Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 90, pp. 1–14.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

BRESSE SYSTEMS WITH LOCALIZED KELVIN-VOIGT

DISSIPATION

GABRIEL AGUILERA CONTRERAS, JAIME E. MUÑOZ-RIVERA

Abstract. We study the effect of localized viscoelastic dissipation for curved

beams. We consider a circular beam with three components, two of them

viscous with constitutive laws of Kelvin-Voigt type, one continuous and the
other discontinuous. The third component is elastic without any dissipative

mechanism. Our main result is that the rate of decay depends on the position

of each component. More precisely, we prove that the model is exponentially
stable if and only if the viscous component with discontinuous constitutive law

is not in the center of the beam. We prove that when there is no exponential
stability, the solution decays polynomially.

1. Introduction

We study a Bresse system that describes the vibrations of a circular arc beam
with three components positioned over intervals I1 =]0, `0[, I2 =]`0, `1[, I3 =]`1, `[.

We denote by Ĩ = I1 ∪ I2 ∪ I3. More precisely, we consider the system

ρ1ϕtt − Sx − lN = 0 in Ĩ × (0,+∞), (1.1)

ρ2ψtt −Mx + S = 0 in Ĩ × (0,+∞), (1.2)

ρ1wtt −Nx + lS = 0 in Ĩ × (0,+∞), (1.3)

where S, M and N stand for the shear force, bending moment, and axial force,
respectively. The constitutive law we use here are:

S = κ(ϕx + ψ + lw) + κ̃(ϕxt + ψt + lwt),

M = bψx + b̃ψxt,

N = K(wx − lϕ) + K̃(wxt − lϕt),

where w, ϕ and ψ are the longitudinal, vertical, and shear angle displacements.
Here, ρ1 = ρA, ρ2 = ρI, κ0 = EA, κ = k′GA, b = EI, and l = R−1, where ρ is
the density of the beam, E the elastic modulus, G the shear modulus, k′ the shear
factor, A is the cross-sectional area, I the second moment of area of the cross-section
and R is the radius of curvature of the beam. Here we assume that all the above
coefficients are positive constants. When the curvature l is zero, the model reduces

2010 Mathematics Subject Classification. 35B40, 35P05, 35Q74.
Key words and phrases. Bresse beam; transmission problem; exponential stability;

localized viscoelastic dissipative mechanism; polynomial stability.
©2021. This work is licensed under a CC BY 4.0 license.

Submitted March 22, 2021. Published November 4, 2021.

1



2 G. AGUILERA C., J. E. MUÑOZ-RIVERA EJDE-2021/90

to the well-known Timoshenko beam system. Therefore we can consider the Bresse
system as an extension of the Timoshenko’s model.

The functions κ̃, b̃, and K̃ are non negative of the following type:

κ̃ = κ0 + κ1, b̃ = b0 + b1, K̃ = K0 +K1,

where κ0, b0 and K0 are discontinuous functions over ]0, `[, positive over ID and
vanishing outside ID. Also κ1, b1 and K1 are C1 functions over ]0, `[, positive over
IC and vanishing outside IC . Figures 1–4 depict typical examples of the functions
κ̃, b̃, and K̃.

`0 `1 `
︸ ︷︷ ︸

IC

IE︷ ︸︸ ︷ ︸ ︷︷ ︸
ID

y = κ̃(x)

Figure 1. The discontinuous component is not centered.

In Figure 1, the discontinuous viscous material is not in the center, so there is
only one discontinuity point in `1. In Figure 2 the discontinuous viscous material
is in the center, we have two points of discontinuity, one at `0 and the second at `1.

`0 `1 `
︸ ︷︷ ︸

IC

IE︷ ︸︸ ︷︸ ︷︷ ︸
ID

y = κ̃(x)

Figure 2. The discontinuous component is centered.

Here we assume that there exist positive constants c, C1, and C2, such that

|b′1|2 ≤ c|b1|, |κ′1|2 ≤ c|κ1|, |K ′1|2 ≤ c|K1|, (1.4)

C1κ1 ≤ b1 ≤ C2κ1. (1.5)

The initial conditions

ϕ(·, 0) = ϕ0, ϕt(·, 0) = ϕ1, ψ(·, 0) = ψ0,

ψt(·, 0) = ψ1, w(·, 0) = w0, wt(·, 0) = w1
(1.6)

are given over (0, `) and we consider the Dirichlet boundary conditions

ϕ(0, t) = ϕ(`, t) = ψ(0, t) = ψ(`, t) = w(0, t) = w(`, t) = 0 in (0, +∞). (1.7)

Additionally, we have the transmission conditions

ϕ(`−i ) = ϕ(`+i ), ψ(`−i ) = ψ(`+i ), w(`−i ) = w(`+i ), (1.8)



EJDE-2021/90 BRESSE SYSTEMS 3

S(`−i ) = S(`+i ), M(`−i ) = M(`+i ), N(`−i ) = N(`+i ). (1.9)

for i = 0, 1. Note that condition (1.9) implies S,M,N ∈ H1(0, `). If we have more

points of discontinuity, the set Ĩ needs to be modified.

IE=Elastic Component IC=Visco Continuous ID=Visco Discontinuous

qPPPPPPi

R = 1/l

ID

IE
IC

qPPPPPPi

R = 1/l

ID

IC
IE

Figure 3. Two possible positions for the components of the beam,
in which the discontinuous part is not centered.

qPPPPPPi

R = 1/l

IC

ID
IE

Figure 4. In this case, the discontinuous part is in the center of
the beam.

It seems to us that the first study about the lack of the exponential stability to lo-
calized viscoelastic system is due to Liu and Liu [13], where the authors proved that
the wave equation with localized Kelvin-Voigt viscoelastic damping (with discontin-
uous constitutive law) is not exponentially stable. See also Cheng, Liu, and Liu [6],
and Tiang and Zhang [21] for similar results for Timoshenko’s model. On the other
hand, Liu and Rao in [14] proved that when the localized viscoelastic damping has
a C1-constitutive law, then the corresponding semigroup is exponentially stable.
Therefore, the regularity of the constitutive law of localized viscoelastic damping
completely changes the asymptotic properties. Concerning Timoshenko system we
have the work [16] where the authors consider the transmission problem of Tim-
oshenko’s beam composed by N components, each of them being either purely
elastic (E), or a Kelvin-Voigt viscoelastic material (discontinuous constitutive law
V), or an elastic material inserted with a frictional damping mechanism (F). The
authors prove that Timoshenko’s model is exponentially stable if and only if all
the elastic components are connected with one component with frictional damping.
Otherwise, there is no exponential stability, but a polynomial decay of the energy
as 1/t2. In [2] the authors obtained similar results for Timoshenko’s model.

Here we consider two types of localized viscoelastic damping, one with continuous
constitutive law and the other with discontinuous constitutive law. We prove that
the exponential stability depends on the order of the viscoelastic components of the
beam. That is, we show that the semigroup is exponentially stable if and only if
the discontinuous component is not in the center of the beam. Furthermore, in case



4 G. AGUILERA C., J. E. MUÑOZ-RIVERA EJDE-2021/90

of lack of exponential stability, we show that the semigroup decays polynomially to
zero.

The remainder part of this paper is organized as follows. In section 2 we establish
the well-posedness of the model. In section 3 we show the exponential stability
provided the discontinuous component is not in the center of the beam. Finally in
section 4 we show the lack of exponential stability.

2. Semigroup approach

The energy associated with system (1.1)-(1.3) is

E(t) =
1

2

∫ `

0

ρ1|ϕt|2 + ρ2|ψt|2 + ρ1|wt|2 + b|ψx|2 + κ|ϕx + ψ + lw|2

+K|wx − lϕ|2 dx.
(2.1)

Multiplying equations (1.1) by ϕt, (1.2) by ψt, and (1.3) by wt, and summing up
these products we arrive at

d

dt
E(t) = −

∫ `

0

b0|ψxt|2 dx+ κ0|ϕxt + ψt + lwt|2 +K0|wxt − lϕt|2 dx ≤ 0. (2.2)

The definition of the energy motivates us to define the phase space H as

H = H1
0 (0, `)× L2(0, `)×H1

0 (0, `)× L2(0, `)×H1
0 (0, `)× L2(0, `)

which is a Hilbert space with the norm

‖U‖2H =

∫ `

0

ρ1|Φ|2+ρ2|Ψ|2+ρ1|W |2+b|ψx|2+κ|ϕx+ψ+lw|2+K|wx−lϕ|2 dx (2.3)

where U = (ϕ,Φ, ψ,Ψ, w,W )t. Because of the Dirichlet boundary condition, the
above norm is equivalent to the usual norm of H. Denoting by ϕt = Φ, ψt = Ψ,
wt = W , system (1.1)-(1.3) can be written as

Ut −AU = 0, U(0) = U0 (2.4)

where

AU =
(
Φ ,

1

ρ1
(Sx + lN) , Ψ ,

1

ρ2
(Mx − S) , W ,

1

ρ1
(Nx − lS)

)
and

D(A) =
{
U ∈ H : Φ,Ψ,W ∈ H1

0 , κϕx + κ̃Φx, bψx + b̃Ψx, Kwx + K̃Wx ∈ H1}.
(2.5)

Note that the operator A is dissipative and

Re〈AU,U〉 = −
∫ `

0

b̃|Ψx|2 dx−
∫ `

0

κ̃|Φx + Ψ + lW |2 dx−
∫ `

0

K̃|Wx − lΦ|2 dx ≤ 0.

The resolvent system λU −AU = F with F = (f1, f2, f3, f4, f5, f6)t ∈ H in terms
of its components is given by

λϕ− Φ = f1, ρ1λΦ− Sx − lN = f2 in Ĩ × (0,+∞), (2.6)

λψ −Ψ = f3, ρ2λΨ−Mx + S = f4 in Ĩ × (0,+∞), (2.7)

λw −W = f5, ρ1λW −Nx + lS = f6 in Ĩ × (0,+∞). (2.8)

Following standard procedures (see [1, 8, 20]) we can show that 0 ∈ ρ(A), which
implies that A is the infinitesimal generator of a contraction semigroup.



EJDE-2021/90 BRESSE SYSTEMS 5

Theorem 2.1. Under the above conditions the operator A is the infinitesimal gen-
erator of a C0 semigroup of contraction. Hence, for any U0 ∈ H there exists only
one mild solution U ∈ C(0, T ;H) of problem (2.4). Also if U0 ∈ D(A), there exists
only one classic solution U ∈ C(0, T ;D(A)) ∩ C1(0, T ;H) of problem (2.4).

3. Asymptotic behavior

The main objective of this section is the characterization of the exponential and
polynomial stabilization of the system. The main tool are theorems due to Prüss
[18], Huang [10], Gearhart[9], and Borichev and Tomilov [5].

Theorem 3.1. Let S(t) be a contraction C0-semigroup, generated by A over a
Hilbert space H. Then, in Prüss [18] is established that there exists C, γ > 0 satis-
fying

‖S(t)‖ ≤ Ce−γt ⇔ iR ⊂ %(A) and ‖(iλ I −A)−1‖L(H) 6M, ∀λ ∈ R. (3.1)

For polynomial stability, Borichev and Tomilov [5] result establish that there exists
C > 0 such that

‖S(t)A−1‖ 6 C

t1/α
⇔ iR ⊂ %(A) and ‖(iλI −A)−1‖ 6M |λ|α, ∀λ ∈ R. (3.2)

Our first step is to prove that the semigroup associated with system (1.1)-(1.3) is
strongly stable. The resolvent equation iλU −AU = F in terms of its components
is given by

iλϕ− Φ = f1, (3.3)

iρ1λΦ− Sx − lN = f2, (3.4)

iλψ −Ψ = f3, (3.5)

iρ2λΨ−Mx + S = f4, (3.6)

iλw −W = f5, (3.7)

iρ1λW −Nx + lS = f6. (3.8)

Theorem 3.2. With the above notation we have iR ⊆ ρ(A).

Proof. Let us denote

N =
{
s ∈ R+ :]− is, is[⊂ ρ(A)

}
. (3.9)

Since 0 ∈ ρ(A), N 6= ∅. Putting σ = supN we have two possibilities: σ = +∞
which implies that iR ⊆ ρ(A), and 0 < σ finite. We will reason by contradiction.
Let us suppose that σ < ∞. Then, exists a sequence {λn} ⊆ R such that λn →
σ < +∞ and

‖(iλnI −A)−1‖L(H) → +∞
Hence, there exists a sequence {fn} ⊆ H such that ‖fn‖H = 1 and ‖(iλnI −
A)−1fn‖H → +∞. Denoting

Ũn = (iλnI −A)−1fn =⇒ fn = iλnŨn −AŨn

and Un = Ũn

‖Ũn‖
, Fn = fn

‖Ũn‖
we obtain

iλnUn −AUn = Fn → 0. (3.10)



6 G. AGUILERA C., J. E. MUÑOZ-RIVERA EJDE-2021/90

Since ‖AUn‖ ≤ C, it follows that Un is bounded in D(A). This implies in particular
that Φn, Ψn and Wn are bounded in H1

0 (0, `) and ϕ, ψ, w are bounded in H2(IE).
Then there exist subsequences such that:

Φn → Φ, Ψn → Ψ, Wn →W strongly in L2(0, `) (3.11)

ϕn,x + ψn + lwn → ϕx + ψ + lw,

ψn,x → ψx, wn,x − lϕn → wx − lϕ, strongly in L2(IE). (3.12)

Taking inner product we obtain

iλn‖Un‖2 − 〈AUn, Un〉 = 〈Fn, Un〉 → 0

and taking real part we arrive at

−Re〈AUn, Un〉 =

∫ `

0

(b|Ψn
x |2 + κ|Φnx + Ψn + lWn|2 +K|Wn

x − lΦn|2) dx→ 0.

This convergence implies

Ψn
x , Φnx + Ψn + lWn, Wn

x − lΦn → 0 strongly in L2(IC ∪ ID). (3.13)

Therefore,

ψnx , ϕ
n
x + ψn + lwn wnx − lϕn → 0 strongly in L2(IC ∪ ID). (3.14)

From (3.11), (3.12) and (3.14) it follows that Un → U strongly in H. Since A is
closed, we conclude that U satisfies

iσU −AU = 0.

Moreover, using the convergences (3.13)-(3.14) and the resolvent system, we obtain
U ≡ 0 over IC ∪ ID. Since IE = [α, β] is linked to IC or ID on α or β, we obtain
that U(α) = 0 or U(β) = 0. So on ]α, β[ we have

−ρ1σ2ϕ+ κ(ϕx + ψ + lw)x − lK(wx − lϕ) = 0,

−ρ2σ2ψ + bψxx + κ(ϕx + ψ + lw) = 0,

−ρ1σ2w +K(wx − lϕ)x + lκ(ϕx + ψ + lw) = 0,

with

ϕ(α) = ψ(α) = ϕx(α) = ψx(α) = w(α) = wx(α) = 0.

Looking the above equation as a second order initial value problem, we obtain
ϕ = ψ = w = 0 over ]α, β[. Hence U ≡ 0 on H, which is a contradiction. This
completes the proof. �

Remark 3.3. From the dissipativity of the operator A, we have∫ `

0

b0|Ψx|2 + κ0|Φx + Ψ + lW |2 +K0|Wx − lΦ|2 dx

= Re(U,F )H ≤ ‖U‖H‖F‖H.
(3.15)

The following lemma is crucial to prove the exponential stability of the system.

Lemma 3.4. Let us suppose that (1.4) and (1.5) hold. Then, the solution of (3.3)-
(3.8) satisfies∫

IC

κ1|λΦ|2 + b1|λΨ|2 dx+K1|λW |2 dx ≤ Cε‖U‖‖F‖+ Cε‖F‖2 + ε‖U‖2.



EJDE-2021/90 BRESSE SYSTEMS 7

Proof. Multiplying (3.4) by iλκ1Φ and integrating over [a, b],∫
IC

ρ1κ1|λΦ|2 dx =

∫
IC

(Sx + lN + f2)iλκ1Φ dx.

Recalling the definitions of S and N , on IC we obtain∫
IC

ρ1κ1|λΦ|2 dx

=

∫
IC

[κ(ϕx + ψ + lw) + κ1(Φx + Ψ + lW )]xiλκ1Φ dx+

∫
IC

f2iλκ1Φ dx

= G + G0 +

∫
IC

f2iλκ1Φ dx.

(3.16)

Where G =
∫
IC

[κ1(Φx + Ψ + lW )]iλ(κ′1Φ + κ1Φx) dx and G0 =
∫
IC

[κ(ϕx + ψ +

lw)]iλ(κ′1Φ + κ1Φx) dx. Then

G =

∫
IC

[κ1(Φx + Ψ + lW )]iλ(κ′1Φ + κ1(Φx + Ψ + lW )) dx

−
∫
IC

[κ1(Φx + Ψ + lW )]iλ(κ1Ψ + κ1lW ) dx.

(3.17)

Taking the real part of the above relation and using (3.15) we obtain

ReG = Re

∫
IC

[κ1(Φx + Ψ + lW )]iλ(κ′1Φ) dx

− Re

∫
IC

[κ1(Φx + Ψ + lW )]iλ(κ1Ψ + κ1lW ) dx

≤ ε‖λΦ‖2 + ε‖λΨ‖2 + ε‖λW‖2 + Cε‖U‖‖F‖.

(3.18)

Similarly, using (3.3), (3.5), (3.7), and (3.15), we obtain

ReG0 = Re

∫
IC

[κ(ϕx + ψ + lw)]iλ(κ′1Φ + κ1Φx) dx

≤ ε
∫
IC

|Φ|2 + |Ψ|2 + |W |2 dx+R.

(3.19)

Thus, substituting (3.18) and (3.19) in (3.16) yields∫
IC

κ1|λΦ|2 dx ≤ ε‖λΦ‖2 + ε‖λΨ‖2 + Cε‖U‖‖F‖, (3.20)

for |λ| > 1. Multiplying (3.6) by iλb1Ψ and multiplying (3.8) by iλK1W and using
the same above procedure we obtain∫

IC

ρ2b1|λΨ|2 dx ≤ ε‖λΨ‖2 + ε‖λW‖2 + Cε‖U‖‖F‖,∫
IC

ρ1K1|λW |2 dx ≤ ε‖λΦ‖2 + ε‖λW‖2 + Cε‖U‖‖F‖.

From the last three inequalities our conclusion follows. �

We introduce the notation

Eϕ =
(κqρ1)′

2
|Φ|2 +

q′

2
|S|2, Iϕ =

κqρ1
2
|Φ|2 +

q

2
|S|2, (3.21)



8 G. AGUILERA C., J. E. MUÑOZ-RIVERA EJDE-2021/90

Eψ =
(bqρ2)′

2
|Ψ|2 +

q′

2
|M |2, Iψ =

bqρ2
2
|Φ|2 +

q

2
|M |2, (3.22)

Ew =
(Kqρ1)′

2
|W |2 +

q′

2
|N |2, Iw =

Kqρ2
2
|W |2 +

q

2
|N |2 (3.23)

E = Eϕ + Eψ + Ew, I = Iϕ + Iψ + Iw (3.24)

and

L =

∫ b

a

E(s) ds−
∫ b

a

(ρ1κqΦΨ + ρ1κqlΦW + ρ1KqlWΦ)dx

+

∫ b

a

(qlSN̄ + qlS̄N + qSM̄)dx.

(3.25)

Taking q(x) = enx−ena

n we have q′(x) = enx � q(x), for n large, hence we obtain

C0

∫ b

a

E dx ≤ L ≤ C1

∫ b

a

E dx (3.26)

Remark 3.5. Recalling the definition of S and M we obtain∫ b

a

|S|2 dx ≤ C
∫ b

a

κ|ϕx + ψ + lw|2 dx+

∫ b

a

|κ̃(Φx + Ψ + lW )|2 dx.

Using the dissipative properties, we have∫ b

a

|S|2 dx ≤ C
∫ b

a

|ϕx + ψ + lw|2 dx+ C‖U‖‖F‖.

Similarly we have ∫ b

a

|M |2 dx ≤ C
∫ b

a

|ψx|2 dx+ C‖U‖‖F‖,∫ b

a

|N |2 dx ≤ C
∫ b

a

|wx − lψ|2 dx+ C‖U‖‖F‖.

Therefore, for n large we have∫ b

a

|Φ|2 + |ϕx+ψ+ lw|2 + |Ψ|2 + |ψx|2 + |W |2 + |wx− lψ|2 dx ≤
∫ b

a

E dx+C‖U‖‖F‖

and∫ b

a

E dx ≤ C
∫ b

a

|Φ|2+|ϕx+ψ+lw|2+|Ψ|2+|ψx|2+|W |2+|wx−lψ|2 dx+C‖U‖‖F‖.

Lemma 3.6. On [a, b] ⊂ IC ∪ IE we have∣∣∣L(s)− I(s)
∣∣b
a

∣∣∣ ≤ Cε‖U‖‖F‖+ Cε‖F‖2 + ε‖U‖2

Also on ID = [a, b] we have∣∣∣L − I(s)
∣∣b
a

∣∣∣ ≤ ε‖U‖2 + Cε|λ|2‖U‖‖F‖2 + C‖F‖2.

Proof. Multiplying (3.4) by qS̄, (3.6) by qM̄ , and (3.8) by qN̄ , we obtain

− ρ1κq

2

d

dx
|Φ|2 − q

2

d

dx
|S|2

= R1 + ρ1κqΦΨ + ρ1κqlΦW − iλρ1qκ̃Φ(Φx + Ψ + lW )− qlS̄N,
(3.27)



EJDE-2021/90 BRESSE SYSTEMS 9

−ρ2bq
2

d

dx
|Ψ|2,−q

2

d

dx
|M |2 = R2 − iλρ2qb̃ΨΨx − qSM̄, (3.28)

− ρ1Kq

2

d

dx
|W |2 − q

2

d

dx
|N |2

= R3 − ρ1KqlWΦ− iλρ1qK̃W (Wx − lΦ)− qlSN̄ .
(3.29)

Summing these equations we arrive at

− ρ1κq

2

d

dx
|Φ|2 − ρ2bq

2

d

dx
|Ψ|2 − ρ1Kq

2

d

dx
|W |2 − q

2

d

dx
|S|2 − q

2

d

dx
|M |2 − q

2

d

dx
|N |2

= R4 + ρ1κqΦΨ + ρ1κqlΦW − ρ1KqlWΦ− qlSN̄ − qlS̄N − qSM̄ + J(x)

where

J(x) = −iλρ1qκ̃Φ(Φx + Ψ + lW )− iλρ2qb̃ΨΨx − iλρ1qK̃W (Wx − lΦ)

and using the notation introduced above,

− d

dx
(I(x)) + E(x)

= R4 + ρ1κqΦΨ + ρ1κqlΦW − ρ1KqlWΦ− qlSN̄ − qlS̄N − qSM̄ + J(x).

Note that when [a, b] ⊂ IC ∪ IE , from Lemma 3.4 we obtain∣∣ ∫ b

a

J(x) dx
∣∣ ≤ Cε‖U‖‖F‖+ Cε‖F‖2 + ε‖U‖2. (3.30)

On ID we obtain∣∣ ∫
ID

J(x) dx
∣∣ ≤ ε‖U‖2 + Cε|λ|2‖U‖‖F‖+ ‖F‖2. (3.31)

After an integration using the above inequalities our conclusion follows. �

Let us denote

E(s) = ρ1|Φ|2 + ρ2|Ψ|2 + ρ1|W |2 + b|ψx|2 + κ|ϕx + ψ + lw|2 +K|wx − lϕ|2.

Theorem 3.7. The semigroup associated with system (1.1)–(1.3) is exponentially
stable if the viscous discontinuous part ID is not in the center of the beam.

Proof. Since ID is not in the center then 0 ∈ ID or ` ∈ ID, hence because of the
boundary conditions, Poincaré inequality is valid for Φ, Ψ and W . So, we have∫

ID

|Ψ|2 dx ≤ Cp
∫
ID

|Ψx|2 dx ≤ C‖U‖‖F‖. (3.32)

Using iλψ = Ψ + f3 and taking λ large we obtain∫ `

`1

|ψx|2 + |Ψ|2 dx ≤ C‖U‖‖F‖+ C‖F‖2

Using Poincaré’s and the triangular inequality we obtain∫
ID

|Φ|2 dx ≤ C
∫
ID

|Φx|2 dx ≤ C
∫
ID

κ|Φx+Ψ+lW |2+|Ψ|2+|lW |2 dx ≤ C‖U‖‖F‖

Similarly for W we have∫
ID

|W |2 dx ≤ C
∫
ID

|Wx|2 dx ≤ C‖U‖‖F‖.



10 G. AGUILERA C., J. E. MUÑOZ-RIVERA EJDE-2021/90

Hence ∫
ID

E(x) dx ≤ C‖U‖‖F‖+ C‖F‖2.

On the other hand, integrating (3.4), (3.6), and (3.8) over [a, b] ⊂ IC we obtain

iλρ1

∫ b

a

Φ dx− S(b−) + S(a+) + l

∫ b

a

N dx =

∫ b

a

f2 dx, (3.33)

iλρ2

∫ b

a

Ψ dx−M(b−) +M(a+) +

∫ b

a

S dx =

∫ b

a

f4 dx, (3.34)

iλρ1

∫ b

a

W dx−N(b−) +N(a+) + l

∫ b

a

S dx =

∫ b

a

f6 dx. (3.35)

From Lemma 3.6 we obtain∣∣ ∫
IC

Φ dx
∣∣ ≤ C

|λ|

(
[S(b−)− S(a+)] +

∣∣ ∫ b

a

lN dx
∣∣+
∣∣ ∫ b

a

f2 dx
∣∣)

≤ C

|λ|

(
‖U‖1/2‖F‖1/2 + ‖U‖+ ‖F‖

)
.

Similarly we arrive at∣∣ ∫
IC

Ψ dx
∣∣ ≤ C

|λ|

(
‖U‖1/2‖F‖1/2 + ‖U‖+ ‖F‖

)
, (3.36)

∣∣ ∫
IC

W dx
∣∣ ≤ C

|λ|

(
‖U‖1/2‖F‖1/2 + ‖U‖+ ‖F‖

)
. (3.37)

So we have ∫
IC

|Ψ|2 dx ≤ C
∣∣∣ ∫ b

a

Ψ dx
∣∣∣2 + C

∫ b

a

b1|Ψx|2 dx

≤ C‖U‖‖F‖+
C

|λ|2
‖U‖2 +

C

|λ|2
‖F‖2,∫

IC

|W |2 dx ≤ C
∣∣∣ ∫ b

a

W dx
∣∣∣2 + C

∫ b

a

b1|Wx|2 dx

≤ C‖U‖‖F‖+
C

|λ|2
‖U‖2 +

C

|λ|2
‖F‖2.

Using (3.3), (3.5), (3.7), and (3.15), we obtain∫
IC

E(x) dx ≤ C‖U‖‖F‖+
C

|λ|2
‖U‖2 + C‖F‖2. (3.38)

Since ID is not in the center of the beam we have IC∪IE = [0, `2] or IC∪IE = [`0, `].
Let us assume the latter case. Using the observability Lemma 3.6 we obtain∫

IC

E(x) dx ≤ CI(a) + C‖U‖‖F‖+
C

|λ|2
‖U‖2 + C‖F‖2

≤ C‖U‖‖F‖+ ε‖U‖2 + C‖F‖2,

for λ large. Using the observability over the interval [a, `] we arrive at∫
IC

E(x) dx ≤ C‖U‖‖F‖+ ε‖U‖2 + C‖F‖2. (3.39)



EJDE-2021/90 BRESSE SYSTEMS 11

From (3.32), (3.39), and (3.39), we obtain

‖U‖2 =

∫ `

0

|Φ|2 + |ϕx + ψ + lw|2 + |Ψ|2 + |ψx|2 + |W |2 + |wx − lϕ|2 dx

≤ C‖U‖‖F‖+ ε‖U‖2 + C‖F‖2,
(3.40)

from where we obtain that ‖U‖ ≤ C‖F‖. So our conclusion follows. �

We conclude this section by showing the polynomial decay when the discontin-
uous viscous part is in the center of the beam. We use the result given in [5].

Theorem 3.8. If the viscoelastic discontinuous part ID is in the center of the beam,
then the semigroup associated with system (1.1)–(1.3) decays polynomially as

‖S(t)U0‖ ≤ Ct−1/2‖U0‖D(A). (3.41)

Proof. According to the hypothesis, we denote ID =]`0, `1[. Using Lemma 3.6,
(3.33), (3.34), and (3.35), for a = `0 and b = `1 we have∫ b

a

|Ψ|2 dx ≤ C
∣∣∣ ∫ b

a

Ψ dx
∣∣∣2 +C

∫ b

a

b1|Ψx|2 dx ≤ C‖U‖‖F‖+
C

|λ|2
‖U‖2 +

C

|λ|2
‖F‖2,

similarly ∫ b

a

|Φ|2 dx ≤ C‖U‖‖F‖+
C

|λ|2
‖U‖2 +

C

|λ|2
‖F‖2∫ b

a

|W |2 dx ≤ C‖U‖‖F‖+
C

|λ|2
‖U‖2 +

C

|λ|2
‖F‖2.

Then ∫
ID

E(x) dx ≤ C‖U‖‖F‖+
C

|λ|2
‖U‖2 + C‖F‖2. (3.42)

On IC , using the same process as in Theorem 3.7 we obtain that estimate (3.38) is
still valid. Let us suppose that `1 ∈ IE . Using Lemma 3.6 on ID =]`0, `1[ we have

I(`+1 ) ≤
∫
ID

E(x) dx+ ε‖U‖2 + Cε|λ|2‖U‖‖F‖+ C‖F‖2. (3.43)

Since S(`−1 ) = S(`+1 ), M(`−1 ) = M(`+1 ), and N(`−1 ) = N(`+1 ), we have∫
IE

E(x) dx ≤ I(`−1 ) + C‖U‖H‖F‖H ≤ ε‖U‖2 + Cε|λ|4‖F‖2. (3.44)

From the above inequality we obtain

‖U‖2 ≤ Cε|λ|4‖F‖2 + ε‖U‖2,
Hence from Borichev-Tomilov Theorem (Theorem 3.1, (3.2)) the polynomial decay
follows. �

4. Lack of exponential stability

Here we assume that ID is in the middle of the beam. The main tool use is the
following theorem due to Rivera et al. [17].

Theorem 4.1. Let H be a Hilbert space and H0 a closed subspace of H. Let S(t) be
a contraction semigroup on H, and S0(t) be a unitary group on H0. If the difference
S(t) − S0(t) is a compact operator from H0 to H, then S(t) is not exponentially
stable.



12 G. AGUILERA C., J. E. MUÑOZ-RIVERA EJDE-2021/90

Another key result for our purposes is the following lemma (see [12]).

Lemma 4.2 (Lions-Aubin). Let be V , H, V0 Banach spaces such that V ⊆ V0 ⊆ H,
where the first embedding is compact. Let

W = {ϕ ∈ Lp([a, b);V ) : ϕt ∈ Lp([a, b];H}.

Then the embedding W ⊆ Lp([a, b];V0) is compact.

Theorem 4.3. If the viscoelastic discontinuous part ID is in the center of the beam,
then the semigroup associated with system (1.1)–(1.3) is not exponentially stable.

Proof. Let us define the spaces:

L0 = {f ∈ L2(0, `) : f
∣∣
[`0,`]

= 0}, V0 = H1
0 (0, `) ∩ L0,

H0 = V0 × L0 × V0 × L0 × V0 × L0.

Let us consider the model on [0, `0]:

ρ1ϕ̃tt − κ(ϕ̃x + ψ̃ + lw̃)x − lK(w̃x − lϕ̃) = 0,

ρ2ψ̃tt − bψ̃xx + κ(ϕ̃x + ψ̃ + lw̃) = 0,

ρ1w̃tt − lK(w̃x − lϕ̃)x + lκ(ϕ̃x + ψ̃ + lw̃) = 0,

ϕ̃(0, t) = ϕ̃(`0, t) = ψ̃(0, t) = ψ̃(`0, t) = w̃(0, t) = w̃(`0, t) = 0.

(4.1)

Let S0 be the semigroup on H0 (null extensions on [`0, `]) associated with (4.1). So
we have

‖S0(t)U0‖2 = ‖U0‖2, ∀U0 ∈ H0. (4.2)

We will prove that S(t)− S0(t) : H0 → H is a compact operator, where

S(t)Um0 = (ϕm, ϕmt , ψ
m, ψmt , w

m, wmt ) ∈ H,

S0(t)Um0 = (ϕ̃m, ϕ̃mt , ψ̃
m, ψ̃mt , w̃

m, w̃mt ) ∈ H0.

Let vm := ϕm − ϕ̃m, ym := ψm − ψ̃m, zm := wm − w̃m. By definition we have

vm(x, t) =

{
ϕm − ϕ̃m, if x ∈ [0, `0],

ϕm, if x /∈ [0, `0],
ym(x, t) =

{
ψm − ψ̃m, if x ∈ [0, `0]

ψm, if x /∈ [0, `0],

zm(x, t) =

{
wm − w̃m, if x ∈ [0, `0],

wm, if x /∈ [0, `0].

Moreover v, y, and z verify

ρ1vtt − κ(vx + y + lz)x − κ̃(vxt + yt + lzt)x − lK(zx − lv)

− lK̃(zxt − lvt) = 0,
(4.3)

ρ2ytt − byxx − b̃yxxt + κ(vx + y + lz) + κ̃(vxt + yt + lzt) = 0, (4.4)

ρ1ztt −−lK(zx − lv)x − lK(zxt − lvt)x + lκ(vx + y + lz)

+ lκ̃(vxt + yt + lzt) = 0.
(4.5)



EJDE-2021/90 BRESSE SYSTEMS 13

Multiplying (4.3) by vt, and (4.4) by wt, and integrating on [0, `], we obtain∫ `

0

(
ρ1|vt|2 + ρ2|yt|2 + ρ1|zt|2 + b|yx|2 + κ|vx + y + lz|2 +K|zx − lv|2

)
dx

= κvxvt
∣∣`0
0

+ byxyt
∣∣`0
0

+ κzxzt
∣∣`0
0
−
∫ L

`0

κ̃|vxt + yt + lzt|2 dx

−
∫ L

`0

b̃|yxt|2 dx−
∫ L

`0

K̃|zxt − lvt|2 dx.

(4.6)

Using the boundary conditions we obtain

κvxvt
∣∣`0
0

+ bwxwt
∣∣`0
0

+ κzxzt
∣∣`0
0

= −κϕ̃x(`−0 , t)ϕt(`
−
0 , t)− bψ̃x(`−0 , t)ψt(`

−
0 , t)−Kw̃x(`−0 , t)wt(`

−
0 , t).

(4.7)

Now, we denote Um(t) = [S(t) − S0(t)]Um0 = (vm, vmt , y
m, ymt , z

m, zmt ). Note that
the left-hand side in (4.6) is ‖Um(t)‖2H. Thus, integrating (4.6) over [0, t], we obtain∫ t

0

‖Um(t)‖2H dt+

∫ t

0

∫ `

`0

κ̃|vmxt + ymt + lzmt |2 + b̃|ymxt|2 + K̃|zmxt − lvmt |2 dx dt

= −
∫ t

0

(κϕ̃mx (`−0 , t)ϕ
m
t (`−0 , t) + bψ̃mx (`−0 , t)ψ

m
t (`−0 , t)) +Kw̃mx (`−0 , t)w

m
t (`−0 , t) dt.

(4.8)
From the observability, we have

ϕ̃mx (`−0 , t)→ ϕ̃x(`−0 , t) weakly in L2(0, T ),

ψ̃mx (`−0 , t)→ ψ̃x(`−0 , t) weakly in L2(0, T ),

w̃mx (`−0 , t)→ w̃x(`−0 , t) weakly in L2(0, T ).

We only need to prove that(
ϕmx (`−0 , t), ψ

m
x (`−0 , t), w

m
x (`−0 , t)

)
→
(
ϕx(`−0 , t), ψx(`−0 , t), wx(`−0 , t)

)
(4.9)

strongly in [L2(0, T )]3, which implies the norm convergence in (4.8). To do that
we use (3.15) and system (1.1)–(1.3) and obtain

ϕmt , ψ
m
t , w

m
t ∈ L2(0, T ;H1(ID)), ϕmtt , ψ

m
tt , w

m
tt ∈ L2(0, T ;H−1(ID)).

Since H1 ⊂ H1−δ ⊂ H−1 for 0 < δ < 1/2, where the first inclusion is a compact
embedding, the inclusion H1−δ ⊂ C(ID) is also compact. From Lemma 4.2, there
exists a subsequence (we still denote with the same symbol) such that

(ϕmt , ψ
m
t , w

m
t )→ (ϕt, ψt, wt)

strong in L2(0, T ;H1−δ(ID) × H1−δ(ID) × H1−δ(ID)). and since the embedding
H1−δ(ID) ⊂ C(ID) is compact, we have

(ϕmt , ψ
m
t , w

m
t )→ (ϕt, ψt, wt) strongly in L2(0, T ;C(ID)× C(ID)× C(ID))

This implies (4.9). Hence inequality (4.8) implies the convergence in norm of Um.

So, S(t)− S̃0(t) is a compact operator. Then our conclusion follows. �

In summary, we have established the following result.

Theorem 4.4. The Bresse system (1.1)-(1.3) is exponentially stable if and only if
the viscoelastic discontinuous part is not in the center of the beam. Otherwise, the
system only has polynomial rate of decay.



14 G. AGUILERA C., J. E. MUÑOZ-RIVERA EJDE-2021/90

Acknowledgments. G. Aguilera C. was supported by ANID-PFCHA grant for
doctoral studies, academic year 2017, No. 21171212. J. Muñoz-Rivera was sup-
ported by CNPq-Brazil Project 310249/2018-0.

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Gabriel Aguilera Contreras

Departamento de Matemática, Universidad del B́ıo B́ıo, Concepción, Chile.

Departamento de Ciencias Básicas, Universidad de Concepción, Los Ángeles, Chile

Email address: ga.aguilerac@gmail.com, gaaguilera@udec.cl

Jaime E. Muñoz-Rivera

Laboratório Nacional de Computaçao Cient́ıfica, Petrópolis, RJ, Brazil.
Departamento de Matemática, Universidad del B́ıo B́ıo, Concepción, Chile

Email address: jemunozrivera@gmail.com


	1. Introduction
	2. Semigroup approach
	3. Asymptotic behavior
	4. Lack of exponential stability
	Acknowledgments

	References