Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 100, pp. 1–14.

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

GLOBAL SOLUTIONS TO A QUASILINEAR HYPERBOLIC

EQUATION

MANUEL MILLA MIRANDA, LUIZ A. MEDEIROS, ALDO T. LOUREDO

Communicated by Jerome A. Goldstein

Abstract. This article concerns the existence and decay of solutions of a
mixed problem for a quasilinear hyperbolic equation which has its motivation

in a mathematical model that describes the nonlinear vibrations of the cross-

section of a bar.

1. Introduction

Milla Miranda et al [16] presented a mathematical model for the small longitu-
dinal vibrations of the cross sections of a bar of length L which is clamped on one
end and the other end is glued in a mass M . This model has the form

u′′(x, t)− ∂

∂x
σ(ux(x, t)) = 0, 0 < x < L, t > 0;

u(0, t) = 0, Mu′′(L, t) + σ(ux(L, t)) = 0, t > 0;

u(x, 0) = u0(x), u′(x, 0) = u1(x), 0 < x < L,

(1.1)

where u(x, t) denotes the displacement of the cross section x of the bar at time t,
and u′ = ∂u

∂t .
To obtain (1.1) we use Hooke’s law τ(x, t) = σ(ux(x, t)) in which τ(x, t) and

ux(x, t) are the tension and the deformation of the bar at (x, t), respectively, and
σ(s) is a real function. The linear version of Problem (1.1) can be found in Timo-
shenko et al [18, p 387].

For a zero Dirichlet boundary conditions in (1.1), there are a lot of papers in-
vestigating the existence and decay of solutions of this problem, among of them
we can mention [2, 4, 5, 11, 12]. MacCamy and Mizel [11] proved that for some
functions σ(s) this problem has solutions that blow up in finite time. Dafermos [3]
consider (1.1) with σ(ux, u

′
x) and the boundary conditions

σ(ux(0, t), u′x(0, t)) = σ0(t), t ∈ [0, T ];

σ(ux(L, t), u′x(L, t)) = σ1(t), t ∈ [0, T ].

Then Dafermos [3] obtained the existence and decay of solutions.

2010 Mathematics Subject Classification. 35L15, 35L20, 35K55, 35L60, 35L70.
Key words and phrases. Quasilinear hyperbolic equation; longitudinal bar;

existence of solutions.
c©2020 Texas State University.

Submitted May 29, 2019. Published September 24, 2020.

1



2 M. MILLA MIRANDA, L. A. MEDEIROS, A. T. LOUREDO EJDE-2020/100

We focus our attention on Problem (1.1) with

σ(s) = |s|ps, with p > 0, (1.2)

M = 1, and internal damping. More precisely, we consider the problem

u′′(x, t)− ∂

∂x

(
|∂u
∂x

(x, t)|p ∂u
∂x

(x, t) +
∂u

∂x
(x, t)

)
= 0, 0 < x < L, t > 0;

u(0, t) = 0, u′′(L, t) + |∂u
∂x

(L, t)|p ∂u
∂x

(L, t) +
∂u

∂x
(L, t) = 0 t > 0;

u(x, 0) = u0(x),
∂u

∂t
(x, 0) = u1(x), 0 < x < L.

(1.3)

We observe that the function σ(s) given in (1.2) is different from the σ(s) consid-
ered in the above papers. Note also that the existence of global solutions of (1.3)
with zero Dirichlet boundary conditions and without internal damping is an open
problem (cf. J. L. Lions [9]). This justifies the introduction of the internal damping
for obtain the existence of global solutions of (1.3).

Tsutsumi [19] and Giorgio and Matarazzo [4] considered Problem (1.3) with zero
Dirichlet boundary conditions. They obtain global solutions for in an n-dimensional
case. Later Maia and Milla Miranda [13] analyzed Problem (1.3) with zero Dirich-
let boundary conditions in an abstract framework. The authors obtained global
solutions and decay of solutions for this problem and generalized the papers [4, 19].

Maia and Milla Miranda [13] found an estimate for (u′′m), where um is an approx-
imate solution of (1.3), to apply the theory of monotone operators. For that, the
eigenvectors of a positive self-adjoint operator of a Hilbert space and the projection
method are used. This approach does not work in Problem (1.3) because of the
boundary conditions (1.3)2

To overcome the above difficulty, the authors in [16] introduced in equation (1.3)1

the internal damping u′xxxx to obtain the existence and decay of solutions of (1.3).
Our objective in this article is not introduce new internal damping in (1.3)1,

but decrease the class of functions σ(s) given in (1.2) to obtain global solutions of
(1.3). More precisely, considering the truncated of functions |s|ps (see Examples in
Section 6), we succeed in to obtain the existence, uniqueness and exponential decay
of solutions of Problem (1.3) in an n-dimensional case.

In our approach to prove the existence of solutions, we use the Faedo-Galerkin
method with a special basis, the theory of monotone operators (cf. J. L. Lions [9]
and Medeiros and Pereira [15]) and results on the trace of non-smooth functions.
The estimate for (u′′m) is obtained thanks to the truncation of the functions |s|ps
and the special basis. In the decay of solutions is used a Liapunov functional (cf.
Komornik and Zuazua [8] and Komornik [7])

We note that it is not usual for hyperbolic problems to have an equation at the
boundary which contains a nonlinear term of the normal derivative and the second
derivative with respect to t, respectively, of the solution.

As far as we know, the only results on the existence of global solutions of (1.3) are
given in the present paper and in Milla Miranda et al [16]. In this case the existence
of solution for the linear case can also be obtained using semigroup theory as in
Goldstein [6].



EJDE-2020/100 QUASILINEAR HYPERBOLIC EQUATIONS 3

2. Notation and main results

Let Ω be open bounded set of Rn whose boundary Γ is constituted of two parts
Γ0 and Γ1 such that Γ = Γ0 ∪ Γ1 and Γ0 ∩ Γ1 = ∅. With ν(x) is denoted the unit
exterior normal at x ∈ Γ1.

The scalar product and norm of L2(Ω) are denoted, respectively, by (u, v) and
|u|. Let

H1
Γ0

= {v ∈ H1(Ω) : v = 0 on Γ0}
equipped with the scalar product

((u, v)) =

n∑
i=1

∫
Ω

∂u

∂xi

∂v

∂xi
dx

and norm ‖u‖ = ((u, u))1/2. Its dual is denoted by H−1
Γ0

(Ω). The notations and
results on Functional Analysis and Sobolev Spaces can be seen in Brezis [1], J. L.
Lions [10] and Medeiros and Milla Miranda [14].

We consider the functions σi : R→ R (i = 1, 2, . . . , n) such that

σi is globally Lipschitz, σi is increasing and σi(0) = 0, i = 1, 2, . . . , n. (2.1)

With the above notation, we introduce the quasilinear hyperbolic problem

u′′ −
n∑
i=1

∂

∂xi

[
σi
( ∂u
∂xi

)
+
∂u′

∂xi

]
= 0 in Ω× (0,∞),

u = 0 in Γ0 × (0,∞),
n∑
i=1

[
σi
( ∂u
∂xi

)
+
∂u′

∂xi

]
νi + u′′ = 0 on Γ1 × (0,∞),

u(0) = u0, u′(0) = u1 in Ω.

(2.2)

Here, u′ = ∂u
∂t . We obtain the following results.

Theorem 2.1. Assume hypotheses (2.1) hold and

u0, u1 ∈ H1
0 (Ω) ∩H2(Ω) with

∂u0

∂ν
=
∂u1

∂ν
= 0 on Γ1. (2.3)

Then, there exists an unique function u with

u ∈ L∞loc(0,∞;H1
Γ0

(Ω)),

u′ ∈ L∞(0,∞, L2(Ω)) ∩ L2(0,∞;H1
Γ0

(Ω)),

u′′ ∈ L∞(0,∞;L2(Ω)) ∩ L2(0,∞;H1
Γ0

(Ω)),

u′′ ∈ L∞(0,∞;L2(Γ1)),

(2.4)

such that u satisfies the equations

u′′ −
n∑
i=1

∂

∂xi

[
σi
( ∂u
∂xi

)
+
∂u′

∂xi

]
= 0 in L2

loc(0,∞;H1
Γ0

(Ω)), (2.5)

n∑
i=1

[
σi
( ∂u
∂xi

)
+
∂u′

∂xi

]
νi + u′′ = 0 in L2

loc(0,∞;H
1/2
Γ0

(Ω)) (2.6)

and the initial conditions

u(0) = u0, u′(0) = u1. (2.7)



4 M. MILLA MIRANDA, L. A. MEDEIROS, A. T. LOUREDO EJDE-2020/100

Let σ̂i(s) =
∫ s

0
σi(τ)dτ , i = 1, 2, . . . , n. The energy functional for (2.2) is

E(t) =
1

2
|u′(t)|2 +

n∑
i=1

∫
Ω

σ̂i
( ∂u
∂xi

)
dx+

1

2
|u′(t)|2L2(Γ1), t ≥ 0.

To state the estimates on the decay of E(t), we introduced some notation and
consider one more hypothesis on σi. We set the notation

|v|2 ≤ a1‖v‖2, ∀v ∈ H1
Γ0

(Ω),

|v|2L2(Γ1) ≤ a2‖v‖2, ∀v ∈ H1
Γ0

(Ω),
(2.8)

in which a1 and a2 are positive constants. We assume that there exist positive
constants bi (i = 1, 2, . . . , n) such that

s2 ≤ biσ̂i(s), ∀s ∈ R, i = 1, 2, . . . , n. (2.9)

Consider the constants

b = max{b1, . . . , bn}, d =
1

2
b(a1 + 1 + a2), (2.10)

ε0 = min{1

2
,

1

2d
}, ε1 = min{ 1

3a1
,

1

3a2
}, (2.11)

η = min{ε0, ε1} (2.12)

Theorem 2.2. Let u be the solution obtained in Theorem 2.1. Assume that (2.9)
is satisfied. Then

E(t) ≤ 3E(0) exp
(
− 2

3
ηt
)
, ∀t ≥ 0. (2.13)

To prove Theorem 2.1, we need some previous results.

3. Results

We denote by ki the Lipschitz constants of σi (i = 1, 2, . . . , n) and by k =
max{ki; i = 1, 2, . . . , n}. In rest of this article we use the notation.

〈Au, v〉 =

n∑
i=1

∫
Ω

σi
( ∂u
∂xi

) ∂v
∂xi

dx, u, v ∈ H1
Γ0

(Ω).

Proposition 3.1. We have

(i) A : H1
Γ0

(Ω)→ H−1
Γ0

(Ω);

(ii) A maps bounded sets of H1
Γ0

(Ω) into bounded sets of H−1
Γ0

(Ω);
(iii) A is monotone;
(iv) A is hemicontinuous.

Proof. We have

|〈Au, v〉| ≤
n∑
i=1

ki

∫
Ω

∣∣ ∂u
∂xi

∣∣∣∣ ∂v
∂xi

∣∣dx ≤ k‖u‖‖v‖.
Thus, Au ∈ H−1

Γ0
(Ω) and

‖Au‖H−1
Γ0

(Ω) ≤ k‖u‖, ∀u ∈ H1
Γ0

(Ω).

This inequality proves (i) and (ii). Item (iii) follows from the fact that each σi is
an increasing function. Item (iv) is proved by using the continuity of each σi and
the Lebesgue Dominated Convergence Theorem. �



EJDE-2020/100 QUASILINEAR HYPERBOLIC EQUATIONS 5

The following result is concerned with the trace of non-smooth functions. Con-
sider the Hilbert space

E(Ω) = {f = (f1, . . . , fn) ∈ (L2(Ω))n : div f ∈ L2(Ω)}

provided with the scalar product

(f, g)E(Ω) =

n∑
i=1

(fi, gi) + (div f, div g).

Note that (D(Ω))n is dense in E(Ω) (cf. Temam [17, Theorem 1.1, p.6]). Take
f ∈ (D(Ω))n and z ∈ H1

Γ0
(Ω). Then

(div f, z) = −
n∑
i=1

(
fi,

∂z

∂xi

)
+

∫
Γ1

( n∑
i=1

fiνi

)
zdΓ,

in which ν(x) = (ν1(x), . . . , νn(x)) is the unit outward normal at x ∈ Γ1. The
above motivates the following result.

Proposition 3.2. The map

E(Ω)→ H−1/2(Γ1), f 7→ γνf = f · ν

is continuous. Also we have

〈γνf, z〉X′×X = 〈f · ν, z〉X′×X = (div f, z) +

n∑
i=1

(
fi,

∂z

∂xi

)
for all z ∈ (D(Ω))n and all z ∈ H1

Γ0
(Ω). Here X = H1/2(Γ1).

Proof. Consider f ∈ (D(Ω))n and z ∈ H1/2(Γ1). By the trace Theorem there exists
w ∈ H1

Γ0
(Ω) such that γ0w = z and

‖w‖ ≤ C‖z‖H1/2(Γ1), (3.1)

in which C is a positive constant independent of w and z. We have

|〈γνf, z〉X′×X | ≤ |(div f, w)|+
n∑
i=1

∣∣(fi, ∂w
∂xi

)∣∣
≤ C1|div f |‖w‖+

( n∑
i=1

|fi|2
)1/2

‖w‖

≤ (C1 + 1)‖f‖E(Ω)‖w‖.

This inequality and (3.1) provide γνf ∈ H−1/2(Γ1) and

‖γνf‖H−1/2(Γ1) ≤ C2‖f‖E(Ω),

where C2 > 0 is a constant independent of f ∈ E(Ω). The proposition follows by
the denseness of (D(Ω))n in E(Ω). �



6 M. MILLA MIRANDA, L. A. MEDEIROS, A. T. LOUREDO EJDE-2020/100

4. Proof of Theorem 2.1

We used the Faedo-Galerkin method with a special basis of H1
Γ0

(Ω). Consider

a basis {w1, w2, . . . , } of H1
Γ0

(Ω) such that u0, u1 ∈ [w1, w2] where [w1, w2] is the
subspace generated by w1 and w2. Let um be an approximate solution of Problem
(2.2), that is,

um =

m∑
i=1

gjm(t)wj

and um be solution of the system

(u′′m, w) +

n∑
i=1

(
σi
(∂um
∂xi

)
,
∂w

∂xi

)
+ ((u′m, w)) + (u′′m, w)L2(Γ1) = 0,

∀w ∈ Vm = [w1, w2, . . . , wm],

um(0) = u0, u′m(0) = u1

(4.1)

First estimate. Setting w = u′m in (4.1)1, we obtain

1

2

d

dt
|u′m|2 +

n∑
i=1

d

dt

∫
Ω

σ̂i
(∂um
∂xi

)
dx+ ‖u′m‖2 +

1

2

d

dt
|u′m|2L2(Γ1) = 0.

Integrating on [0, t], 0 < t < tm, we obtain

1

2
|u′m(t)|2 +

n∑
i=1

∫
Ω

σ̂i
(∂um(t)

∂xi

)
dx+

∫ t

0

‖u′m(τ)‖2dτ +
1

2
|u′m(t)|2L2(Γ1)

=
1

2
|u1|2 +

n∑
i=1

∫
Ω

σ̂i
(∂u0

∂xi

)
dx+

1

2
|u1|2L2(Γ1).

(4.2)

Remark 4.1. We have

|σ̂i(s)| ≤ ki
s2

2
, ∀s ∈ R, i = 1, 2, , . . . , n.

Therefore, ∫
Ω

σ̂i
(∂u0

xi

)
dx ≤ ki

2

∫
Ω

(∂u0

xi

)
dx, i = 1, 2, . . . , n.

Taking into account Remark 4.1 in (4.2), we obtain

1

2
|u′m(t)|2 +

n∑
i=1

∫
Ω

σ̂i

(∂um(t)

∂xi

)
dx+

∫ t

0

‖u′m(τ)‖2dτ +
1

2
|u′m(t)|2L2(Γ1)

≤ C, ∀m, ∀t ∈ [0,∞).

(4.3)

We denote by C > 0 the various constants independent of m and t ∈ [0,∞).

Second estimate. Differentiate the approximate equation (4.1)1 with respect to
t then set w = u′′m. We obtain

1

2

d

dt
|u′m|2 +

n∑
i=1

(
σ′i
(∂um
∂xi

)∂u′m
∂xi

,
∂u′′m
∂xi

)
+ ‖u′′m‖2 +

1

2

d

dt
|u′′m|2L2(Γ1) = 0. (4.4)

We have∣∣ ∫
Ω

σ′i
(∂um
∂xi

)∂u′m
∂xi

,
∂u′′m
∂xi

dx
∣∣ ≤ ki|∂u′m

∂xi
| |∂u

′′
m

∂xi
| ≤ 1

2
k2|∂u

′
m

∂xi
|2 +

1

2
|∂u
′′
m

∂xi
|2.



EJDE-2020/100 QUASILINEAR HYPERBOLIC EQUATIONS 7

Thus ∣∣ ∫
Ω

σ′i
(∂um
∂xi

)∂u′m
∂xi

,
∂u′′m
∂xi

dx
∣∣ ≤ 1

2
k2‖u′m‖2 +

1

2
‖u′′m‖2, ∀m, ∀t ∈ [0,∞).

Combining this inequality with (4.4), then integrating on [0, t] and using estimate
(4.3), we obtain

1

2
|u′′m(t)|2 +

1

2

∫ t

0

‖u′′m(τ)‖2dτ +
1

2
|u′′m(t)|2L2(Γ1)

≤ 1

2
k2C +

1

2
|u′′m(0)|2 +

1

2
|u′′m(0)|2L2(Γ1), ∀m, ∀t ∈ [0,∞).

(4.5)

Next, we estimate the two last terms of the second member of (4.5).

Third estimate. Make t = 0 in the approximate equation (4.1)1 and then set
w = u′′m(0). We find

|u′′m(0)|2 + |u′′m(0)|2L2(Γ1)

= −
n∑
i=1

(
σi
(∂u0

∂xi

)
,
∂u′′m(0)

∂xi

)
−

n∑
i=1

(∂u1

∂xi
,
∂u′′m(0)

∂xi

)
.

(4.6)

Since u0 ∈ H1
0 (Ω) ∩H2(Ω), we have ∂u0

∂xi
= νi

∂u0

∂ν on Γ1. Also from (2.3), we have

∂u0

∂ν = 0 on Γ1. Then ∂u0

∂xi
= 0 on Γ1 and therefore σi

(
∂u0

∂ν

)
= 0 on Γ1. Thus by

Gauss’ Theorem (
σi
(∂u0

∂xi

)
,
∂u′′m(0)

∂xi

)
= −

(
σ′i
(∂u0

∂xi

)∂2u0

∂x2
i

, u′′m(0)
)
.

This implies ∣∣ n∑
i=1

(
σi
(∂u0

∂xi

)
,
∂u′′m(0)

∂xi

)∣∣ ≤ k|4u0||u′′m(0)|. (4.7)

In a similar way, we obtain∣∣ n∑
i=1

(∂u1

∂xi
,
∂u′′m(0)

∂xi

)∣∣ ≤ |4u1||u′′m(0)|. (4.8)

Taking into account (4.7) and (4.8) in (4.6), we obtain

|u′′m(0)|2 + |u′′m(0)|2L2(Γ1) ≤ C, ∀m.

This inequality and (4.5) provide

1

2
|u′′m(t)|2 +

1

2

∫ t

0

‖u′′m(τ)‖2dτ +
1

2
|u′′m(t)|2L2(Γ1) ≤ C, ∀m, ∀t ∈ [0,∞). (4.9)

By estimate (4.3) and the equality um(t) =
∫ t

0
u′m(τ)dτ + u0, we obtain that

(um) is bounded in L∞loc(0,∞;H1
Γ0

(Ω)).

This estimate, Proposition 3.1 and part (ii) imply

(Aum) is bounded in L∞loc(0,∞;H−1
Γ0

(Ω)). (4.10)



8 M. MILLA MIRANDA, L. A. MEDEIROS, A. T. LOUREDO EJDE-2020/100

Estimates (4.3), (4.9)-(4.10) provide a subsequence of (um), still denoted by
(um), and a function u such that

um → u weak star in L∞loc(0,∞;H1
Γ0

(Ω)),

Aum → χ weak star in L∞loc(0,∞;H−1
Γ0

(Ω)),

u′m → u′ weak star in L∞(0,∞;L2(Ω)),

u′m → u′ weak in L2(0,∞;H1
Γ0

(Ω)),

u′′m → u′′ weak star in L∞(0,∞;L2(Ω)),

u′′m → u′′ weak in L2(0,∞;H1
Γ0

(Ω)),

u′m → u′ weak star in L∞(0,∞;L2(Γ1)),

u′′m → u′′ weak star in L∞(0,∞;L2(Γ1)).

(4.11)

The above convergences allow us to pass the limit in the approximate equation
(4.1)1 and obtain∫ ∞

0

(u′′, z)dt+

∫ ∞
0

〈χ, z〉dt+

∫ ∞
0

((u′, z))dt+

∫ ∞
0

(u′′, z)L2(Γ1)dt = 0, (4.12)

for all z ∈ L2
loc(0,∞;H1

Γ0
(Ω)), z with compact support.

Convergence of (Aum). In this part, we use the method of the monotone operator
(cf. J.L. Lions [9] and Medeiros and Pereira [15]). Fix an arbitrary T > 0. As A is
monotone, we have∫ T

0

〈Av −Aum, v − um〉dt ≥ 0, ∀v ∈ L1(0, T ;H1
Γ0

(Ω)).

Then by convergence (4.11), we find that∫ T

0

〈Av, v − u〉dt−
∫ T

0

〈χ, v〉dt+ lim sup

∫ T

0

〈Aum, um〉dt ≥ 0. (4.13)

By the approximate equation (4.1)1, we obtain∫ T

0

〈Aum, um〉dt

= −(u′m(T ), um(T )) + (u1, u0) +

∫ T

0

|u′m|2dt−
1

2
‖um(T )‖2 +

1

2
‖u0‖2

− (u′m(T ), um(T ))L2(Γ1) + (u1, u0)L2(Γ1) +

∫ T

0

|u′m|2L2(Γ1) = 0.

(4.14)

Now we will find the limit of first and third term of the second member of the last
equality. By convergences (4.11)1, (4.11)3, the compact embedding of H1

Γ0
(Ω) in

L2(Ω) and the Aubin-Lions Compactness Theorem, we have

um(T )→ u(T ) in L2(Ω).

Note that convergences (4.11)3 and (4.11)5 provide

u′m(T )→ u′(T ) weak in L2(Ω).

Convergences (4.11)4 and (4.11)5 and the compactness embedding of H1
Γ0

(Ω) in

L2(Ω) imply
u′m → u′ in L2(0, T ;L2(Ω)).



EJDE-2020/100 QUASILINEAR HYPERBOLIC EQUATIONS 9

The above three convergences provide

− (u′m(T ), um(T )) +

∫ T

0

|u′m(t)|2dt→ −(u′(T ), u(T )) +

∫ T

0

|u′(t)|2dt. (4.15)

On the other hand, convergences (4.11)1 and (4.11)3 imply

um(T )→ u(T ) weak in H1
Γ0

(Ω).

Thus

lim sup
(
− 1

2
‖um(T )‖2

)
≤ −1

2
‖u(T )‖2. (4.16)

By convergences (4.11)1, (4.11)4 and noting that the embedding of H1/2(Γ1) in
L2(Γ1) is compact, we obtain

um(T )→ u(T ) in L2(Γ1).

Also (4.11)7 and (4.11)8 imply

u′m(T )→ u′(T ) weak in L2(Γ1)

and (4.11)4, (4.11)8 imply

u′m → u′ in L2(0, T ;L2(Γ1)).

The las two convergences provide

− (u′m(T ), um(T ))L2(Γ1) +

∫ T

0

|u′m(t)|2L2(Γ1)dt

→ −(u′(T ), u(T ))L2(Γ1) +

∫ T

0

|u′(t)|2L2(Γ1)dt.

(4.17)

From (4.14), (4.15), (4.16) and (4.17) we obtain

lim sup

∫ T

0

〈Aum, um〉dt

≤ −(u′(T ), u(T )) + (u1, u0) +

∫ T

0

|u′(t)|2dt− 1

2
‖u(T )‖2

+
1

2
‖u0‖2 − (u′(T ), u(T ))L2(Γ1) + (u1, u0)L2(Γ1) +

∫ T

0

|u′(t)|2L2(Γ1)dt.

(4.18)

Make z = u1(0,T ) in (4.12), where 1(0,T ) is the characteristic function of the interval
(0, T ). We obtain∫ T

0

〈χ, u〉dt =− (u′(T ), u(T )) + (u1, u0) +

∫ T

0

|u′(t)|2dt− 1

2
‖u(T )‖2 +

1

2
‖u0‖2

− (u′(T ), u(T ))L2(Γ1) + (u1, u0)L2(Γ1) +

∫ T

0

|u′(t)|2L2(Γ1)dt.

Comparing this equality with (4.18), we derive

lim sup

∫ T

0

〈Aum, um〉dt ≤
∫ T

0

〈χ, u〉dt.

Taking into account the last inequality in (4.13), we find∫ T

0

〈Av, v − u〉dt−
∫ T

0

〈χ, v − u〉dt ≥ 0, ∀v ∈ L1(0, T ;H1
Γ0

(Ω)).



10 M. MILLA MIRANDA, L. A. MEDEIROS, A. T. LOUREDO EJDE-2020/100

This inequality and the hemicontinuity of A provide

χ = Au in L∞(0, T ;H−1
Γ0

(Ω)).

By diagonalization process and noting that T > 0 was arbitrary, this equality
implies

χ = Au in L∞loc(0,∞;H−1
Γ0

(Ω)).

Thus equation (4.12) becomes∫ ∞
0

(u′′, z)dt+

∫ ∞
0

〈Au, z〉dt+

∫ ∞
0

((u′′, z))dt+

∫ ∞
0

(u′′, z)L2(Γ1)dt = 0, (4.19)

for all z ∈ L2
loc(0,∞;H1

Γ0
(Ω)), z with compact support.

Taking z ∈ D(Ω× (0,∞)) in (4.19) and noting that u′′ belongs to L2(0,∞;H1
Γ0

),
we obtain equation (2.5).

Consider f = (f1, f2, . . . , fn), where

fi = σi
( ∂u
∂xi

)
+
∂u′

∂xi
, i = 1, 2, . . . , n.

Then by (2.5) we obtain f ∈ [L2
loc(0,∞;L2(Ω))]n and div f ∈ L2

loc(0,∞;L2(Ω)).

Therefore by Proposition 3.2, we find γνf ∈ L2
loc(0,∞;H−1/2(Γ1)).

Multiply both sides of (2.5) by z, z ∈ L2
loc(0,∞;H1

Γ0
(Ω)) of compact support,

and then integrate. We obtain∫ ∞
0

(u′′, z)dt+

∫ ∞
0

〈Au, z〉dt+

∫ ∞
0

((u′′, z))dt−
∫ ∞

0

〈γνf, γ0z〉dt = 0.

On the other hand, equation (4.19) implies∫ ∞
0

(u′′, z)dt+

∫ ∞
0

〈Au, z〉dt+

∫ ∞
0

((u′′, z))dt+

∫ ∞
0

(u′′, z)L2(Γ1)dt = 0.

Comparing the last two equations, we obtain

γνf + u′′ = 0 in L2
loc(0,∞;L2(Γ1)).

Then the regularity of u′′ given by (4.11)6, allows us to obtain equation (2.6).
Convergence (4.11) say us that u belong to class (2.4). The verification of the ini-

tial conditions (2.7) follows by convergences (4.11). Thus the proof of the existence
of solutions is concluded.

Uniqueness. Let u and v be in the class (2.4) that satisfy (2.5)-(2.7). Consider
w = u− v. Introduce the notation

Biu = σi
( ∂u
∂xi

)
+
∂u′

∂xi
, i = 1, 2, . . . , n.

For short notation, we write
∑

instead of
∑n
i=1. By equation (2.5), we obtain

(w′′, w′)−
(∑ ∂

∂xi
Biu−

∑ ∂

∂xi
Biv, w

′
)

= 0

Then, by Proposition 3.2 and (2.6), we have

(w′′, w′) +
∑([

σi
( ∂u
∂xi

)
− σi

( ∂v
∂xi

)]
,
∂w′

∂xi

)
+ ‖w′‖2 +

∫
Γ1

w′′w′dΓ = 0,



EJDE-2020/100 QUASILINEAR HYPERBOLIC EQUATIONS 11

that is,
1

2

d

dt
|w′|2 + ‖w′‖2 +

1

2

d

dt
|w′|2L2(Γ1)

= −
∑[(

σi
( ∂u
∂xi

)
− σi

( ∂v
∂xi

)
,
∂w′

∂xi

)]
.

(4.20)

Modifying the last term of this expression, we have∑[(
σi
( ∂u
∂xi

)
− σi

( ∂v
∂xi

)
,
∂w′

∂xi

)]
≤ k

∑
| ∂w
∂xi
| |∂w

′

∂xi
| ≤ k‖w‖‖w′‖.

Fix an arbitrary real number T > 0. Taking into account the last inequality in
(4.20) and then integrating on [0, s], 0 < s ≤ T , we obtain

1

2
|w′(s)|2 +

∫ s

0

‖w′(τ)‖2dτ +
1

2
|w′(s)|2L2(Γ1) ≤ k

∫ s

0

‖w(τ)‖‖w′(τ)‖dτ. (4.21)

From the equality w(τ) =
∫ τ

0
w′(σ)dσ, we derive

‖w(τ)‖2 ≤ τ
∫ τ

0

‖w′(σ)‖2dσ.

Thus by using this inequality and Cauchy-Schwarz inequality in (4.21), we derive

1

2
|w′|2 +

∫ s

0

‖w′(τ)‖2dτ +
1

2
|w′|2L2(Γ1) ≤ ks

∫ s

0

‖w′(τ)‖2dτ.

Choose 0 < s0 ≤ T such that ks0 ≤ 1. Then the last inequality implies

1

2
|w′(s)|2 +

1

2
|w′(s)|2L2(Γ1) ≤ 0, for 0 ≤ s ≤ s0.

Thus,

w(s) = 0, w′(s) = 0, ∀s ∈ [0, s0].

We apply the above arguments to the interval [s0, T ]. Since s0 does not depend on
T , we obtain

w(s) = 0, w′(s) = 0, ∀s ∈ [s0, 2s0].

After a finite number of steps, we prove that w(t) = 0 for all t ∈ [0, T ]. As T > 0
was arbitrary, we conclude that u = v on [0,∞).

5. Proof of Theorem 2.2

Let u be the solution given by Theorem 2.1. Multiplying both sides of equation
(2.5) by u′, we obtain

d

dt

[1

2
|u′(t)|2 +

n∑
i=1

∫
Ω

σ̂i
( ∂u
∂xi

)
dx+

1

2
|u′(t)|2L2(Γ1)

]
= −‖u′(t)‖2;

that is,
d

dt
E(t) = −‖u′(t)‖2. (5.1)

Also multiply both sides of equation (2.5) by u. We find

d

dt

[
(u′(t), u(t)) +

1

2
‖u(t)‖2 + (u′(t), u(t))L2(Γ1)

]
= |u′(t)|2 −

n∑
i=1

∫
Ω

σi
(∂u(t)

∂xi

)∂u(t)

∂xi
dx+ |u′(t)|2L2(Γ1);



12 M. MILLA MIRANDA, L. A. MEDEIROS, A. T. LOUREDO EJDE-2020/100

that is,

d

dt
ρ(t) = |u′(t)|2 −

n∑
i=1

∫
Ω

σi
(∂u(t)

∂xi

)∂u(t)

∂xi
dx+ |u′(t)|2L2(Γ1), (5.2)

where

ρ(t) = (u′(t), u(t)) +
1

2
‖u(t)‖2 + (u′(t), u(t))L2(Γ1), t ≥ 0.

Consider ε > 0. We introduce the perturbed energy

Eε(t) = E(t) + ερ(t), t ≥ 0.

Relation between Eε(t) and E(t). We have

|Eε(t)− E(t)| = ε|ρ(t)|. (5.3)

We obtain

|ρ(t)| ≤ 1

2
|u′(t)|2 +

1

2
(a1 + 1 + a2)‖u(t)‖2 +

1

2
|u(t)|2L2(Γ1), t ≥ 0,

where a1 and a2 were introduced in (2.8). Then by hypothesis (2.9), we have

|ρ(t)| ≤ 1

2
|u′(t)|2 + d

n∑
i=1

∫
Ω

σ̂i
(∂u(t)

∂xi

)
dx+

1

2
|u(t)|2L2(Γ1).

Consider ε0 = min{ 1
2 ,

1
2d}. Then

ε|ρ(t)| ≤ 1

2
E(t), ∀0 < ε ≤ ε0. (5.4)

From (5.3) and (5.4) it follows that

1

2
E(t) ≤ Eε(t) ≤

3

2
E(t), ∀t ≥ 0, ∀0 < ε ≤ ε0. (5.5)

Boundedness of E′ε(t). From (5.1) and (5.2), we obtain

E′ε(t) = −‖u′(t)‖2 + ε
[
|u′(t)|2 −

n∑
i=1

∫
Ω

σi
(∂u(t)

∂xi

)∂u(t)

∂xi
dx

+ |u′(t)|2L2(Γ1)

]
.

(5.6)

By (2.8) we deduce that

− ‖u′(t)‖2 ≤ − 1

2a1
|u′(t)|2 − 1

2a2
|u′(t)|2L2(Γ1). (5.7)

Since σi is an increasing continuous function, we have

σ̂i(s) ≤ sσi(s), ∀s ∈ R.

Thus

−
∑∫

Ω

σi
(∂u(t)

∂xi

)∂u(t)

∂xi
dx ≤ −

n∑
i=1

∫
Ω

σ̂i
(∂u(t)

∂xi

)
dx. (5.8)

Taking into account (5.7) and (5.8) in (5.6), we have

E′ε(t) ≤ −
( 1

2a1
− ε
)
|u′(t)|2 − ε

n∑
i=1

∫
Ω

σ̂i
(∂u(t)

∂xi

)
dx−

( 1

2a2
− ε
)
|u′(t)|2L2(Γ1).



EJDE-2020/100 QUASILINEAR HYPERBOLIC EQUATIONS 13

Take ε1 = min{ 1
3a1

, 1
3a2
}. Then the above inequality implies

E′ε(t) ≤ −
ε

2
|u′(t)|2 − ε

n∑
i=1

∫
Ω

σ̂i
(∂u(t)

∂xi

)
dx− ε

2
|u′(t)|2L2(Γ1);

that is,

E′ε(t) ≤ −εE(t), for 0 < ε ≤ ε1. (5.9)

Consider η > 0 defined in (2.12). Then by (5.9) and (5.5), we obtain

E′η(t) ≤ −2η

3
Eη(t),

and therefore

Eη(t) ≤ Eη(0) exp
(
− 2

3
ηt
)
.

This inequality and (5.5) provide inequality (2.13).

6. Examples

In what follows we will give examples of functions that satisfy the hypotheses
considered in Section 1. Consider real numbers p and Li with p ≥ 1 and Li > 1.
The function

σi(s) =


Lpi s, s > Li

|s|ps, −Li ≤ s ≤ Li
Lpi s, s < −Li.

satisfies hypothesis (2.1). The function

σi(s) =



Lpi s, s > Li

|s|ps, 1 < s ≤ Li
s, −1 ≤ s ≤ 1

|s|ps, −Li ≤ s < −1

Lpi s, s < −Li
satisfies hypotheses (2.1) and (2.9).

References

[1] H. Brezis; Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer,

New York, 2011.
[2] J. C. Clements; On the existence and uniqueness of solutions of the equation utt− ∂

∂xi
σ(uxi )−

∆Nut = f , Canad. Math. Bull., 16(2) (1975), 181-187.
[3] M. Dafermos; The mixed initial boundary value problem for equation of nonlinear one di-

mensional viscoelasticity, J. Differ. Eq., 6(1) (1969), 71-86.

[4] G. Giorgi, G. Matarazzo; An existence theorem for nonlinear evolution equation in viscoelas-
ticity, Ann. Univ. Ferrara- Sez. VII- Sc. Mat., XXVI (1980), 113-124.

[5] J. M. Greenberg, R. C. MacCamy, V. L. Mizel; On the existence uniqueness and stability

of solutions of the equation σ′(ux)uxxx + λuxtx = ρ0utt, J. Math. and Mech., 17 (1968),
707-728.

[6] J. A. Goldstein; Semigroups and second order differential equations, J. Funct. Anal., 4(1)

(1969), 50-70.
[7] V. Komornik; Exact Controllability, The Multiplier Method, John Wiley & Sons and Masson,

1994.

[8] V. Komornik, E. Zuazua; A direct method for boundary stabilization of the wave equation,
J. Math. Pure Appl., 69 (1990), 33-54.



14 M. MILLA MIRANDA, L. A. MEDEIROS, A. T. LOUREDO EJDE-2020/100

[9] J. L. Lions; Quelques méthodes de résolutions des problèmes aux limites non lineaires, Dunod,

Paris, 1969.

[10] J. L. Lions; Problèmes aux limites dans les équations aux derivées partielles. Oeuvres Choisies
de Jacques Louis Lions, Vol. I, EDP Sciences Ed., Paris (2003), 431-588.

[11] R. C. MacCamy, V.J. Mizel; Existence and nonexistence in large of solution of quasilinear

wave equation, Arc. Rat. Mech. and Analysis 25 (1967), 299-309.
[12] R. C. MacCamy; Existence uniqueness and stability of solutions of the equation utt =

∂
∂x
σ(ux) + λ(ux)utt, Indiana Univ. Math. J., 20(3) (1970/71), 231–238.

[13] S. Maia, M. Milla Miranda; Existence and decay of solutions of an abstract second order

nonlinear problem, J. Math. Analysis Appl., 358 (2009), 445-456
[14] L. A. Medeiros, M. Milla Miranda; Espaços de Sobolev (Iniciação aos Problemas Eĺıticos

Não Homogêneos), IM-UFRJ, Rio de Janeiro, RJ, 2011.

[15] L. A. Medeiros, D. C. Pereira; Problemas de Contorno para Operadores Diferenciais Parciais
Não Lineares, IM-UFRJ, Rio de Janeiro, RJ, 1990.

[16] M. Milla Miranda, L. A. Medeiros, A.T. Louredo; Global solutions for a nonlinear model for

longitudinal vibrations of a bar, to appear.
[17] R. Temam; Navier-Stokes Equation, Studies in Mathematics ans its Applications, V.2, North-

Holland Publishing Company, Amsterdam, 1979.

[18] S. Timoshenko, D. H. Young, W. Weaver Jr.; Vibration problems in Enginearing, J. Wiley
& Sons, New York., 1974.

[19] M. Tsutsumi; Some nonlinear evolution equations of second order, Proc. Japan Acad. 47

(1971), 950-955.

Manuel Milla Miranda

Universidade Estadual da Paráıba, DM, PB, Brazil

Email address: mmillamiranda@gmail.com

Luiz A. Medeiros

Universidade Federal do Rio de Janeiro, IM, RJ, Brazil
Email address: luizadauto@gmail.com

Aldo T. Louredo

Universidade Estadual da Paráıba, DM, PB, Brazil
Email address: aldolouredo@gmail.com


	1. Introduction
	2. Notation and main results
	3. Results
	4. Proof of Theorem 2.1
	First estimate
	Second estimate
	Third estimate
	Convergence of (Aum)
	Uniqueness

	5. Proof of Theorem 2.2
	Relation between E(t) and E(t)
	Boundedness of E'(t)

	6. Examples
	References