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Unilateral Problem for a Viscoelastic Beam Equation Type p-Laplacian with Strong Damping

and Logarithmic Source

Ducival C. Pereira1, Geraldo M. de Araújo2, Carlos A. Raposo3,∗

1Department of Mathematics, State University of Pará, Belém, PA, 66113-200, Brazil
ducival@uepa.br

2Department of Mathematics, Federal University of Pará, Belém, PA, 66075-110, Brazil
gera@ufpa.br

3Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Brazil
∗Correspondence: raposo@ufsj.edu.br

Abstract. In this manuscript, we investigate the unilateral problem for a viscoelastic beam equationof p-Laplacian type. The competition of the strong damping versus the logarithmic source term isconsidered. We use the potential well theory. Taking into account the initial data is in the stabilityset created by the Nehari surface, we prove the existence and uniqueness of global solutions by usingthe penalization method and Faedo-Galerkin’s approximation.

1. Introduction
We denote the p-Laplacian operator by ∆pu = div

(
|∇u|p−2∇u

), which can be extended to amonotone, bounded, hemicontinuos and coercive operator between the spaces W 1,p
0 (Ω) and its dualby

−∆p : W 1,p
0 (Ω)→ W−1,q(Ω), 〈−∆pu, v〉p =

∫
Ω

|∇u|p−2∇u · ∇v dx.

In [3] the authors establish existence of global solution to the problem
utt + ∆2u − ∆pu +

∫ t

0

g(t − s)∆u(s)ds − ∆ut + f (u) = 0 in Ω× R+, (1.1)
u = ∆u = 0 on Γ× R+, (1.2)

u(x, 0) = u0, ut(x, 0) = u1 in Ω, (1.3)
where Ω is a bounded domain of Rn with smooth boundary Γ = ∂Ω.Equations of the type (1.1) are related to models of elastoplastic microstructure flows. Asconsidered by An and Peirce [1, 2], they are essentially of the form

utt + uxxxx − a(u2
x )x = 0.

Received: 13 Nov 2021.
Key words and phrases. Unilateral problem; viscoelastic beam equation type p-Laplacian; logarithmic source.1

https://adac.ee
https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 2A more general equation,
utt + ∆2u − div(σ(|∇u|2)∇u)− ∆ut + h1(ut) + h2(u) = h3(x),

was considered by Yang et al [22–24]. They studied de existence of attractors and their Hausdorffdimensions. Another related equation is
utt + ∆2u − div(f0(∇u)) + kut = ∆(f1(u))− f2(u),

which was considered by Chueshov and Lasiecka [12]The problem (1.1), with its memory term ∫ t

0

g(t− s)∆u(s)ds , can be regarded as a fourth-orderviscoelastic plate equation with a lower order perturbation of the p-Laplacian type. This kind ofproblem can be also regarded as an elastoplastic flow equation with some kind of memory effect.We observe that for viscoelastic plate equation, it is usual consider a memory of the form∫ t

0

g(t − s)∆2u(s)ds,

see for instance [10]. However, because the main dissipation of the system (1.1) is given by strongdamping −∆ut , here we consider a weaker memory, acting only on ∆u. There is a large literatureabout stability in viscoelasticity. We refer the reader to [11,13].A nonlinear perturbation of problem (1.1) is given by
utt + ∆2u − ∆pu +

∫ t

0

g(t − s)∆u(s)ds − ∆ut + f (u) ≥ 0. (1.4)
Variational inequality theory was introduced by Hartman and Stampacchia (1966) [14] as a toolfor the study of partial differential equations with applications principally in mechanics.In [7] the authors investigated the unilateral problem associated with this perturbation, thatis, a variational inequality given for (1.4) (see [16]). Making use of the penalization method andGalerkin’s approximations, they established existence and the uniqueness of strong solutions.The unilateral problem is very interesting because, in general, dynamic contact problems arecharacterized by nonlinear hyperbolic variational inequalities. Variational inequality theory wasintroduced by Hartman and Stampacchia (1966) [14] as a tool for the study of partial differentialequations with applications principally in mechanics. Bensoussan and Lions (1982) [9] used vari-ational inequalities initially in the study of stochastic control. In [5] was obtained a variationalinequality for the Navier-Stokes operator with variable viscosity. In [6] was studied the contactproblem on the Oldroyd model of viscoelastic fluids. By using results from the theory of monotoneoperators, was established the existence of weak solutions. In [8] was studied the problem forparabolic variational inequalities with Volterra type operators. The authors proved the existenceand the uniqueness of the solution. For contact problems on elasticity and finite element method,see Kikuchi-Oden [15] and reference therein. In [18] was studied the unilateral problem for the

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 3Klein-Gordon operator with the nonlinearity of Kirchhoff-Carrier type. By using an appropriate pe-nalization was shown the existence and uniqueness of solutions for the perturbed equation. In [19]was considered the unilateral problem for a nonlinear wave equation with p-Laplacian operatorand source term. By using an appropriate penalization, authors obtained a variational inequalityfor the equation perturbed and then the existence of solutions was proved.In this work, we propose to investigate the existence and uniqueness of solutions for the vari-ational inequality associated with the problem (1.4) with the source term f (u) = −|u|r−2u ln |u|.More precisely, we investigate the existence and uniqueness of solutions for the unilateral problem
utt + ∆2u − ∆pu +

∫ t

0

g(t − s)∆u(s)ds − ∆ut ≥ |u|r−2u ln |u| in Ω×R+, (1.5)
u(x, 0) = u0(x), ut(x, 0) = u1(x) in Ω, (1.6)

u(x, t) = ∆u(x, t) = 0 on Γ× R+. (1.7)
This work is organized as follows: In section 2 we introduce the notation and some well-knownresults. In section 3 we introduce the potential theory suitable for our problem. In section 4 definestrong solution to the boundary value problem (1.5)-(1.7) and present the theorem of existence ofstrong solution. In section 5 we apply the penalization method. The existence of global solutionsis given by using Faedo-Galerkin approximation. Finally, in Section 6 we prove the result ofuniqueness.

2. Preliminaries
Let Ω be a bounded domain in Rn with the boundary Γ of class C2. For T > 0, we denote by Qthe cylinder Ω×(0, T ), with lateral boundary Σ = Γ×(0, T ). By 〈·, ·〉 we will represent the dualitypairing between a Banach space X and X ′, X ′ being the topological dual of the space X , and by

C we denote various positive constants. The inner product in H1
0(Ω) and L2(Ω) , respectively, willbe denoted by (∇·,∇·), (·, ·). The norm in Lp(Ω) will be denoted by | · |p .The inequality (1.5) must be satisfied in the following sense. Let

K = {v ∈ H1
0(Ω); v ≥ 0 a.e. in Ω}

be a closed and convex subset of H1
0(Ω), the unilateral problem consists to find a solution u(x, t)satisfying∫

Q

(utt + ∆2u − ∆pu +

∫ t

0

g(t − s)∆u(s)ds − ∆ut − |u|r−2u ln |u|)(v − ut) ≥ 0, (2.1)
for all v ∈ K with ut(x, t) ∈ K a.e. on [0, T ] and the initial and boundary data

u = ∆u = 0 in Γ× (0, T ), (2.2)
u(x, 0) = u0, ut(x, 0) = u1 in Ω. (2.3)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.5 4To study the existence and uniqueness of the problem (1.5)-(1.7), let us consider the followinghypotheses:
H1. Suppose that 

2 ≤ p , if n = 1, 2

2 ≤ p ≤
2n − 2

n − 2
, if n ≥ 3.H2. With respect to the power r , let us suppose that

2 < r < +∞ , if n = 1, 2

2 < r <
2n

n − 2
, if n ≥ 3.

H3. With respect to the function g : [0,+∞)→ R, we will assume that g ∈ C1[0, T ] and
g(0) > 0, I = 1− µ

∫ ∞
0

g(s)ds > 0,

where µ > 0 is the embedding constant for |∇u| ≤ √µ |∆u|, for all u ∈ H1
0(Ω) ∩H2(Ω).

H4. There exists a constant k1 > 0 such that
g′(t) ≤ −k1g(t), ∀t ≥ 0.

By H1 we have
H1

0(Ω) ∩H2(Ω) ↪→ W
1,2(p−1)
0 (Ω) ↪→ H1

0(Ω) ↪→ L2(Ω).

The Lemmas below will be a important role in this manuscript.
Lemma 2.1. (Sobolev Poincaré inequality) Let p be a number with 2 < p < ∞ if n = 1, 2 or
2 ≤ p ≤

2n

n − 2
if n ≥ 3, then there exists a constant C > 0 such that

|u|p ≤ C|∇u|,∀u ∈ H1
0(Ω)

Lemma 2.2. (Technical lemma) For v ∈ C1(0, T ;H1
0(Ω)), we have∫

Ω

∫ t

0

g(t − s)∇v · ∇vtdsdx =
1

2
(g′ � ∇v)(t)−

1

2
g(t)|∇v(t)|2

−
1

2

d

dt

[
(g � ∇v)(t)−

(∫ t

0

g(s)ds

)
|∇v(t)|2

]
,

where (g � ∇u)(t) =

∫ t

0

g(t − s)|∇u(s)−∇u(t)|2ds .
Proof. Differentiating the term (g � ∇u)(t) we arrive to the above inequality. �

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 53. Potential well
In this section, we use the potential well theory, a power full tool in the study of the globalexistence of solution in partial differential equation. See Payne-Sattinger [17]. It is well-knownthat the energy of a PDE system, in some sense, splits into kinetic and potential energy. The sourceterm induces potential energy in the system that acts in opposition to the effect of the stabilizingmechanism. In this sense, it is possible that the energy from the source term destabilizes all thesystem and produces a blow-up in a finite time. To provide a global solution, we are able toconstruct a stability set corresponding to the source term created from the Nehari Manifold, see Y.Ye [20]. In the stability set, there exists a valley or a well of the depth d created in the potentialenergy. If d is strictly positive, then we find that, for solutions with the initial data in the goodpart of the potential well, the potential energy of the solution can never escape the potential well.In general, the energy from the source term causes the blow-up in a finite time. However, thegood part of the potential well is an invariant set where it remains bounded. As a result, the totalenergy of the solution remains finite for any time interval [0, T ], providing the global existence ofthe solution.For the model considered here, the total energy is given by

E(t) =
1

2

[
|ut(t)|2 + |∆u(t)|2 +

2

p
|∇u(t)|pp + (g � ∇u)(t)

−
(∫ t

0

g(s)ds

)
|∇u(t)|2 +

2

r2
|u(t)|rr

−
2

r

∫
Ω

|u(t)|r ln |u(t)|dx
] (3.1)

and satisfies
d

dt
E(t) ≤ −|∇ut(t)|2. (3.2)

From (H3) we get
I(t) =

1

2

[
|ut(t)|2 +

(
1− µ

∫ t

0

g(s)ds

)
|∆u(t)|2 +(g � ∇u(t)+

2

p
|∇u(t)|pp

+
2

r2
|u(t)|rr −

2

r

∫
Ω

|u(t)|r ln |u(t)|dx
]
.

(3.3)
Then, we introduce the functional J : H1

0(Ω) ∩H2(Ω)→ R defined by
J(u) =

1

2

[(
1− µ

∫ t

0

g(s)ds

)
|∆u|2 +

2

p
|∇u(t)|pp + (g � ∇u)(t) +

2

r2
|u(t)|rr

−
2

r

∫
Ω

|u(t)|r ln |u(t)|dx
]
.

(3.4)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.5 6For u ∈ H1
0(Ω) ∩H2(Ω), we have
J(λu) =

λ2

2

(
1− µ

∫ t

0

g(s)ds

)
|∆u|2 +

λp

p
|∇u(t)|pp +

λ

2
(g � ∇u)(t)

+
λr

r2
|u(t)|rr −

λr

r

∫
Ω

|u(t)|r ln |u(t)|dx. (3.5)
Associated with J , we have the well-known Nehari Manifold given by

N def
=

{
u ∈ H1

0(Ω) ∩H2(Ω) \ {0};
[
d

dλ
J(λu)

]
λ=1

= 0

} (3.6)
or equivalently,

N =

{
u ∈ H1

0(Ω) ∩H2(Ω)) \ {0};
(

1− µ
∫ t

0

g(s)ds

)
|∆u|2

+ |∇u(t)|pp +
1

2
(g � ∇u)(t) =

∫
Ω

|u(t)|r ln |u(t)|dx
}
. (3.7)

We define as in the Mountain Pass theorem due to Ambrosetti and Rabinowitz [4]
d
def
= inf

u∈(H1
0(Ω)∩H2(Ω)\{0}

sup
λ≥0

J(λu).

similar to the result in [21] one has 0 < d = inf
u∈N

J(u).Now, we introduce
W = {u ∈ H1

0(Ω) ∩H2(Ω); J(u) < d} ∪ {0}

and partition it into two sets W = W1 ∪W2 as follows
W1 =

{
u ∈ W ;

(
1− µ

∫ t

0

g(s)ds

)
|∆u|2

+
2

p
|∇u(t)|pp + (g � ∇u)(t) +

2

r2
|u(t)|rr >

2

r

∫
Ω

|u(t)|r ln |u(t)|dx
}
∪ {0}

(3.8)
and

W2 =

{
u ∈ W ;

(
1− µ

∫ t

0

g(s)ds

)
|∆u|2

+
2

p
|∇u(t)|pp + (g � ∇u)(t) +

2

r2
|u(t)|rr <

2

r

∫
Ω

|u(t)|r ln |u(t)|dx
}
.

(3.9)
So, we define by W1 the set of stability for the problem (1.5)-(1.7), and before starting the sectionof existence and uniqueness of solution, we will prove thatW1 is invariant set for sub-critical initialenergy.

Proposition 1. Let u0 ∈ W1 and u1 ∈ H1
0(Ω). If E(0) < d then u(t) ∈ W1.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.5 7

Proof. Let T > 0 be the maximum existence time. From (3.2) we get
E(t) ≤ E(0) < d, for all t ∈ [0, T ).

and then,
1

2

∫
Ω

|ut(t)|2 dx + J(u(t)) < d, for all t ∈ [0, T ). (3.10)
Arguing by contradiction, we suppose that there exists a first t0 ∈ (0, T ) such that I(u(t0)) = 0and I(u(t)) > 0 for all 0 ≤ t < t0, that is,(

1− µ
∫ t

0

g(s)ds

)
|∆u(t0)|2 +

2

p
|∇u(t0)|pp + (g � ∇u)(t0) +

2

r2
|u(t0)|rr

=
2

r

∫
Ω

|u(t0)|r ln |u(t0)|dx

From the definition of N , we have that u(t0) ∈ N , which leads to
J(u(t0)) ≥ inf

u(t)∈N
J(u(t)) = d.

We deduce
1

2

∫
Ω

|ut(t0)|2 dx + J(u(t0)) ≥ d,

which contradicts with (3.10). Then u(t) ∈ W1 for all t ∈ [0, T ). �

4. Existence of strong solutions
Next, we shall state the main results of this paper.

Theorem 4.1. Consider the space

H3
Γ(Ω) = {u ∈ H3(Ω)|u = ∆u = 0 on Γ}.

If u0 ∈ W1 ∩H3
Γ(Ω), J(u0) < d, u1 ∈ H1

0(Ω) and the hypothesis (H1)-(H4) holds, then there exists
a function u : Ω× (0, T )→ R such that

u ∈ L∞(0, T ; (H1
0(Ω) ∩H2(Ω))) ∩ L∞(0, T ;H3

Γ(Ω)), (4.1)
ut ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1

0(Ω) ∩H2(Ω)), (4.2)
utt ∈ L∞(0, T ;H−1(Ω)), (4.3)
ut(t) ∈ K a.e. in [0, T ], (4.4)

∫ T

0

[
〈utt , v − ut〉+ (∆2u, v − ut)− (∆pu, v − ut)

+

(∫ t

0

g(t − s)∆u(s)ds, v − ut
)
− (∆ut , v − ut)

− (|u|r−2u ln |u|, v − ut)
]
≥ 0,

(4.5)

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Eur. J. Math. Anal. 10.28924/ada/ma.2.5 8

for all v ∈ L2(0, T ;H1
0(Ω)), v(t) ∈ K a.e. in t and initial data

u(0) = u0, ut(0) = u1.

The proof of Theorem 4.1 is given in Section 5 by the penalization method. It consists in con-sidering a perturbation of the problem (1.5) adding a singular term called penalization, dependingon a parameter ε > 0. We solve the mixed problem in Q for the penalization operator and theestimates obtained for the local solution of the penalized equation, allow to pass to limits, when εgoes to zero, in order to obtain a function u which is the solution of our problem. First of all, letus consider the penalization operator
β : H1

0(Ω) −→ H−1(Ω)

associated to the closed convex set K, cf. Lions [16], p. 370. The operator β is monotonous,hemicontinuous, takes bounded sets of H1
0(Ω) into bounded sets of H−1(Ω), its kernel is K and

β : L2(0, T ;H1
0(Ω)) −→ L2(0, T ; (H−1(Ω))

is monotone and hemicontinous. The penalized problem associated with the variational inequality(1.5)-(1.7), consists in given 0 < ε < 1, find uε satisfying
uεtt + ∆2uε − ∆pu

ε +

∫ t

0

g(t − s)∆uε(s)ds − ∆uεt

+
1

ε
(β(uεt ))− |uε|r−2uε ln |u| = 0, in Q (4.6)

and
uε(x, 0) = uε0(x), uεt (x, 0) = uε1(x) in Ω.

uε(x.t) = ∆uε(x, t) = 0 on ∂Ω× R+.
(4.7)

Definition 4.2. Suppose that uε0 ∈ W1, J(uε0) < d , uε1 ∈ H1
0(Ω) and hypothesis (H1) − (H4)

holds. A strong solution to the boundary value problem (4.6)-(4.7) is a function uε such that

uε ∈ L∞(0, T ;H1
0(Ω) ∩H2(Ω)),

uεt ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1
0(Ω)),

uεtt ∈ L2(0, T ; (H1
0(Ω) ∩H2(Ω))′)

satisfying for all w ∈ H1
0(Ω) ∩H2(Ω)

d

dt
(uεt (t), w) + (∆uε(t),∆w) + (−∆pu

ε(t), w) +

∫ t

0

g(t − s)(∆uε(s), w)ds

+(∇uεt (t),∇w) +
1

ε
(β(uεt (t)), w)−(|uε(t)|r−2uε(t) ln |uε(t)|), w) = 0

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Eur. J. Math. Anal. 10.28924/ada/ma.2.5 9

and initial data

uε(0) = uε0, u
ε
t (0) = uε1.

The solution of problem (4.6)-(4.7) is given by the following theorem:
Theorem 4.3. Assume that hypotheses (H1)− (H4) holds,

uε0 ∈ W1, J(uε0) < d and uε1 ∈ H1
0(Ω), (4.8)

then, for each 0 < ε < 1, there exists a function uε strong solution of (4.6)-(4.7).

5. Penalization method
In order to prove Theorem 4.1, we first prove the penalized Theorem 4.3. The existence ofglobal solutions will be given by using Faedo-Galerkin method. First we consider the approximateproblem. Then we obtain the a priori estimates needed to passage to the limit in the approximatesolutions.

5.1. Approximate problem. Let {wj} be the Galerkin basis given by eigenfunctions of ∆2 withboundary condition u = ∆u = 0 on Γ × R+ and let Vm ⊂ N be the subspace spanned by thevectors w1, w2, ..., wm.. Consider
uεm(t) =

m∑
j=1

gεjm(t)wj

solution of approximate problem
(uεmtt (t), w)+(∆uεm(t),∆w)+(−∆pu

εm(t), w)+

∫ t

0

g(t − s)(∆uεm(t), w)ds

−(|uεm(t)|r−2uεm(t) ln |uεm|, w) + (∇uεm(t),∇w) +
1

ε
(β(uεmt )(t), w) = 0 (5.1)

with initial conditions
uεm(0) = uε0m → uε0 strongly in H2(Ω) ∩H1

0(Ω), (5.2)
uεmt (0) = uε1m → uε1 strongly in L2(Ω). (5.3)

The system of ordinary differential equation (5.1) in the variable t has a local solution uεm(t)defined in [0, tm[, 0 < tm ≤ T . In the next step obtain priori estimates for the solution uεm(t) thatpermits us to extend this solution to the whole interval [0, T ].

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Eur. J. Math. Anal. 10.28924/ada/ma.2.5 105.2. First estimate. We consider w = uεmt in (5.1) to obtain
d

dt

[
1

2
|uεmt (t)|2 +

1

2
|∆uεm(t)|2 +

1

p
|∇uεm(t)|pp +

1

r2
|uεm(t)|pp

−
1

r

∫
Ω

|uεm(t)|r ln |uεm(t)|dx
]

+ |∇uεmt (t)|2 +
1

ε
(β(uεmt (t)), uεmt (t))

=

∫ t

0

g(t − s)(∇uεm(s),∇uεmt (t))ds.

(5.4)

We have (β(uεmt (t)), uεmt (t)) ≥ 0. Then from Lemma 2.2 and (H4)

1

2

d

dt

[
|uεmt (t)|2 + |∆uεm(t)|2 +

2

p
|∇uεm(t)|pp + (g � ∇uεm)(t)

−
(∫ t

0

g(s)ds

)
|∇uεm(t)|2 +

2

r2
|uεm(t)|rr

−
2

r

∫
Ω

|uεm(t)|r ln |uεm(t)|dx
]

+ |∇uεmt (t)|2+

≤
1

2
(g′ � ∇uεm)(t)−

1

2
g(t)|∇uεm(t)|2 ≤ 0.

(5.5)

Let
Eεm(t) =

1

2

[
|uεmt (t)|2 + |∆uεm(t)|2 +

2

p
|∇uεm(t)|pp + (g � ∇uεm)(t)

−
(∫ t

0

g(s)ds

)
|∇uεm(t)|2 +

2

r2
|uεm(t)|rr

−
2

r

∫
Ω

|uεm(t)|r ln |uεm(t)|dx
]
.

(5.6)

So, by (5.5) and (5.8), we have
d

dt
Eεm(t) ≤ −|∇uεmt (t)|2.

Integrating from 0 to t , t ≤ tm, we obtain
Eεm(t) +

∫ t

0

|∇uεmt (t)|2 ≤ Eεm(0). (5.7)
By (H3), it follows

1

2

[
|uεmt (t)|2 +

(
1− µ

∫ t

0

g(s)ds

)
|∆uεm(t)|2 +(g � ∇uεm)(t)+

2

p
|∇uεm(t)|pp

+
2

r2
|uεm(t)|rr −

2

r

∫
Ω

|uεm(t)|r ln |uεm(t)|dx
]

+

∫ t

0

|∇uεmt |2ds ≤ Eεm(t) ≤ Eεm(0) =
1

2
|uε1m|2 + C1J(uε0m),

(5.8)

where C1 > 0 is a positive constant, independent of m and t .

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 11We have J(u0εm) < d and by (5.3), there exists a constant C2 > 0 such that
|uεmt (t)|2 +

(
1− µ

∫ t

0

g(s)ds

)
|∆uεm(t)|2 +(g � ∇uεm)(t)+

2

p
|∇uεm(t)|pp

+
2

r2
|uεm(t)|rr −

2

r

∫
Ω

|uεm(t)|r ln |uεm(t)|dx

+

∫ t

0

|∇uεmt |2ds ≤ C2.

(5.9)

From (3.7) and (5.9) we get
∆uεm ⇀ ∆uε in L∞(0, T ;L2(Ω)), (5.10)

uεm ⇀ uε in L∞(0, T ;H1
0(Ω) ∩H2(Ω)), (5.11)

−∆pu
εm ⇀ χ in L2(0, T ;H−1(Ω)), (5.12)

uεmt ⇀ uεt in L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1
0(Ω)), (5.13)

β(uεmt ) ⇀ ψ in L2(0, T ;H−1(Ω)). (5.14)
Follows from(5.11), (5.13) and Aubin-Lions Theorem, for any T > 0,

uεm → uε in L2(0, T ;H1
0(Ω)), strong and a.e. in Q. (5.15)

Now, we prove that χ(t) = −∆pu
ε(t). We consider x, y ∈ R, p ≥ 2. Then the elementaryinequality ∣∣|x |p−2x − |y |p−2y

∣∣ ≤ C (|x |p−2 + |y |p−2
)
|x − y | (5.16)is a consequence of the mean value theorem. Using (5.16) and Hölder generalized inequality with

p − 2

2(p − 1)
+

1

2
+

1

2(p − 1)
= 1,

we deduce for θ ∈ D(0, T ) and v ∈ Vm,∣∣∣∣∫ T

0

〈(−∆uεmp (t))− (−∆uεp(t)), v〉pθ(t)dt

∣∣∣∣
=

∣∣∣∣∫ T

0

∫
Ω

(
|∇uεm(t)|p−2∇uεm(t)− |∇uε(t)|p−2∇uε(t)

)
∇v dx θ(t) dt

∣∣∣∣
≤ C|θ|∞

∫ T

0

∫
Ω

(
|∇uεm(t)|p−2 + |∇uε(t)|p−2

)
|∇uεm(t)−∇uε(t)||∇v | dx dt

≤ C1

∫ T

0

(
|∇uεm(t)|p−2

2(p−1)
+ |∇uε(t)|p−2

2(p−1)

)
|∇uεm(t)−∇uε(t)||∇v |2(p−1) dt

≤ C2

∫ T

0

|∇uεm(t)−∇uε(t)| dt

(5.17)

where C1 and C2 are positive constants independent of m and t .

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 12Now, from estimate (5.10) and (5.11), we have
d

dt
|∇uεm(t)−∇uε(t)|2 ≤ 2|∆(uεm(t)− uε(t))||∇(uεmt (t)− uεt (t))| ≤ C3,

where C3 is a constant independent of m and t .
So,

|uεm(t)− uε(t)|H1
0(Ω) ∈ H1[0, T ] ↪→ C[0, T ],

whence ∇uεm(t)→ ∇uε(t) a. e. in [0, T ]. Therefore, χ = −∆pu
ε.

Now, we observe that Sobolev inequality∫
Ω

||uεm(t)|r−2uεm(t) ln |uεm(t)||2dx ≤ |uεm(t)|2r2r ≤ C2r |∇uεm(t)|2r

≤ µrC2r |∆uεm(t)|r ≤ C4,

where C4 is a constant independent of m and t.Then
(|uεm|r−2uεm ln |uεm|) is bounded in L2(0, T ;L2(Ω)) = L2(Q). (5.18)

Using continuity of function s → |s|r−2s ln |s| and (5.15) we have
|uεm|r−2uεm ln |uεm| → |uε|r−2uε ln |uε| a.e. in Q. (5.19)

By (5.18), (5.19) and applying Lions Lemma (Lemma 1.3, page 12, [16]), we get
|uεm|r−2uεm ln |uεm|⇀ |uε|r−2uε ln |uε| weakly in L2(0, T ;L2(Ω)). (5.20)

5.3. Second estimate. Let us consider the initial data
uε0 ∈ H3

Γ(Ω), uε1 ∈ H1
0(Ω) and uεm0 = ∆uεm0 = 0 on Γ. (5.21)

We consider w = −∆uεmt in approximate equation (5.1).Then we have
d

dt

{
1

2
|∇uεmt (t)|2 +

1

2
|∇∆uεm(t)|2

}
+ 〈∆puεmt (t),∆uεmt (t)〉

+|∆uεmt (t)|2 +
1

ε
(β(uεmt (t),−∆uεmt (t))

= (|uεm(t)|r−2uεm ln |uεm(t)|,−∆uεmt (t))

+

∫ t

0

g(t − s)(∆uεm(s),∆uεmt (t))ds.

Now,
〈∆puεm(t),∆uεmt (t)〉 =

d

dt
〈∆puεm(t),∆uεm(t)〉 − J1,

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 13where
J1 =

∫
Ω

{
(p − 2)|∇uεm(t)|p−4(∇uεm(t) · ∇uεmt (t))∇uεm(t)

+|∇uεm(t)|p−2∇uεmt (t)
}
· ∇∆uεm(t)dx.

Then
d

dt

{
1

2
|∇uεmt (t)|2 +

1

2
|∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉

}
+|∆uεmt (t)|2 +

1

ε
(β(uεmt (t)),−∆uεmt (t)) = J1 + J2 + J3.

(5.22)
where

J2 =

∫
Ω

|uεm(t)|r−2uεm ln |uεm(t)|∆uεmt (t)

and
J3 =

∫ t

0

g(t − s)(∆uεm(s),∆uεmt (t))ds.

Let us the right hand side of (5.22). We denote by C a generic positive constant not dependingon m, t . By estimate (5.9) and
p − 2

2(p − 1)
+

1

2(p − 1)
+

1

2
= 1,

|J1| ≤ (p − 1)

∫
Ω

|∇uεm(t)|p−2|∇uεmt (t)||∇∆uεm(t)|dx

≤ (p − 1)|∇uεm(t)|p−2
2(p−1)

|∇uεmt (t)|2(p−1)|∇∆uεm(t)|

≤ C|∇uεmt (t)|2(p−1)|∇∆uεm(t)|.

How H1
0(Ω) ∩H2(Ω) ↪→ W 1,2

0 (Ω), we have
|∇uεmt (t)|22(p−1) ≤ µ2|∆uεmt (t)|2,

where µ2 > 0 is the corresponding embedding constant. Then
|J1| ≤

1

2
|∆uεmt (t)|2 + C|∇∆uεm(t)|2. (5.23)

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 14Let Ω1 = {x ∈ Ω : |uεm(t)| < 1} and Ω2 = {x ∈ Ω : |uεm(t)| ≥ 1}. By (5.9) and Sobolevinequality
|J2| ≤

∫
Ω1

||uεm(t)|r−2uεm ln |uεm(t)|∆uεmt (t)|dx

+

∫
Ω2

||uεm(t)|r−2uεm ln |uεm(t)|∆uεmt (t)|dx

≤ (e(r − 1))−1

∫
Ω

|∆uεmt (t)|dx + (e(r − 1))−1

∫
Ω

|uεm(t)|r−1|∆uεmt (t)|dx

≤ 2(e(r − 1))−2 +
1

8
|∆uεmt (t)|2 + 2(e(r − 1))−2|uεm(t)|2(r−1)

2(r−1)
+

1

8
|∆uεmt (t)|2

≤ 2(e(r − 1))−2 +
1

4
|∆uεmt (t)|2 + 2C(e(r − 1))−2|∇uεm(t)|2(r−1)

≤ C +
1

4
|∆uεmt (t)|2 (5.24)

where we have used |x r−1 ln x | ≤ (e(r − 1))−1 for 0 < x < 1 and
ln x ≤ (e(r − 1))−1x r−1, if x ≥ 1.
Remark 5.1. We note from the Cauchy-Schwarz inequality and Fubini’s Theorem follows

‖g � ∇u‖L2(Q) ≤ ‖g‖L1(0,∞)‖∇u‖L2(Q)

Again from estimate (5.9) and remark 5.1
|J3| ≤

(∫ t

0

g(t − s)|∆uεm(t)|ds
)
|∆uεmt (t)| (5.25)

≤ C‖g‖L1(R+)|∆uεmt (t)|

≤ C +
1

4
|∆uεmt (t)|2.

Follows from (5.22)-(5.25) that
d

dt

[
1

2
|∇uεmt (t)|2 +

1

2
|∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉

]
+

1

2
|∆uεmt (t)|2 +

1

ε
(β(uεmt (t)),−∆uεmt (t))

≤ C + C|∇∆uεm(t)|2.

(5.26)
Now, observe that

|〈∆puεm(t),∆uεm(t)〉| ≤
∫

Ω

|∇uεm(t)|p−1|∇∆uεm(t)|dx

≤ |∆uεm(t)|p−1
2(p−1)

|∇∆uεm(t)| (5.27)
≤ C + |∇∆uεm(t)|2,

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 15and then
C + |∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉 ≥ 0.

Therefore, there exists C0 > 0 such that
d

dt

[
1

2
|∇uεmt (t)|2 +

1

2
|∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉

]
+

1

2
|∆uεmt (t)|2 +

1

ε
(β(uεmt (t)),−∆uεmt (t))

≤ C0 + C0|∇∆uεm(t)|2 + 〈∆puεm(t),∆uεm(t)〉.

(5.28)
Taking into account that (β(uεmt (t),−∆uεmt (t)) ≥ 0, (5.21), integrating from 0 to t and applyingGronwall inequality, we obtain

|∇uεmt (t)|2 + |∇∆uεm(t)|2 +

∫ t

0

|∆uεmt (t)|2 ≤ C, (5.29)
then

uεm ⇀ uε in L∞(0, T ;H3
Γ(Ω)), weakly star. (5.30)

uεmt ⇀ uεt in L2(0, T ;H1
0(Ω) ∩H2(Ω)), weakly (5.31)

∆uεm ⇀ ∆uε in L∞(0, T ;H1
0(Ω)), weakly star. (5.32)

5.4. Third estimate. Let Pm be the ortogonal projection Pm : L2(Ω)→ Vm, that is
Pmφ =

m∑
n=1

(φ,wj)wj , φ ∈ L2(Ω).

Remark 5.2. By remark 5.1, we observe that if

ψ ∈ L2(0, T ;H1
0(Ω)) then

∫ t

0

g(t − s)ψ(s)ds ∈ L2(0, T ;H−1(Ω))

and by (5.12) −∆pu
εm ∈ L2(0, T ; (H−1(Ω)).

We obtain using the notation and ideas of Lions [16], pages 75-76, remark 5.2 and estimatesabove that
uεmtt ⇀ uεtt in L2(0, T ; (H−1(Ω)), weakly. (5.33)

(5.31), (5.33) and Aubin-Lions compactness Theorem imply that there exists a subsequence from
(uεmt ), still denoted by (uεmt ), such that

uεmt → uεt strongly in L2(0, T ;H1
0(Ω)) and a.e. in Q. (5.34)

Now, we are in position to prove Theorem 4.1.

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 165.5. Strong solution. Let v ∈ L2(0, T ;H1
0(Ω)) be v(t) ∈ K a. e. for t ∈ (0, T ). From (4.6)1follows that ∫ T

0

(uεtt , v − uεt )dt +

∫ T

0

(∆2uε, v − uεt )dt +

∫ T

0

(−∆pu
ε, v − uεt )dt

+

∫ T

0

(∫ t

0

g(t − s)∆uε(s)ds, v − uεt
)
dt +

∫ T

0

(−∆uεt , v − uεt )dt

−
∫ T

0

(|uε|r−2uε ln |uε|, v − uεt )dt =
1

ε

∫ T

0

(β(uεt ), uεt − v) dt

=
1

ε

∫ T

0

(β(uεt )− βv, uεt − v) dt ≥ 0,

(5.35)

because v ∈ K (β(v) = 0) and β is monotone.From (5.11), (5.12), (5.15), (5.20), (5.30), (5.31), (5.33), (5.34) and the Bannach-Steinhauss The-orem, it follows that there exists a subsequence (uε)0<ε<1, such that it converge to u as ε → 0,that is
uε ⇀ u in L∞(R+;H1

0(Ω) ∩H2(Ω)), (5.36)
−∆puε ⇀ −∆pu in L2(0, T ;H−1(Ω), (5.37)

uε → u in L2(0, T ;H1
0(Ω))and a.e. in Q, (5.38)

uε ⇀ u in L∞(0, T ;H3
Γ(Ω)), (5.39)

uεt ⇀ ut in L2(0, T ;H1
0(Ω) ∩H2(Ω)), (5.40)

uεtt ⇀ utt in L2(0, T ;H−1(Ω)), (5.41)
|uε|r−2uε ln |uε|⇀ |u|r−2u ln |u| in L2(0, T ;L2(Ω)), (5.42)

uεt → ut in L2(0, T ;H1
0(Ω)) and a.e. in Q. (5.43)

The convergences above are sufficient to pass to the limit in (5.35) with ε > 0 to conclude that(4.5) is valid. To complete the proof of Theorem 4.1, it remains to show that ut(t) ∈ K a.e.In the position, we observe that using convergences (5.10)-(5.16) and (5.30)-(5.32), making m →
∞ in (5.1), we can find uε such that

uεtt + ∆2uε − ∆pu
ε +

∫ t

0

g(t − s)∆uε(s)ds

− ∆uεt − |uε|r−2uε ln |uε|+
1

ε
β(uεt ) = 0 in L2(0, T ;H−1(Ω). (5.44)

Then,
β(uεt ) = ε[−uεtt − ∆2uε + ∆pu

ε −
∫ t

0

g(t − s)∆uε(s)ds + ∆uεt + |uε|r−2uε ln |uε|]. (5.45)
So,

β(uεt )→ 0 in D′(0, T ;H−1Ω).

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 17From (5.45) it follows that
β(uεt ) is bounded in L2(0, T ;H−1(Ω)),

therefore
β(uεt ) ⇀ 0 weak in L2(0, T ;H−1Ω). (5.46)

On the other hand we deduce from (5.45) that
0 ≤

∫ T

0

(β(uεt ), uεt ) dt ≤ ε C. (5.47)
Thus ∫ T

0

(β(uεt ), uεt )dt −→ 0. (5.48)
We have that ∫ T

0

(β(uεt )− β(ϕ), uεt − ϕ) dt ≥ 0, ∀ϕ in L2(0, T ;H1
0(Ω)),

because β is a monotonous operator. Thus,∫ T

0

(β(uεt ), uεt ) dt −
∫ T

0

(β(uεt ), ϕ) dt −
∫ T

0

(β(ϕ), uεt − ϕ) dt ≥ 0. (5.49)
From (5.40), (5.46) and (5.48) we obtain∫ T

0

(β(ϕ), ut(t)− ϕ) dt ≤ 0. (5.50)
Taking ϕ = ut − λv , with v ∈ L2(0, T ;H1

0(Ω)) and λ > 0, we deduce using the hemicontinuityof β that
β(ut(t)) = 0, (5.51)

and this implies that ut(t) ∈ K a. e.
6. Uniqueness

Let u1, u2 two solutions of (4.5) , w = u2− u1 and t ∈ (0, T ). Because ut ∈ L2(0, T ;H1
0(Ω), wecan talking u1

t (resp. u2
t ) in the inequality (4.5) relative to v2 (resp. v1) and adding up the resultswe obtain

−
∫ t

0

(wtt , wt)ds −
∫ t

0

(∆2w,wt)ds +

∫ t

0

(∆pu
1, wt)ds −

∫ t

0

(∆pu
2, wt)ds

+

∫ t

0

(∫ t

0

g(t − s)∆w(s)ds, wt

)
ds +

∫ t

0

(∆wt , wt)ds

−
∫ t

0

(|u1|r−2u1 ln |u1|, wt)ds +

∫ t

0

(|u2|r−2u2 ln |u2|, wt)ds ≥ 0,

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 18thus, we have
1

2

∫ t

0

d

dt

(
|wt(t)|2 + |∆w(t)|2

)
ds +

∫ t

0

|∇wt(t)|2ds

≤
∫ t

0

〈∆pu1(t)− ∆pu
2(t), wt(t)〉ds +

∫ t

0

∫ t

0

g(t − s)(∇w(s),∇wt(t))dsdσ∫ t

0

(
|u1(t)|r−2u1(t) ln |u1(t)| − |u2(t)|r−2u2(t) ln |u2(t)|, wt(t)

)
ds.

By Lemma 2.2, we derive
1

2

∫ t

0

d

dt

{
|wt(t)|2 + |∆w(t)|2 −

(∫ t

0

g(s)ds

)
|∇w(t)|2 + (g � ∇w)(t)

}
ds

+

∫ t

0

|∇wt(t)|2ds ≤
∫ t

0

|〈∆pu1(t)− ∆pu
2(t), wt(t)〉|ds

+

∫ t

0

∫
Ω

(
|u1(t)|r−2u1(t) ln |u1(t)| − |u2(t)|r−2u2(t) ln |u2(t)|, wt(t)

)
dxds.

(6.1)

From Mean Value Theorem,
|〈∆pu1(t)− ∆pu

2(t), wt(t)〉|

≤ C
(
|∇u1(t)|p−2

2(p−1)
+ |∇u2(t)|p−2

2(p−1)

)
|∇w(t)|2(p−1)|∇wt(t)|

≤ C|∆w(t)|2 +
1

4
|∇wt(t)|2, (6.2)

for some constant C > 0, and∫ t

0

∫
Ω

(
|u1(t)|r−2u1(t) ln |u1(t)| − |u2(t)|r−2u2(t) ln |u2(t)|, wt(t)

)
dxds

≤
∫ t

0

∫
Ω

|θu1(t) + (1− θ)u2(t))|r−2|w(t)||wt(t)|dxds

+(r − 1)

∫ t

0

∫
Ω

|θu1(t) + (1− θ)u2|r−2 ln |θu1(t)

+(1− θ)u2(t)||w(t)|wt(t)|dxds = I1 + I2, 0 < θ < 1.

(6.3)

Hence, from the Hölder inequality and Sobolev inequality, we have∫
Ω

|θu1(t) + (1− θ)u2(t))|r−2|w(t)||wt(t)|dx

≤ |θu1(t) + (1− θ)u2(t)|r−2
n(r−2)

|w(t)| 2n
n−2
|wt(t)|

≤ Cr−2
1 C2C3|∆w(t)||∇wt(t)| ≤ C|∆w(t)|2 +

1

4
|∇wt(t)|2, (6.4)

where C1 , C2 and C3 are constants satisfying
|θu1(t) + (1− θ)u2(t)|r−2

n(r−2)
| ≤ C1|θu1(t) + (1− θ)u2(t)|,

|w(t)| 2n
n−2
≤ C|w(t)| ≤ C2|∆w(t)| and |w(t)| ≤ C3|∇w(t)|.

https://doi.org/10.28924/ada/ma.2.5


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 19

Also we used the condition n(p − 2) <
2n

n − 2
.

Now, using the calculation similar to (5.24), it follows that∫
Ω

|θu1(t) + (1− θ)u2|r−2 ln |θu1(t) + (1− θ)u2(t)|ndx

≤ (e(r − 2)−n)|Ω|+ (e(r − 2))−n|θu1(t) + (1− θ)u2(t)|n(r−2)
n(r−2)

≤ (e(r − 2)−n)|Ω|+ (e(r − 2))−1C4|θu1(t) + (1− θ)u2(t)|n(r−2) ≤ C.

(6.5)
Inserting (6.5) into I2, we have

I2 = (r − 1)

∫ t

0

∫
Ω

|θu1(t) + (1− θ)u2|r−2 ln |θu1(t)

+(1− θ)u2(t)||w(t)|wt(t)|dxds

≤ (r − 1)

∫ t

0

(∫
Ω

||θu1(t) + (1− θ)u2|r−2 ln |θu1(t) + θu2(t)||ndx
) 1

n

×|wt(t)||w(t)| 2n
n−2
ds ≤ C|∆w(t)|2 +

1

4
|∇wt(t)|2.

(6.6)

By (6.1), (6.2), (6.4) and (6.6) we get∫ t

0

d

dt

{
|wt(t)|2 + |∆w(t)|2 −

(∫ t

0

g(s)ds

)
|∇w(t)|2 + (g � ∇w)(t)

}
ds

+

∫ t

0

|∇wt(t)|2ds ≤ C
∫ t

0

(|∆w(t)|2 + |∇wt(t)|2)ds.

(6.7)
Putting,

Φ(t) = |wt(t)|2 + |∆w(t)|2 −
(∫ t

0

g(s)ds

)
|∇w(t)|2 + (g � ∇w)(t)

and using (H3), we have
|∆w(t)|2 −

(∫ t

0

g(s)ds

)
|∇w(t)|2 ≥ I|∆w(t)|2 ≥ 0.

As (g � ∇w)(t) ≥ 0, we have from (6.7) that ∫ t

0

d

dt
Φ(t) ≤ CΦ(t) and because Φ(0) = 0, followsfrom the Gronwall lemma that

|wt(t)|2 + I|∆w(t)|2 ≤ Φ(t) ≤ 0,

which proves that w = 0 in H1
0(Ω) ∩H2(Ω).

References
[1] L. An, A. Pierce, The effect of microstructure on elastic-plastic models, SIAM J. Appl. Math. 54(3) (1994) 708-730.

https://doi.org/10.1137/S0036139992238498[2] L. An, A. Pierce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math. 55(1)(1995) 136-155. https://doi.org/10.1137/S0036139993255327[3] A. Andrade, M. A. Jorge Silva, T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memoryterms, Math. Meth. Appl. Sci. 35(4) (2012) 417-426. https://doi.org/10.1002/mma.1552

https://doi.org/10.28924/ada/ma.2.5
https://doi.org/10.1137/S0036139992238498
https://doi.org/10.1137/S0036139993255327
https://doi.org/10.1002/mma.1552


Eur. J. Math. Anal. 10.28924/ada/ma.2.5 20

[4] A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. FunctionalAnalysis 14(4) (1973) 349-381. https://doi.org/10.1016/0022-1236(73)90051-7[5] G. M. Araújo, S. B. Menezes, On a variational inequality for the Navier-stokes operator with variable viscosity,Commun. Pur. Appl. Anal. 1(3) (2006) 583-596. https://doi.org/10.3934/cpaa.2006.5.583[6] G. M. Araújo, S. B. Menezes, A. O. Marinho, On a variational inequality for the equation of motion of oldroyd fluid,Electron J. Differential Equations 69 (2009) 1-16. http://ejde.math.txstate.edu[7] G. M. Araújo, M. A. F. Araújo, D. C. Pereira, On a variational inequality for a plate equation with p-Laplacian andmemory terms, Appl. Anal. 1 (2020) 1-14. https://doi.org/10.1080/00036811.2020.1766028[8] M. Bokalo, O. Sus, Evolutionary variational inequalities with volterra type operators, Mathematics and Statistics7(5) (2019) 182-190. https://doi.org/10.13189/ms.2019.070504[9] A. Bensoussan, J. L. Lions, Contrôle Impulsionnel et Inèquations quasi Variationnelles, Math. Models MethodsInform. Sci. 11, Gauthier-Villars, Paris, 1982.[10] M. M. Cavalcanti, V. N. Domigos Cavalcanti, T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli withnonlocal dissipation in general domains, Differ. Integral. Equ. 17(5-6) (2004) 495-510.[11] M. M. Cavalcanti, H. P. Oquendo, Frictional versus viscoelastic damping in a semi linear wave equation, SIAM J.Control. Optim. 14(4) (2003) 1310-1324. https://doi.org/10.1137/S0363012902408010[12] I. Chueshov, I. Lasiecka, Existence and uniqueness of weak solutions and attractors global for a class of nonlinear 2DKirchhoff-Boussinesq models, Discret. Contin. Dyn. S. 15(3) (2006) 777-809. https://doi.org/10.3934/dcds.
2006.15.777[13] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal. 37 (1970) 297-308. https:
//doi.org/10.1007/BF00251609[14] P. Hartman, G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115 (1966)271-310. https://doi.org/10.1007/BF02392210[15] N. Kikuchi, J. T. Oden, Contacts Problems in Elasticity: A Study of Variational Inequalities and Finite ElementMethods, SIAM, Philadelphia, 1988.[16] J. L. Lions, Quelques Méthodes de Resolution des Problémes Aux Limites non Linéaires, Dunod, Paris, 1969.[17] L. E. Payne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22(1975) 273-303. https://doi.org/10.1007/BF02761595[18] C. A. Raposo, D. C. Pereira, G. Araújo, A. Baena, Unilateral problems for the Klein-Gordon operator with nonlinearityof Kirchhoff-Carrier type, Electron J. Differential Equations 137 (2015) 1-14. https://ejde.math.txstate.edu/
Volumes/2015/137/abstr.html[19] C. A. Raposo, D. C. Pereira, C. H. Maranhão, Unilateral problem for a nonlinear wave equation with p-Laplacianoperator, J. Appl. Anal. Comput. 11(1) (2021) 546-555. https://doi.org/10.11948/20200147[20] Y. Ye, Global existence and asymptotic behavior of solutions for a class of nonlinear degenerate wave equations,Differ. Equ. Nonlinear Mech. 2007 (2007) 1-9. https://doi.org/10.1155/2007/19685[21] M. Willem, Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications 24, BirkhouserBoston Inc. Boston, MA, 1996.[22] Y. Zhijian, Longtime behavior for a nonlinear wave equation arising in elastoplastic flow, Math. Meth. Appl. Sci.32(9) (2009) 1082-1104. https://doi.org/10.1002/MMA.1080[23] Y. Zhijian, Global attractores and their Hausdorff dimensions for a class of Kirchhoff models, J. Math. Phys. 51(3)(2010) 032701. https://doi.org/10.1063/1.3303633[24] Y. Zhijian, J. Baoxia, Global attractor for a class of Kirchhoff models, J. Math. Phys. 50(3) (2009) 032701. https:
//doi.org/10.1063/1.3085951

https://doi.org/10.28924/ada/ma.2.5
https://doi.org/10.1016/0022-1236(73)90051-7
https://doi.org/10.3934/cpaa.2006.5.583
http://ejde.math.txstate.edu
https://doi.org/10.1080/00036811.2020.1766028
https://doi.org/10.13189/ms.2019.070504
https://doi.org/10.1137/S0363012902408010
https://doi.org/10.3934/dcds.2006.15.777
https://doi.org/10.3934/dcds.2006.15.777
https://doi.org/10.1007/BF00251609
https://doi.org/10.1007/BF00251609
https://doi.org/10.1007/BF02392210
https://doi.org/10.1007/BF02761595
https://ejde.math.txstate.edu/Volumes/2015/137/abstr.html
https://ejde.math.txstate.edu/Volumes/2015/137/abstr.html
https://doi.org/10.11948/20200147
https://doi.org/10.1155/2007/19685
https://doi.org/10.1002/MMA.1080
https://doi.org/10.1063/1.3303633
https://doi.org/10.1063/1.3085951
https://doi.org/10.1063/1.3085951

	1. Introduction
	2. Preliminaries
	3. Potential well
	4. Existence of strong solutions
	5. Penalization method
	5.1. Approximate problem 
	5.2. First estimate
	5.3. Second estimate 
	5.4. Third estimate 
	5.5. Strong solution

	6. Uniqueness
	References

