Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 17, pp. 1–24.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.17

SIGNORINI’S PROBLEM FOR THE BRESSE BEAM MODEL

WITH LOCALIZED KELVIN-VOIGT DISSIPATION

JAIME E. MUNOZ RIVERA, CARLOS A. DA COSTA BALDEZ,
SEBASTIÃO M. S. CORDEIRO

Abstract. We prove the existence of a global solution to Signorini’s problem

for the localized viscoelastic Bresse beam model (circular arc) with continuous
and discontinuous constitutive laws. We show that when the constitutive law

is continuous, the solution decays exponentially to zero, and when the consti-

tutive law is discontinuous the solution decays only polynomially to zero. The
method we use for proving our result is different the others already used in

Signorini’s problem and is based on approximations through a hybrid model.

Also, we present some numerical results using discrete approximations in time
and space, based on the finite element method on the spatial variable and the

implicit Newmark method to the discretized the temporal variable.

1. Introduction

In this work we consider the Signorini problem for Bresse model. The beam is
configured over a circular arch of length ` over the interval [0, `] ⊂ R,

ρ1ϕtt = Sx + lN, (1.1)

ρ2ψtt = Mx − S, (1.2)

ρ1ωtt = Nx − lS, (1.3)

where
S = κ(ϕx + ψ + lω) +K(x)(ϕxt + ψt + lωt),

M = bψx +B(x)ψxt,

N = κ0(ωx − lϕ) +K(x)(ωxt − lϕt).
(1.4)

The functions ϕ,ψ and ω are the transversal displacement, rotatory angle, and
longitudinal displacement, respectively. The coefficient are ρ1 = ρA, ρ2 = ρI,
κ = kGA, b = EI, κ0 = EA, l = R−1. Where k is correction factor, E is the Young
modulus, G is shear modulus. Moreover ρ, A, I, and R represent the density of the
body, area of the cross-section, and radius of curvature of the beam, respectively.
We assume the above coefficients are constant.

2020 Mathematics Subject Classification. 74H45, 74H20, 74H15, 93D20, 74K10, 74M15.
Key words and phrases. Bresse beams; dynamic vibrations; contact problem;

localized dissipation; asymptotic behavior; numerical experiment.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted October 12, 2023. Published February 13, 2024.

1



2 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

We consider the initial conditions

ϕ(x, 0) = ϕ0(x), ψ(x, 0) = ψ0(x), ω(x, 0) = ω0(x), ∀x ∈ (0, `)

ϕt(x, 0) = ϕ1(x), ψt(x, 0) = ψ1(x), ωt(x, 0) = ω1(x) ∀x ∈ (0, `).
(1.5)

and Dirichlet boundary conditions

ϕ(0, t) = ψ(0, t) = ω(0, t) = 0, ψ(`, t) = 0 ∀t > 0. (1.6)

On the other hand, at x = ` we consider the Signorini’s conditions and ϕ:

ω(`, t) ≤ g1, ∀t > 0

g2 ≤ ϕ(`, t) ≤ g3, ∀t > 0.
(1.7)

where g1, g2 and g3 are the gaps to the obstacle, see Figure 1. We have the following
conditions

S(`, t)


≥ 0 if ϕ(`, t) = g2,

= 0 if g2 < ϕ(`, t) < g3,

≤ 0 if ϕ(`, t) = g3,

N(`, t)

{
≤ 0 if ω(`, t) = g,

= 0 if ω(`, t) < g.
(1.8)

Figure 1. Beam subject to a constraint at the free x = `−end.

To ensure that in (1.8) only one condition occurs at the same time, we impose
that

S(`, t)g2 − ϕ(`, t)]+[ϕ(`, t)− g3]+ = 0 and N(`, t)[ω(`, t)− g1]+ = 0, (1.9)

where h+ = max{h, 0} is the positive part of function h.
Here we consider two cases, first when the model (1.1)-(1.9) has a continuous

constitutive law, and when the model has a discontinuous constitutive law. In the
continuous case we assume that the functions K,B ∈ C1([0, `]) are positive on the
interval ]`0, `1[ and vanish outside this interval. Furthermore, we assume that there
are positive constants c, C1 and C2 such that

|B′|2 ≤ cB; |K ′|2 ≤ cK (1.10)

C1K ≤ B ≤ C2K. (1.11)

When the constitutive law is discontinuous we assume that K,B ∈ C1([`0, `1]) are
positive functions vanishing outside ]`0, `1[. The typical graph of B and K for the



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 3

continuous and the discontinuous case are given in figures 2 and 3. In both cases
the viscoelastic component is localized over the interval ]`0, `1[. This interval will
be denoted by IC when the constitutive law is continuous and ID in the discontinue
case.

IC =]`0, `1[Viscoelastic component IE =]0, `0[∪]`1, `[Elastic Component

`0 `1 `

y = K(x) (y = B(x))

Figure 2. Typical example of y = K(x) (y = B(x)).

ID =]`0, `1[Viscoelastic component IE =]0, `0[∪]`1, `[Elastic Component

`0 `1 `

y = K(x) (y = B(x))

Figure 3. Typical example of y = K(x) (y = B(x)).

The Signorini problem for the wave equation was studied by Kim [8], there
the author proved the existence of at least one solution, by using the Divergent-
Rotational Lemma. Similarly, Andrews et al [2] considered the one-dimensional
contact problem for the Euler Bernoulli beam model. They showed the existence of
a global solution. Kuttler and Shillor [9] considered the contact problem between
two stops to viscoelastic Euler Bernoulli beam equation. Numerical aspects of
the problem were considered in Dumont and Paoli [5] and Coppeti and Elliot [4].
Uniqueness has not been proven so far.

The main result of this article shows the existence of a global solution to the
Signorini problem (1.1)-(1.9). Moreover we prove the exponential stability of the
system provided the constitutive law is continuous, and satisfy conditions (1.10)-
(1.11). When the constitutive law is discontinuous we show the lack of exponential
stability and that the solution decays polynomially as t−1/2.

The rest of this article is organized as follows. In section 2 introduce the semi-
group associated with the hybrid model. In section 3, we show the exponential
decay in case of continuous constitutive law and the polynomial stability in the
discontinuous case. In section 4 we show the lack of exponential stability for the
discontinuous case. In section 5 we introduce the penalized problem as a Lipschitz
perturbation of the semigroup. In Section 6 we show the existence of solution of
Signorini’s problem. Finally, in section 7 we show some numerical results.



4 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

2. Existence: the hybrid-penalized method

To prove the existence of weak solutions to Signorin’s problem we use the hybrid-
penalized method introduced in [12]. That is given ε > 0, we consider the linear
hybrid model

ρ1ϕtt − Sx − lN = 0, in (0, L)× (0,∞)

ρ2ψtt −Mx + S = 0, in (0, L)× (0,∞)

ρ1ωtt −Nx + lS = 0, in (0, L)× (0,∞)

(2.1)

where S, M and N are given in (1.4). Here we consider dynamic boundary condition
on ϕ and ω,

ϕ(`, t) = u(t), ω(`, t) = z(t),

where the functions u and z are defined by the coupled system of ordinary differ-
ential equations

εutt + εut + εu+ Sε(`, t) = 0,

εztt + εzt + εz +N ε(`, t) = 0,
(2.2)

together with the stationary boundary condition

ϕ(0, t) = ψ(0, t) = ω(0, t) = 0, ψ(`, t) = 0, ∀t > 0.

System (2.1) coupled with the ordinary differential equation (2.2) is called hybrid
system. The initial conditions are given by

ϕ(x, 0) = ϕ0(x), ψ(x, 0) = ψ0(x), ω(x, 0) = ω0(x),

ϕt(x, 0) = ϕ1(x), ψt(x, 0) = ψ1(x), ωt(x, 0) = ω1(x),

(u(0), ut(0), z(0), zt(0)) = (u0, u1, z0, z1) ∈ C4.

System (2.1)-(2.2) is the linear version of the penalized problem. This procedure
allow us to arrive to the semi linear penalized problem (normal compliance) by using
Lipstchitz perturbations to the hybrid model (2.1)-(2.2). To follows this ideas we
introduce the notation

Φ = ϕt, Ψ = ψt, W = ωt, ut = U, zt = Z.

Let us denote by
U := (ϕ,Φ, ψ,Ψ, ω,W, u, U, z, Z)>.

The phase space considered here is

H := V0 × L2(0, `)×H1
0 (0, `)× L2(0, `)× V0 × L2(0, `)× C4. (2.3)

where
V0 = {u ∈ H1(0, `);u(0) = 0}.

H is a Hilbert space with the norm

‖U‖2H =

∫ `

0

[
κ|ϕx + ψ + lω|2 + ρ1|Φ|2 + b|ψx|2 + ρ2|Ψ|2 + κ0|ωx

− lϕ|2 + ρ1|W |2
]
dx+ ε(|u|2 + |U |2 + |z|2 + |Z|2).

(2.4)

Let us denote the operator

AU =
(

Φ,
Sx
ρ1

+
lN

ρ1
,Ψ,

Mx

ρ2
− S

ρ2
,W,

Nx
ρ1
− lS

ρ1
, U,−[U + u+

1

ε
S(`, t)],

Z,−[Z + z +
1

ε
N(`, t)]

)>
.

(2.5)



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 5

The domain of A is

D(A) :=
{
U ∈ H;

Φ
W
Ψ

 ∈ V 2
0 ×H1

0 ,

 κϕx +KΦx,
bψx +BΨx,
κ0ωx +KWx

 ∈ [H1(0, `)]3
}
. (2.6)

The set D(A) is the typical domain generated by the Kelvin-Voigh operators.
Among its main properties we have that the domain depends on the differential
operator and that the family of resolvent operators are not compact. System (2.1)-
(2.2) can be rewritten as

Ut = AU , U(0) = U0, (2.7)

where U0 := (ϕ0, ϕ1, ψ0, ψ1, ω0, ω1, u0, u1, z0, z1)>. A straightforward calculation
gives

Re〈AU ,U)H = −
∫ `

0

K|Φx + Ψ + lW |2 +B|Ψx|2 +K|Wx − lΦ|2 dx

− ε|U |2 − ε|Z|2.
(2.8)

Therefore A is a dissipative operator. The resolvent equation is

iλU − AU = F,

taking the inner product with U over H and then taking the real part we obtain

−Re(AU ,U)H = Re(F,U)H.

From (2.8) we obtain∫ `

0

K(x)|Φx+Ψ+lW |2+B(x)|Ψx|2+K(x)|Wx−lΦ|2 dx+ε|U |2+ε|Z|2 = Re(F,U)H.

(2.9)
The resolvent equation in terms of its components is

iλϕ− Φ = f1, (2.10)

iλΦ− Sx − lN = f2, (2.11)

iλψ −Ψ = f3, (2.12)

iλΨ−Mx + S = f4, (2.13)

iλω −W = f5, (2.14)

iλW −Nx + lS = f6, (2.15)

iλu− U = f7, (2.16)

iλU + U + u+
1

ε
S(`, t) = f8, (2.17)

iλz − Z = f9, (2.18)

iλZ + Z + z +
1

ε
N(`, t) = f10. (2.19)

Our next step is to show that A is the infinitesimal generator of a contraction
semigroup. To do that use the following result which is a consequence of Pazy
result [16, Theorem 4.6] and the Lumer-Phillips Theorem.

Lemma 2.1. Let A be dissipative with 0 ∈ %(A). If H is reflexive then A is the
infinitesimal generator of a semigroup of contractions.



6 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

Proof. Since %(A) is an open set, there exists ε > 0 such that ε ∈ %(A). This
implies that each λ > 0 belongs to %(A). In particular we have that R(I−A) = H.

Using [16, Theorem 4.6], we conclude that D(A) = H. By using the Lumer-Phillips
Theorem our conclusion follows. �

Theorem 2.2. The operator A is the infinitesimal generator of a C0 semigroup T
of contractions.

Proof. Since A is dissipative and because of Lemma 2.1 it is sufficient to show that
0 ∈ %(A). In fact, we take F ∈ H and show that there exist only one U ∈ D(A)
such that −AU = F . Let us denote

F = (f1, f2, f3, f4, f5, f6, f7, f8, f9, f10)>,

U = (ϕ,Φ, ψ,Ψ, ω,W, u, U, z, Z) ∈ D(A).

For λ = 0 the resolvent system (2.10)-(2.19) can be written as

Φ = f1 ,Ψ = f3 ,W = f5 , U = f7, Z = f9,

−κ(ϕx + ψ + lω)x − lκ0(ωx − lϕ) = F1,

−bψxx + κ(ϕx + ψ + lω) = F2,

−κ0(ωx − lϕ)x + κl(ϕx + ψ + lω) = F3,

(2.20)

where

F1 = ρ1f2 + [K(x)(fx,1 + f3 + lf4)]x + lK0(x)(f5,x − lf1),

F2 = ρ2f4 + (B(x)f3,x)x −K(x)(fx,1 + f3 + lf4),

F3 = ρ1f6 + [K0(x)(f5,x − lf1)]x − lK(x)(fx,1 + f3 + lf4)

satisfying the following boundary conditions

ϕ(0) = 0, ϕ(`) +
1

ε
S(`) = f8 − f7,

ψ(0) = 0, ψ(`) = 0,

ω(0) = 0, ω(`) +
1

ε
N(`) = f10 − f9.

Let us introduce the space V = V0 ×H1
0 × V0. Denoting Ui = (ϕi, ψi, ωi) ∈ V we

conclude that the bilinear form a : V → C,

a(U1,U2)

=

∫ `

0

κ(ϕ1
x + ψ1 + lω1)(ϕ2

x + ψ2 + lω2) + bψ1
xψ

2
x + κ0(ω1

x − lϕ1)(ω2
x − lϕ2) dx

+ εϕ1(`)ϕ2(`) + εω1(`)ω2(`),

is coercive and continuous over V. Note that the function

F(U) =

∫ `

0

F1ϕ+ F2ψ + F3ω dx+ ε(f8 − f7)ϕ(`) + ε(f10 − f9)ω(`), (2.21)

belongs to V∗. So, Lax-Milgran’s Lemma guarantees that there exists only one
weak solution to problem

a(U, Ũ) = F(Ũ), ∀ Ũ ∈ K. (2.22)

Using system (2.20) we conclude the solution U ∈ D(A). Hence 0 ∈ %(A). �

As a consequence of Theorem 2.2 we have the following result.



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 7

Theorem 2.3. For each U0 ∈ H there exist a unique mild solution to problem
(2.1)-(2.2). Moreover if the initial data U0 ∈ D(A) there exist a strong solution
satisfying

U ∈ C1(0, T ;H) ∩ C(0, T ;D(A)).

3. Asymptotic behavior

Here we show that the semigroup associated with the hybrid system (2.1)-(2.2)
is exponentially stable provided the constitutive law is continuous and satisfies sat-
isfies (1.10)-(1.11). When the constitutive law is discontinuous, the corresponding
semigroup is polynomially stable, with rate t−1/2. Our main tool is the characteri-
zation due to Prüss [18] and Borichev and Tomilov [3].

Theorem 3.1. Let S(t) = eA be a contraction C0-semigroup over a Hilbert space
H. Then, in Prüss [18] is established that there exist C and , γ > 0 satisfying

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

For polynomial stability, Borichev and Tomilov [3] results 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 starting point to show the exponential stability is to show the strong sta-
bility.

Lemma 3.2. The operator A defined by (2.5) and (2.6) satisfies iR ⊂ %(A).

Proof. Let us consider the set

M = {s ∈ R+ :]− is, is[⊂ ρ(A)}.
Since 0 ∈ ρ(A), M 6= ∅, the supremum σ = supM can be finite or infinite. If
σ = +∞ then iR ⊆ ρ(A) and we have nothing to prove. We will prove that the
finite case is not possible. 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 → +∞. Noting

Ũn = (iλnI −A)−1fn ⇒ fn = iλnŨn −AŨn

and Un = Ũn
‖Ũn‖

, Fn = fn
‖Ũn‖

we obtain

iλnUn −AUn = Fn → 0. (3.3)

Taking the inner product,

iλn‖Un‖2 − 〈AUn,Un〉 = 〈Fn,Un〉 → 0

and taking real part,

−Re〈AUn,Un〉 =

∫ `

0

(B|Ψn
x |2 +K|Φnx + Ψn + lWn|2 +K|Wn

x − lΦn|2) dx

+ |Un|2 + |Zn|2 → 0,

(3.4)

Ψn
x ; Φnx + Ψn + lWn; Wn

x − lΦn → 0 strong in L2(]`0, `1[). (3.5)



8 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

Therefore,

ψnx ; ϕnx + ψn + lωn; ωnx − lϕn → 0 strong in L2(]`0, `1[), (3.6)

(Un, Zn)→ (0, 0). (3.7)

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 strong in L2(0, `) (3.8)

ϕn,x + ψn + lωn → ϕx + ψ + lω, (3.9)

ψn,x → ψx, ωn,x − lϕn → ωx − lϕ, strong in L2(IE), (3.10)

where IE =]0, `0[∪]`1, `[. From (2.16) and (3.7) we obtain u = z = 0; therefore
ϕ(0) = ω(0) = 0. From (3.8), (3.10) and (3.6), it follows that Un → U strongly in
H. Since A is closed, we conclude that U satisfies

iσU −AU = 0.

Moreover, using the convergences (3.5)-(3.6) and the resolvent system, we conclude
that the solution vanishes over ]`0, `1[ so we have that

ϕ(`0) = ψ(`0) = ϕx(`0) = ψx(`0) = ω(`0) = ωx(`0) = 0.

So, over ]0, `0[∪]`1, `[ the solution satisfies

−ρ1σ
2ϕ+ κ(ϕx + ψ + lω)x − lκ0(ωx − lϕ) = 0,

−ρ2σ
2ψ + bψxx + κ(ϕx + ψ + lω) = 0,

−ρ1σ
2ω + κ0(ωx − lϕ)x + lκ(ϕx + ψ + lω) = 0.

Looking the above equation as a second-order final-value problem over ]0, `0[,
we obtain ϕ = ψ = ω = 0 over ]0, `0[. Using a similar argument we conclude that
ϕ = ψ = ω = 0 over ]`1, `[. Hence U ≡ 0 on H, which is a contradiction. This
completes the proof. �

The next Lemma plays an important role in the proof of exponential stability.

Lemma 3.3. Assume that hypothesis (1.10) and (1.11) hold. Then for each ε > 0
there exists Cε > 0 such that the solution of (2.10)-(2.19) satisfies∫

IC

K|λΦ|2 +B|λΨ|2 dx+K|λW |2 dx ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε|U‖2H.

Proof. Multiplying (2.11) by iλKΦ and integrating over IC = [`0, `1],∫
IC

ρ1K|λΦ|2 dx =

∫
IC

(Sx + lN + f2)iλKΦ dx. (3.11)



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 9

Recalling the definition of S and N over IC we obtain∫
IC

ρ1K|λΦ|2 dx

=

∫
IC

[κ(ϕx + ψ + lω) +K(Φx + Ψ + lW )]xiλKΦ dx

+

∫
IC

l[κ0(ωx − lϕ) +K(Wx − lΦ)]iλKΦ dx+

∫
IC

f2iλKΦ dx,

= G + G0 + G1 + G2 +

∫
IC

f2iλKΦ dx.

(3.12)

Here

G =

∫
IC

[K(Φx + Ψ + lW )]iλ(K ′Φ +KΦx) dx,

G0 =

∫
IC

[κ(ϕx + ψ + lω)]iλ(K ′Φ +KΦx) dx,

G1 =

∫
IC

`[κ0(ωx − `ϕ)]iλKΦ dx,

G2 =

∫
IC

`[K(Wx − `Φ)]iλKΦ dx,

where IC =]`0, `1[ and K(`0) = K(`1) = 0. Estimating G we have

G =

∫
IC

[K(Φx + Ψ + lW )]iλ(K ′Φ +K(Φx + Ψ + lW )) dx

−
∫
IC

[K(Φx + Ψ + lW )]iλ(KΨ +KlW ) dx.

(3.13)

Taking the real part in (3.13) and using (2.9) we obtain

ReG = Re

∫
IC

[K(Φx + Ψ + lW )]iλ(K ′Φ) dx

− Re

∫
IC

[K(Φx + Ψ + lW )]iλ(KΨ + κ1lW ) dx,

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

(3.14)

Similarly, using (2.10), (2.12), (2.14), and (2.9) we obtain

ReG0 = Re

∫
IC

[κ(ϕx + ψ + lω)]iλ(K ′Φ +KΦx) dx,

≤ ε
∫
IC

|Φ|2 + |Ψ|2 + |W |2 dx+R,

(3.15)

ReG1 ≤ ε
∫
IC

|Φ|2 + |Ψ|2 + |W |2 dx+R, (3.16)

ReG2 ≤ ε
∫
IC

|Φ|2 + |Ψ|2 + |W |2 dx+R. (3.17)

Substituting (3.14) and (3.15) into (3.12) yields∫
IC

κ1|λΦ|2 dx ≤ ε‖λΦ‖2 + ε‖λΨ‖2 + Cε‖U‖H‖F‖H (3.18)



10 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

for |λ| > 1. Repeating the above procedure multiplying equation (2.13) by iλb1Ψ
and equation (2.15) by iλK1W we arrive at∫

IC

ρ2B|λΨ|2 dx ≤ ε‖λΨ‖2 + ε‖λW‖2 + Cε‖U‖H‖F‖H,∫
IC

ρ1K|λW |2 dx ≤ ε‖λΦ‖2 + ε‖λW‖2 + Cε‖U‖H‖F‖H.

Summing the last three inequalities our conclusion follows. �

Remark 3.4. Lemma 3.3 is also valid for discontinuous constitutive law. That is
for any θ vanishing out side of ID =]`0, `1[ satisfying

|θ′|2 ≤ c|θ| (3.19)

it is valid that∫
ID

θ|λΦ|2 + θ|λΨ|2 dx+ θ|λW |2 dx ≤ Cε‖U‖H‖F‖H + Cε‖U‖2H + ε‖F‖2H.

The proof is identical as Lemma 3.3.

We introduce the following notation:

Eϕ =
(κqρ1)′

2
|Φ|2 +

q′

2
|S|2, Iϕ =

κqρ1

2
|Φ|2 +

q

2
|S|2, (3.20)

Eψ =
(bqρ2)′

2
|Ψ|2 +

q′

2
|M |2, Iψ =

bqρ2

2
|Φ|2 +

q

2
|M |2, (3.21)

Ew =
(κ0qρ1)′

2
|W |2 +

q′

2
|N |2, Iw =

κ0qρ2

2
|W |2 +

q

2
|N |2, (3.22)

E = Eϕ + Eψ + Eω, I = Iϕ + Iψ + Iω (3.23)

and

L =

∫ b

a

E(s) ds−
∫ b

a

(ρ1κqΦΨ + ρ1κqlΦW + ρ1KqlWΦ)dx

+

∫ b

a

(qlSN̄ + qlS̄N + qSM̄)dx.

Taking q(x) = enx−ena

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

C0

∫ b

a

E dx ≤ L ≤ C1

∫ b

a

E dx. (3.24)

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

a

|S|2 dx ≤ c
∫ b

a

κ|ϕx + ψ + lω|2 dx+

∫ b

a

|K(Φx + Ψ + lW )|2 dx.

Using the dissipative properties,∫ b

a

|S|2 dx ≤ c
∫ b

a

|ϕx + ψ + lω|2 dx+ c‖U‖H‖F‖H.

Similarly ∫ b

a

|M |2 dx ≤ c
∫ b

a

|ψx|2 dx+ c‖U‖H‖F‖H,



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 11

and ∫ b

a

|N |2 dx ≤ c
∫ b

a

|ωx − lψ|2 dx+ c‖U‖H‖F‖H.

From where it follows that, for large values of n,∫ b

a

|Φ|2 + |ϕx + ψ + lω|2 + |Ψ|2 + |ψx|2 + |W |2 + |ωx − lψ|2 dx

≤
∫ b

a

E dx+ c‖U‖H‖F‖H,∫ b

a

E dx ≤ c
∫ b

a

|Φ|2 + |ϕx + ψ + lω|2 + |Ψ|2 + |ψx|2 + |W |2 + |ωx − lψ|2 dx

+ c‖U‖H‖F‖H.

Lemma 3.6. If the constitutive law is continuous, then for each [a, b] ⊂]0, `[ we
have ∣∣∣L(s)− I(s)

∣∣b
a

∣∣∣ ≤ Cε‖U‖H‖F‖H + Cε‖F‖H + ε‖U‖2H.

If the constitutive law in discontinuous, then for each [a, b] ⊂]0, `[ we have∣∣∣L − I(s)
∣∣b
a

∣∣∣ ≤ +Cε|λ|2‖U‖H‖F‖H + Cε‖F‖H + ε‖U‖2H.

Proof. Multiplying (2.11) by qS̄, (2.13) by qM̄ , and (2.15) by qN̄ , we obtain

− ρ1κq

2

d

dx
|Φ|2 − q

2

d

dx
|S|2

= R1 + ρ1κqΦΨ + ρ1κqlΦW − iλρ1qKΦ(Φx + Ψ + lW )− qlS̄N,
(3.25)

−ρ2bq

2

d

dx
|Ψ|2 − q

2

d

dx
|M |2 = R2 − iλρ2qBΨΨx − qSM̄, (3.26)

− ρ1Kq

2

d

dx
|W |2 − q

2

d

dx
|N |2

= R3 − ρ1KqlWΦ− iλρ1qKW (Wx − lΦ)− qlSN̄ .
(3.27)

Summing identities (3.25), (3.26), (3.27) 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) = −iλρ1qKΦ(Φx + Ψ + lW )− iλρ2qBΨΨx − iλρ1qKW (Wx − lΦ). (3.28)

Note that J vanishes outside of ]`0, `1[. Recalling the definition of I and E we
obtain

− d

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

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

Hence, when the constitutive law is continuous, Lemma 3.3 implies∣∣ ∫ `

0

J(x) dx
∣∣ ≤ Cε‖U‖H‖F‖H + Cε‖F‖H + ε‖U‖2H. (3.29)



12 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

Instead when the constitutive law is discontinuous we obtain∣∣ ∫ `

0

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

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 + lω|2 + κ0|ωx − lϕ|2.

Theorem 3.7. The semigroup associated with system (2.1)-(2.2) is exponentially
stable provided the constitutive law is continuous and satisfies (1.10)-(1.11). Also
if the constitutive law is discontinuous, then solution decays polynomially to zero as

‖S(t)U0‖ ≤ Ct−1/2‖U0‖D(A). (3.31)

Proof. From (2.9), (2.16), and (2.18), we obtain

ε(|u|2 + |U |2 + |z|2 + |Z|2) ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H. (3.32)

From (2.9), (2.10), (2.12), and (2.14), we obtain∫
IC

K|ϕx+ψ+ lω|2 +B|ψx|2 +K|ωx− lϕ|2 dx ≤ Cε‖U‖H‖F‖H+Cε‖F‖2H. (3.33)

On the other hand, from Lemma 3.3 we have∫
IC

K|Φ|2 +B|Ψ|2 dx+K|W |2 dx ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H.

For λ large and each interval ]a, b[⊂ IC we have∫ b

a

E(x) dx ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H. (3.34)

Using the observability Lemma 3.6 over the interval ]a, b[ we obtain

E(a) + E(b) ≤ c
∫ b

a

E(x) dx+ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2

≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H.
(3.35)

Using Lemma 3.6 over the interval ]0, a[ and ]a, `[ and the above inequalities we
obtain ∫ a

0

E(x) dx ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H,∫ `

a

E(x) dx ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H.

From the above inequalities we obtain

‖U‖2H =

∫ `

0

E(x) dx+ε(|u|2+|U |2+|z|2+|Z|2) ≤ Cε‖U‖H‖F‖H+Cε‖F‖2H+ε‖U‖2H,

which implies that ‖U‖H ≤ C‖F‖H. From where the exponential stability holds.
Finally, we consider the discontinuous case. Note that (3.33) is also valid for

discontinuous functions B and K. From Remark 3.4,∫
ID

θ|λΦ|2 + θ|λΨ|2 dx+ θ|λW |2 dx ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H.



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 13

Therefore for each interval ]a, b[⊂ ID we have∫ b

a

E(x) dx ≤ Cε‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H. (3.36)

Using the observability Lemma 3.6 for discontinuous constitutive law over the in-
terval ]a, b[ we obtain

E(a) + E(b) ≤ Cε|λ|2‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H. (3.37)

Reasoning as above we conclude that

‖U‖2H =

∫ `

0

E(x) dx+ ε(|u|2 + |U |2 + |z|2 + |Z|2)

≤ Cε|λ|2‖U‖H‖F‖H + Cε‖F‖2H + ε‖U‖2H.

From where we obtain that ‖U‖H ≤ |λ|2‖F‖H, which implies the polynomial sta-
bility. �

4. Lack of exponential stability

In this section we prove that the semigroup associated with system (2.1)-(2.2)
is not exponentially stable when the viscoelastic constitutive law is discontinuous.
To do that we use [14, Theorem 5.1 ].

Theorem 4.1. Let T be a contractions semigroup over H and T0 a group with
unitary norm, that is ‖T0(t)U‖ = ‖U0‖, defined in H0 ⊂ H. If the operator
T (t) − T0(t) is a compact operator, then the semigroup T (t) is not exponentially
stable.

Another key result for our purpose is given in the following Lemma.

Lemma 4.2 (Lions-Aubin [10, Theorem 5.1]). Let be V , H, V0 Banach spaces such
that V ⊆ V0 ⊆ H, where the first embedding is compact. Let ϕ ∈ Lp([a, b];V ), ϕt ∈
Lp([a, b];H). Denoting

W = {ϕ ∈ Lp([a, b);V ) : ϕt ∈ Lp([a, b];H}.
Then the embedding W ⊆ Lp([a, b];V0) is compact.

Finally, we establish the observability inequality to the evolution Bresse system.
To do it let us denote

Iϕ(x, t) = |ϕ̃t(x, t)|2 + |ϕ̃x(x, t) + ψ̃(x, t) + lω̃(x, t)|2,

Iψ(x, t) = |ψ̃t(x, t)|2 + |ψ̃x(x, t)|2,
Iω(·, t) = |ω̃t(·, t)|2 + |ω̃x(x, t)− lϕ̃(x, t)|2.

Lemma 4.3 ([19, Lemma 2.1]). Let us suppose that there exists a solution to Bresse
system, bounded by the initial energy associated with the model

ρ1ϕ̃tt − κ(ϕ̃x + ψ̃ + lω̃)x − lK(ω̃x − lϕ̃) = 0 in ]a, b[×]0, T [,

ρ2ψ̃tt − bψ̃xx + κ(ϕ̃x + ψ̃ + lω̃) = 0 in ]a, b[×]0, T [,

ρ1ω̃tt − lK(ω̃x − lϕ̃)x + lκ(ϕ̃x + ψ̃ + lω̃) = 0 in ]a, b[×]0, T [.

(4.1)

Then there exists a positive constant satisfying∫ T

0

Iϕ(a, t) + Iϕ(b, t) + Iψ(a, t) + Iψ(b, t) + Iω(a, t) + Iω(b, t) dt ≤ cE(0)



14 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

where E(t) =
∫ b
a
Iϕ(x, t) + Iψ(x, t) + Iω(x, t) dx.

Now we are in a position to show the lack of exponential stability.

Theorem 4.4. If the constitutive law is discontinuous then the semigroup T (t) is
not exponentially stable.

Proof. Let us denote by T̃0 the group associated with system (4.1) for ]a, b[=]0, `0[
satisfying the boundary conditions

ϕ̃(0, t) = ϕ̃(`0, t) = ψ̃(0, t) = ψ̃(`0, t) = ω̃(0, t) = ω̃(`0, t) = 0 (4.2)

over the phase space

H̃ = H1
0 (0, `0)× L2(0, `0)×H1

0 (0, `0)× L2(0, `0).

Note that ‖T̃0(t)U0‖2 = ‖U0‖2 for all U0 ∈ H̃. Let us consider the spaces

L0 = {f ∈ L2(0, `) : f
∣∣∣
[`0,`]

= 0}, V0 = H1
0 (0, `) ∩ L0,

H0 = V0 × L0 × V0 × L0 × V0 × L0 × {0} × {0} × {0} × {0}.

Let us denote by T0 the group over H0 (null extensions on [`0, `]) associated with
(4.1)-(4.2) that is

T0(t)Ũ0 = (ϕ̃, ϕ̃t, ψ̃, ψ̃t, w̃, w̃t,0,0,0,0).

So we have

‖T0(t)Ũ0‖2 = ‖Ũ0‖2, ∀Ũ0 ∈ H0 (4.3)

Now, we prove that T (t)− T0(t) : H0 → H is a compact operator, where

T Um0 = (ϕm, ϕmt , ψ
m, ψmt , ω

m, ωmt , u, ut, z, zt) ∈ H,

T0(t)Um0 = (ϕ̃m, ϕ̃mt , ψ̃
m, ψ̃mt , ω̃

m, ω̃mt ,0,0,0,0)) ∈ H0

Let vm := ϕm − ϕ̃m, ym := ψm − ψ̃m, ζm := ωm − ω̃m. 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],

ζm(x, t) =

{
ωm − ω̃m, if x ∈ [0, `0]

ωm, if x /∈ [0, `0].

Moreover v, y and ζ satisfies

ρ1vtt − κ(vx + y + lz)x −K(vxt + yt + lζt)x

− lK(ζx − lv)− lK(ζxt − lvt) = 0,
(4.4)

ρ2ytt − byxx − (Byxt)x + κ(vx + y + lζ) +K(vxt + yt + lζt) = 0, (4.5)

ρ1ζtt − κ0(ζx − lv)x − lK(ζxt − lvt)x
+ lκ(vx + y + lζ) + lK(vxt + yt + lζt) = 0.

(4.6)



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 15

Multiplying (4.4) by vt, (4.5) by wt, and integrating over [0, `], we obtain

d

dt
‖Um(t)‖2H +

∫ `

`0

K|vxt + yt + lζt|2 +B|yxt|2 +K|ζxt − lvt|2 dx+ ε|ut|2 + ε|zt|2

= −S̃m(`−0 , t)ϕt(`
−
0 , t)− M̃m(`−0 , t)ψt(`

−
0 , t)− Ñm(`−0 , t)wt(`

−
0 , t)

(4.7)
where

Um(t) = [T (t)− T0(t)]Um0 = (vm, vmt , y
m, ymt , ζ

m, ζmt , u, ut, z, zt).

Integrating (4.7) over [0, t], we obtain

‖Um(t)‖2H +

∫ t

0

∫ `

`0

κ̃|vmxt + ymt + lζmt |2 + b̃|ymxt|2 + K̃|ζmxt − lvmt |2 dx dt

= −
∫ t

0

(S̃m(`−0 , t)ϕt(`
−
0 , t) + M̃m(`−0 , t)ψt(`

−
0 , t) + Ñm(`−0 , t)wt(`

−
0 , t)) dt.

(4.8)

Using Lemma 4.3 on the interval ]0, `0[, we conclude that S̃m. M̃m, Ñm are
bounded, therefore there exists a subsequence (we still denote in the same way)
such that

S̃m(`−0 , t)→ S̃(`−0 , t) weak in L2(0, T ),

M̃m(`−0 , t)→ M̃(`−0 , t) weak in L2(0, T ),

Ñm(`−0 , t)→ Ñ(`−0 , t) weak in L2(0, T ).

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

m
t (`−0 , t), ω

m
t (`−0 , t)

)
→
(
ϕt(`

−
0 , t), ψt(`

−
0 , t), ωt(`

−
0 , t)

)
(4.9)

strong in [L2(0, T )]3, which implies the norm convergence in (4.8). To do that we
use (2.9) and system (2.1) to obtain

ϕmt , ψ
m
t , ω

m
t ∈ L2(0, T ;H1(ID)), ϕmtt , ψ

m
tt , ω

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, then Lemma 4.2 implies that there exists a subsequence (we still denote
in the same way) such that

(ϕmt , ψ
m
t , ω

m
t )→ (ϕt, ψt, ωt) 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 , ω

m
t )→ (ϕt, ψt, ωt) strong in L2(0, T ;C(ID)× C(ID)× C(ID)).

Since ID = [`0, `1] the above convergence implies (4.9). Hence (4.8) implies the

convergence in norm of Um, since H is a Hilbert space we obtain that T (t)− T̃0(t)
is a compact operator. The proof is now complete. �

5. Penalized problem

Here we establish the well possedness and the asymptotic behavior of the abstract
semi linear problem. We introduce a local Lipschitz F function defined over a
Hilbert space H. We assume that for each ball BR = {W ∈ H : ‖W‖H ≤ R}, there

exists a function globally of Lipschitz type F̃R, such that

F(0) = 0, F(U) = F̃R(U), ∀U ∈ BR (5.1)



16 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

and additionally, that there exists a positive constant κ0 such that∫ t

0

(
F̃R(U(s)), U(s)

)
H ds ≤ κ0‖U(0)‖2H, ∀U ∈ C([0, T ];H). (5.2)

Under the above conditions we have the following theorem proved in [13].

Theorem 5.1. Let {T (t)}t≥0 be a C0 semigroup of contraction, exponentially or
polynomially stable semigroup with infinitesimal generator A over the phase space
H. Let F locally Lipschitz on H satisfying conditions (5.1) and (5.2). Then there
exists a global solution to

Ut − AU = F(U), U(0) = U0 ∈ H, (5.3)

that decays exponentially or polynomially respectively.

Let us consider the semilinear system

ρ1ϕtt − Sx − lN = 0,

ρ2ψtt −Mx + S = 0,

ρ1ωtt −Nx + lS = 0,

εutt + εut + εu+ Sε(`, t) = −1

ε

[
(u− g3)+ − (g2 − u)+

]
,

εztt + εzt + εz +N ε(`, t) = −1

ε
(z − g1)+.

(5.4)

Let us denote

F(U) = (0, 0, 0, 0, 0, 0, f1(u), 0, f2(z))

where

f1(u) = −1

ε

[
(u− g3)+ − (g2 − u)+

]
and f2(z) = −1

ε
(z − g1)+.

Since f1, f2 are Lipschitz functions, it follows that F is Lipschitz on H satisfying
condition (5.2). Under these conditions we have the following result.

Theorem 5.2. The semigroup defined by (5.4) is exponentially or polynomially
stable provided K(x), B(x) and K0(x) are differentiable or discontinuous functions,
respectively.

Proof. Note that the above system can be written as

Ut − AU = F(U), U(0) = U0.

Where A is given by (2.5). From Theorem 3.7 and Theorem 5.1 our conclusion
follows. �

6. Signorini’s problem

Here we show the main result of this article. First we introduce the space

H0 = K × L2(0, `)×H1
0 (0, `)× L2(0, `)× V × L2(0, `)

where

K = {u ∈ V0; g2 ≤ u(`) ≤ g3}, V = {w ∈ V0;w(`) ≤ g1}.

Theorem 6.1. For any initial data (ϕ0, ϕ1, ψ0, ψ1, ω0, ω1) ∈ H there exist a weak
solution to Signorin’s problem (1.1)-(1.8) which decays as establish in Theorem 5.2.



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 17

Proof. Multiplying, equation (5.4)1-(5.4)6 by ϕεt, ψ
ε
t , ω

ε
t , ut, vt, and zt, respectively.

Integrating over (0, `), we obtain

d

dt
Eε(t) = −

∫ `

0

K|ϕεxt+ψεt+lωεt |2+B|ψεxt|2+K|ωεtx−lϕεt|2 dx−ε|uεt|2−ε|zεt |2 (6.1)

where

2Eε(t) = Eε(t) +
1

ε
Nε(t) + ε(|uε|2 + |uεt|2 + |zε|2 + |zεt |2),

2Eε(t) =

∫ `

0

[
ρ1|ϕεt|2 + ρ2|ψεt |2 + ρ1|ωεt |2 + κ|ϕεx + ψε + lωε|2 + b|ψεx|2

+ κ0|ωεx − lϕε|2
]
dx

Nε(t) := |(ωε(`, t)− g1)+|2 + |(ϕε(`, t)− g3)+|2 + |(g2 − ϕε(`, t))+|2.

Taking (ϕ0, ω0) ∈ K × V, and integrating we obtain

Eε(t) +
1

ε
Nε(t) + ε(|uε|2 + |uεt|2 + |zε|2 + |zεt |2) ≤ Eε(0), (6.2)

Nε(t) ≤ εEε(0)→ 0, as ε→ 0. (6.3)

Hence denoting the limit (ϕε, ψε, ωε)→ (ϕ,ψ, ω) we obtain

ω(`, t) ≤ g1, g2 ≤ ϕ(`, t) ≤ g3. (6.4)

Using Lemma 4.3 we obtain that ϕεt(`, t), ω
ε
t (`, t), S

ε(`, t), and N ε(`, t) are
bounded in L2(0, T ), so is utt and ztt. Using (5.4)4 we obtain∫ T

0

[
εuεtt+εu

ε
t+εu

ε+Sε(L, t)
]
[v−uε] dt = −1

ε

∫ T

0

[
(uε−g2)+−(g1−uε)+

]
[v−uε] dt.

For all v ∈ L2(0, T ;K) ∩ H1(0, T ;L2(0, L)). Where K = {w ∈ H1(0, L) : g1 ≤
w(L) ≤ g2}. It is no difficult to show that

lim
ε→0

∫ T

0

[
εuεtt + εuεt + εuε

]
[v − uε] dt = 0.

In fact, from (5.4)4 εu
ε
tt is bounded by a constant depending on ε, in L2(0, T ). More-

over, Lemma 4.3 implies that uεt = ϕεt(`, t) is also uniformly bounded in L2(0, T ).
Therefore uεt is a continuous function, uniformly bounded in L∞(0, T ). Integrating
by parts we obtain∫ T

0

εuεtt[u(t)− uε] dt = ε uεt[v(t)− uε]|T0 −
∫ T

0

εuεt[vt(t)− uεt] dt→ 0.

Hence,

lim
ε→0

∫ T

0

Sε(L, t)[v(t)− uε(t)] dt

= lim
ε→0

∫ T

0

−1

ε

[
(uε − g2)+ − (g1 − uε)+

]
[v(t)− uε(t)] dt.

Since ∫ T

0

(uε − g2)+[v(t)− uε(t)] dt



18 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

=

∫ T

0

(uε − g2)+[u(ε(t)− g2] dt−
∫ T

0

(uε − g2)+(uε − g2) dt,

=

∫ T

0

(vε − g2)+[uε(t)− g2] dt−
∫ T

0

(uε − g2)+(uε − g2)+ dt ≤ 0.

For all g1 ≤ v(L, t) ≤ g2. Similarly we obtain

−
∫ T

0

[(g1 − uε)+[v(t)− uε(t)] dt ≤ 0.

Therefore, from the last two inequalities we obtain∫ T

0

1

ε

[
(uε − g2)+ − (g1 − uε)+

]
[v(t)− uε(t)] dt ≤ 0, ∀ε > 0.

For all v ∈ H1(0, T ;L2(0, L)) such that g1 ≤ v(L, t) ≤ g2. Taking the limit ε → 0
we obtain ∫ T

0

S(L, t)[v(L, t)− ϕ(L, t)] dt ≥ 0, ∀v ∈ L2(0, T ;K). (6.5)

Inequality (6.5) implies condition (1.8). Using similar ideas we arrive at∫ T

0

N(L, t)[w(L, t)− ω(L, t)] dt ≥ 0, ∀w ∈ L2(0, T ;V).

From where condition (1.8) follows. Then the proof of existence is now complete.
To show the asymptotic behavior we use Theorem 5.2 and obtain

E(t) ≤ CE(0)e−γt,

where

E(t) =
1

2

∫ `

0

[
ρ1|ϕt|2 +ρ2|ψt|2 +ρ1|ωt|2 +κ|ϕx+ψ+ lω|2 +b|ψx|2 +κ0|ωx− lϕ|2

]
dx.

So, using the semicontinuity of the norm and noting that N (0) = 0, we obtain

E(t) ≤ lim inf
ε→0+

Eε(t) ≤ C
{

lim
ε→0+

Eε(0)
}
e−γt ≤ CE(0)e−γt

where C is a positive constant independent of parameter ε. Thus, we conclude the
exponential stability of the Signorini’s problem. Similarly we obtain the polynomial
stability. �

7. Numerical results

In this section we show numerical results for the penalized system (2.1)-(2.2).
Here, we use the well-known Newmark’s methods [6, 15].

7.1. Variational formulation. Letting u = [ϕ,ψ, ω]>, from (2.1) we obtain the
variational problem

(uεtt(t), ũ) + a1(uε(t), ũ) + a2(uεt(t), ũ) = 0, ∀ũ ∈ V (7.1)

where V = V0 ×H1
0 × V0 and uε satisfy the initial conditions

(uε(0), ũ) = (uε0, ũ), (uεt(0), ũ) = (uε1, ũ). (7.2)

Here ai : V × V 7→ R are functionals defined by

(uεtt(t), ũ) = ρ1(ϕεt, u1) + ρ2(ψεt , u2) + ρ1(ωεt , u3),



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 19

a1(uε(t), ũ) = k(ϕεx + ψε + lωε, u1,x + u2 + lu3) + b(ψεx, u2,x)

+ k0(ωεx − lϕε, u3,x − lu1)− Sε(`, t)u1(L)−N ε(`, t)u3(L),

a2(uεt(t), ũ) =

∫ `

0

K(x)(ϕεtx + ψεt + lωεt )(u1,x + u2 + lu3) dx+

∫ `

0

B(x)ψεtxu2,x dx

+

∫ `

0

K0(x)(ωεtx − lϕεt)(u3,x − lu1) dx.

Here (·, ·) is the inner product in L2(0, `).

7.2. Algorithms and numerical approximation. To have the full discretization
of (7.1)–(7.2) we first consider a partition of the interval Ω = (0, `), Xh = {0 =
x0 < x1 < · · · < xN = `}, Ωj+1 = (xj , xj+1), and Ωi ∩ Ωj = ∅ if i 6= j, and

Ω = ∪Nee=1Ωe, where Ne is the number of the elements of the partition. Then we
define the finite-dimensional subspaces Sh1 = {u ∈ C(0, `);u

∣∣
Ωe
∈ P1(Ωe)} where

P1 is the set linear polinomials defined in Ωe,

Vh = {vh ∈ Sh1 ; vh(0) = 0} and Uh = {uh ∈ Sh1 ;uh(0) = uh(`) = 0}.

Considering uh = [ϕh, ψh, ωh]>, the approximation is characterized as the finite-
dimensional problem in R3Ne ,

(uh,εtt (t), ũh) + a1(uh,ε(t), ũh) + a2(uh,εt (t), ũh) = 0, ∀ũh ∈ Vh × Uh × Vh, (7.3)

where uh,ε(t) satisfies the initial conditions

(uh,ε(0), ũh) = (uh,ε0 , ũh), (uh,εt (0), ũh) = (uh,ε1 , ũh). (7.4)

In matrix form, dynamical problem (7.3)-(7.4) can be written as

Md̈(t) + K(d(t)) + Cḋ(t) = 0,

d(0) = d0, ḋ(0) = d1,

where, d(t) is the vector of displacement nodal generalized at time t. M, C are
matrices associated with the functionals: (uεtt(t), ũ) and a2(uεt(t), ũ) respectively.
K(d(t)) is the vector of consistent nodal elastic stiffness at time t.

Taking a partition P on the interval [0, T ] of M intervals of length ∆t such that
0 = t0 < t1 < · · · < tM = T , with tn+1 − tn = ∆t and considering the non-linearity
the numerical scheme becomes

Md̈n+1 + Cḋn+1 + Kdn+1 = K̃(dn+1),

dn+1 = dn + ∆tḋn +
∆t2

2
[(1− 2β)d̈n + 2βd̈n+1],

ḋn+1 = ḋn + ∆t[(1− γ)d̈n + γd̈n+1]

with

K̃(dn+1) = −1

ε

(
0, . . . , 0, (d3Ne−2(t)−g3)+−(g2−d3Ne−2(t))+, 0, (d3Ne

(t)−g1)+)
)>
.

Here, β and γ are two parameters that govern the stability and accuracy of the
method. The matrices, from the above system, are obtained from the finite element
method standard assembly, M = ∪Nee=1m

e, K = ∪Nee=1(keS +keM +keN ), for instance,



20 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

considering linear functions in the interpolation functions, we obtain the elementary
matrices

me =


ρ1h/3 0 0 ρ1h/6 0 0

0 ρ2h/3 0 0 ρ2h/6 0
0 0 ρ1h/3 0 0 ρ1h/6

ρ1h/6 0 0 ρ1h/3 0 0
0 ρ2h/6 0 0 ρ2h/3 0
0 0 ρ1h/6 0 0 ρ1h/3

 ,

keM =
b

h


0 0 0 0 0 0
0 1 0 0 −1 0
0 0 0 0 0 0
0 0 0 0 0 0
0 −1 0 0 1 0
0 0 0 0 0 0

 ,

keS =


k/h −k/2 −kl/2 −k/h −k/2 −kl/2
−k/2 kh/3 klh/3 k/2 kh/6 klh/6
−kl/2 klh/3 kl2h/3 kl/2 klh/6 kl2h/6
−k/h k/2 kl/2 k/h k/2 kl/2
−k/2 kh/6 klh/6 k/2 kh/3 klh/3
−kl/2 klh/6 kl2h/6 kl/2 klh/3 kl2h/2

 .

Remark 7.1. Generally to penalized models, in particular to Bresse beams, oc-
curs a typical numerical problem, the shear locking, for more details see [11]. To
overcome this problem alternatives was performed in Hughes et al [7], Prathap and
Bhashyam [17] and Abimael et al [11] and references therein.

To obtain computational results, we use the implemented code in the Language
C. The graphics were developed using GNUplot. In all experiments we use the
following parameters to Newmark’s method: β = 1

4 and γ = 1
2 . The finite element

mesh h = 0.01 m and length of the beam ` = 1 m.
Let Iv viscoelastic component in I = [0, `]. Here we are consider the following

case.

7.3. Cases Iv. We consider the localized viscoleastic damping functions Iv =
[`0, `1] and Iv = [`0, `].

f1(x) =

{
c0(x− `0)4(x− `1)4 if x ∈ Iv,
0 if x ∈ I \ Iv.

f2(x) =

{
c0(x− `0)2 if x ∈ Iv
0 if x ∈ I \ Iv.

7.4. Linear case - without contact: g1 = +∞, g2 = −∞, g3 = +∞.

7.4.1. Experiment. We take a rectangular arch beam with Iv = [0.3, 0.6], thickness
and width equal 0.08 m, ρ = 2700 kg/m3, κ = 5/6, r = 0.3 (Poisson ratio),



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 21

Figure 4. Dissipation functions: continuous (blue lines) and dis-
continuous (red lines) cases.

E = 69 · 109 N/m2, R = 2 m and ∆t = 1 s. We use the initial conditions

ϕ0 =


x2, x ∈ [0, 0.15],

−3(x− 0.15)2 − 0.3(x− 0.15)− 0.0225, x ∈ [0.15, 0.3]

2(x− 0.3)2 − 0.6(a− 0.3), x ∈ [0.3, 0.6]

−2(x− 0.6)2 − 0.8(a− 0.6), x ∈ [0.6, 1],

ϕ1 =

{
1 x ∈ (0, 0.15),

0 x ∈ [0.15, 1].

and ψ0 = ψ1 = ω0 = ω1 = 0. See Figures 5 and 6.

Figure 5. Evolution of the rotation angle of filaments: ψh,ε(x, t).

7.5. Nonlinear case - penalized problem.

7.5.1. Experiment: contact problem: g2 = −0.3, g3 = 0.3. We consider a rectangu-
lar arch beam with ρ1 = 0.2, ρ2 = 0.16, ρ3 = 0.3, k = 0.064, b = 1.0, and κ0 = 0.2.
The initial conditions ϕ(x, 0) = 0, ϕt(x, 0) = x, ψ(x, 0) = 0, ψt(x, 0) = sin(π2x),

ω(x, 0) = ωt(x, 0) = 0. Furthermore, we consider ∆t = 10−2 s. and the penalization
parameter ε = 10−1. See Figure 7.



22 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

Figure 6. Asymptotic behavior of the numerical energy
Eh(t, ϕh,ε, ψh,ε, ωh,ε) at time 100 s. Here we compared the numer-
ical experiments to continuous case (left side) versus discontinuous
case (right side), where we obtain the exponential and polynomial
stability, respectively.

Figure 7. Beam’s oscillations at the end x = `: ϕh,ε(`, t) and the
asymptotic behavior of the energy at 100 s we performed the ex-
periment for Iv = [0.6, 1] and viscoelastic damping function differ-
entiable (blue line and discontinuous (red line) cases, respectively,
see Theorem 5.2.

Acknowledgments. This research was supported by CNPq project 310249/2018-
0 and by project UBB 2020108 IF/I.



EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 23

References

[1] Avalos, G. G.; Muñoz Rivera, J. E.; Vera Villagran, O.; Stability in Thermoviscoelasticity
with Second Sound, Appl. Math Optim., Volume 1, No. 1 (2019), 1-2. doi: 10.1007/s00245-

018-9495-8

[2] Andrews, K. T.; Shillor, M.; Wright, S.; On the dynamic vibrations of a elastic beam
in frictional contact with a rigid obstacle, Journal of Elasticity, 42 (1996) 1-30. doi:

10.1007/BF00041221

[3] Borichev, A.; Tomilov, Y.; Optimal polynomial decay of functions and operator semigroups,
Math. Annalen, 347 (2009) 455-478. doi: 10.1007/s00208-009-0439-0

[4] Copetti, M. I. M.; Elliot, C. M.; A one dimensional quasi-static contact problem in linear
thermoelasticity, Eur. J. Appl. Math., 4 (1993) 151-174.doi: /10.1017/S0956792500001042

[5] Dumont, Y.; Paoli, L.; Vibrations of a beam between obstacles: convergence of a fully

discretized appproximation. ESAIM: Math. Model. Numer. Anal., 40 (2006) 705-734. doi:
10.1051/m2an:2006031

[6] Hughes, T. J. R.; The finite Element Method: Linear Static and Dynamic Finite Elelment

Analysis, Dover Publications Inc., 2000.
[7] Hughes, T. J. R.; Taylor, R. L.; Kanoknukulcahi, W.; A simple and efficient finite ele-

ment method for plate bending, Internat. J. Numer. Met. Engrg., 11 (1977) 1529-1543. doi:

10.1002/nme.1620111005
[8] Kim, J. U.; A boundary thin obstacle problem for a wave equation, Comm. Partial Diff.

Equations, Vol. 14, (8-9), pages 1011-1026, (1989). doi:10.1080/03605308908820640

[9] Kutter, K. L.; Shillor, M.; Vibrations of a Beam Between two stops, Dynamics of Continuous
and Discrete and Impulsive Systems. Ser. B Applications and Algorithms 8 (2001) 93-110.

[10] Lions, J. L.; Quelques méthodes de résolution des problemes aux limites non linéaires, Dunod,
Paris, 1969.

[11] Loula, A. F. D.; Franca, L. P.; Hughes, T. J .R.; Miranda, I.; Stability, convergence and

Accuracy of a New Finite Element Method for the Circular Arch Problem, Comp. Meth.
Appl. Mechanics and Engineering, 63 (1987) 281-303. doi: 10.1016/0045-7825(87)90074-0

[12] Muñoz Rivera, J. E.; da Costa Baldez, C. A.; The hybrid-penalized method for the Timo-

shenko’s beam, Journal of Mathematical Analysis and Applications, 458 (2018) 1274-1291.
doi: 10.1016/jmaa:2017.10.22

[13] Muñoz Rivera, J. E.; da Costa Baldez, C. A.; Stability for a boundary contact problem in

thermoelastic Timoshenkos beam, Z. Angew. Math. Phys. Volume L, No. 1, 1-18 (2021). doi:
10.1007/s00033-020-01

[14] Muñoz Rivera, J. E.; Villagran, O. V.; Sepulveda, M.; Stability to localized viscoelastic trans-

mission problem, Communications in Partial Differential Equations, 43:5 (2018) 821-838. doi:
10.1080/03605302.2018.1475490

[15] Newmark, M. N.; A method of computation for structural dynamics, J. Engrg. Mech. ,85

(1959) 67-94.
[16] Pazy A.; Semigroup of linear operators and applications to partial diferential equations,

Springer-Verlag. New York, 1983.
[17] Prathap, G.; Bhashyam, G. R.; Reduced integration and the shear-flexible beam element,

Internat.J. Numer. Methods Engrg., 18 (1982), 195-210. doi: 10.1002/nme.1620180205
[18] Prüss, J.; On the spectrum of C0-semigroups, Trans. AMS. 284 (1984) 847-857. doi:

10.2307/1999112

[19] Rifo, S.; Vera , O.; Munoz Rivera, J. E.; The lack of exponential stability of the hybrid Bresse

system, Journal of Mathematical Analysis and Applications, Volume 436, No. 1 (2015), 1-15.
doi: 10.1016/j.jmaa.2015.11.041

Jaime E. Muñoz Rivera
National Laboratory for Scientific Computation, Petrópolis, Rio de Janeiro, Brazil.

Department of Mathematics, Universidad del B́ıo B́ıo, Faculty of Science, 1202, Con-

cepción, Chile
Email address: jemunozrivera@gmail.com



24 J. E. MUNOZ RIVERA, C. A. DA COSTA BALDEZ, S. M. S. CORDEIRO EJDE-2024/17

Carlos A. da Costa Baldez

Department of Mathematics, Federal University of Pará, Bragança, Pará, Brazil.

Postgraduate Program in Mathematics and Statistics (PPGME), ICEN - Federal Uni-
versity of Pará, Pará, Brazil

Email address: baldez@ufpa.br

Sebastião M. S. Cordeiro

Department of Mathematics, Federal University of Pará, Abaetetuba, Pará, Brazil.

Doctorate Program in Mathematics (PDM), UFPA-UFAM
Email address: sebastiao@ufpa.br


	1. Introduction
	2. Existence: the hybrid-penalized method
	3. Asymptotic behavior
	4. Lack of exponential stability
	5. Penalized problem
	6. Signorini's problem
	7. Numerical results
	7.1. Variational formulation
	7.2. Algorithms and numerical approximation
	7.3. Cases Iv
	7.4. Linear case - without contact: g1=+, g2=-, g3=+.
	7.5. Nonlinear case - penalized problem
	Acknowledgments

	References