
En(t)dt is bounded for every n ∈ N,

and also,
∫ T

0

∫ L0

0

[
|un

x|
2 + |ψn

x |2
]

dx dt → 0 as n → ∞, (4.2)

and ∫ T

0

∫ L

L0

[
|vn

x |
2 + |φn

x |
2
]

dx dt → 0 as n → ∞. (4.3)

Now we observe thatEn(t) > 0 and that
∫ T

0
En(t)dt is bounded. HenceEn(t)

is bounded and we can take a subsequence of(un, ψn), (vn, φn) (for which we

use the same notations) such that

un ⇀ u in L∞(0, T,H 2 ∩V ),

ψn ⇀ ψ in L∞(0, T,H 2 ∩V ),

vn ⇀ v in L∞(0, T,V ),

φn ⇀ φ in L∞(0, T,V ).

Applying the lemma 5 we conclude that forr < 1

un → u in C([0, T]; Hr (0, L0)),

ψn → ψ in C([0, T]; Hr (0, L0)),

vn → v in C([0, T]; Hr (L0, L)),

φn → φ in C([0, T]; Hr (L0, L)).

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“main” — 2007/6/27 — 17:40 — page 230 — #16

230 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

It follows from (4.1) that ∫ T

0
B(L0, t) = 1. (4.4)

We observe that the convergences (4.2) and (4.3) result in

ux = 0 almost everywhere in (0, L0)× (0, T),

ψx = 0 almost everywhere in (0, L0)× (0, T),

vx = 0 almost everywhere in (L0, L)× (0, T),

φx = 0 almost every where in (L0, L)× (0, T).

Now, applying Poincare’s inequality we obtain
∫ T

0
|u(L0)|

2dt ≤ c2
p

∫ T

0

∫ L0

0
|ux|

2dxdt = 0,

∫ T

0
|ux(L0)|

2dt ≤ c2
p

∫ T

0

∫ L0

0
|uxx|

2dxdt = 0,

∫ T

0
|ψ(L0)|

2dt ≤ c2
p

∫ T

0

∫ L0

0
|ψx|

2dxdt = 0,

∫ T

0
|v(L0)|

2dt ≤ c2
p

∫ T

0

∫ L

L0

|vx|
2dxdt = 0,

∫ T

0
|vx(L0)|

2dt ≤ c2
p

∫ T

0

∫ L

L0

|vxx|
2dxdt = 0,

∫ T

0
|φ(L0)|

2dt ≤ c2
p

∫ T

0

∫ L

L0

|φx|
2dxdt = 0.

This estimates implies ∫ T

0
B(L0, t) = 0,

which is a contradiction to (4.4). This completes the proof of the lemma.

We are now ready to prove the main result of this paper, that is, the exponential

decay of the energy associated to the transmission problem for the Timoshenko

System with frictional dissipation. This is the content of the next section.

5 Exponential decay

Theorem 2. Let (u, ψ, v, φ) be a strong solution to the transmission problem

for the Timoshenko System defined by(2.1)-(2.6). Then there exist positive

Comp. Appl. Math., Vol. 26, N. 2, 2007



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C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 231

constantsC andw such that

E(t) ≤ C E(0)e−ωt .

Proof. We start defining

E(t) = N3E1(t)+ E2(t)+ E3(t).

It follows from lemmas 2, 3, and 4

d

dt
E(t) ≤ − C0 E(t)− C1 N3

{ ∫ L0

0

[
|ux|

2 + |ψx|
2
]

dx

+
∫ L

L0

[
|vx|

2 + |φx|
2
]

dx

}
.+ B(L0, t).

Now, integrating this inequality over(0, T) and using the Lemma of Compact-

ness we obtain

E(T)− E(0) ≤ − C0

∫ T

0
E(t) dt + η

∫ T

0
E(t) dt

− C1 N3

{ ∫ T

0

∫ L0

0

[
|ux|

2 + |ψx|
2
]

dx dt

+
∫ T

0

∫ L

L0

[
|vx|

2 + |φx|
2
]

dx dt

}

+ Cη

{ ∫ T

0

∫ L0

0

[
|ux|

2 + |ψx|
2
]

dxdt

+
∫ T

0

∫ L

L0

[
|vx|

2 + |φx|
2
]

dxdt

}
.

If we choose and fixη < C0 andN3 such thatN3C1 > Cη, we get

E(T)− E(0) ≤ − C2

∫ T

0
E(t)dt. (5.1)

SinceE(t) decreases, we have

T E(T) ≤
∫ T

0
E(t)dt. (5.2)

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“main” — 2007/6/27 — 17:40 — page 232 — #18

232 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

Using (5.2) in (5.1) we obtain

E(T)− E(0) ≤ − T C2E(T).

Now observe that for sufficiently largeN we have

N

2
E(t) ≤ E(t) ≤ 2 N E(t), (5.3)

from what follows that

E(T)− E(0) ≤ −
C2

2N
T E(T),

or else

E(T) ≤ α E(0) with α =
[
1 +

C2

2N

]−1

.

Note thatα does not depend on the initial data, and hence, by using the semigroup

property we have

E(t + T) ≤ α E(t) for every t > 0. (5.4)

For t > 0, there exists a naturaln and a realr , 0 ≤ r < T such thatt = nT + r .

This is equivalent to

n =
t

T
−

r

T
.

Now, using the inequalities (5.3) and (5.4)n times we obtain

E(t) ≤ αn E(r )

≤ 2Nαn E(r ).

Observing once more thatE(t) decreases we have

E(t) ≤ 2Nα(
t
T − r

T ) E(0)

≤ 2Nα−1 E(0) e−w t ,

wherew = − ln
(
α

1
T

)
.

Finally, using (5.3) we obtain

E(t) ≤ 4α−1 E(0) e−w t ,

and conclude the proof.

We can extend the previous theorem to the weak solutions by using simple

density argument and the laws of semi-continuity for the energy functional. In

this direction we have the following corollary.

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“main” — 2007/6/27 — 17:40 — page 233 — #19

C.A. RAPOSO, W.D. BASTOS and M.L. SANTOS 233

Corollary 1. Under the hypothesis of the previous theorem, there exist positive

constantsC andw, such that

E(t) ≤ C E(0)e−wt ,

for every weak solution(u, ψ, v, φ) of the system(2.1)-(2.6).

6 Concluding remarks

During the past several decades, many authors have studied the same physical

phenomenon for the Timoshenko system formulated into different mathematical

models. Our approach to this problem is important not only from mathematical

but mainly from the physical point of view with applications in Mechanics,

amongst other branches of science. The system studied here is a model for

vibrating beams subjected to two frictional mechanisms. More precisely, we

proved that the presence of two frictional damping acting in a natural way on

a small part of the beam, is enough to stabilize the whole beam. Moreover,

it stabilizes quickly (at exponential rate). To the best of our knowledge, our

result is the first in this direction. In this sense, this work generalizes the results

previously obtained for Timoshenko’s system where attrition acting in the whole

beam was considered.

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234 A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM

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