Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 137, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
UNILATERAL PROBLEMS FOR THE KLEIN-GORDON
OPERATOR WITH NONLINEARITY OF
KIRCHHOFF-CARRIER TYPE
CARLOS RAPOSO, DUCIVAL PEREIRA, GERALDO ARAUJO, ANTONIO BAENA
Abstract. This work concerns the unilateral problem for the Klein-Gordon
operator
∂2u
− M (|∇u|2 )∆u + M1 (|u|2 )u − f.
L=
∂t2
Using an appropriate penalization, we obtain a variational inequality for a
perturbed equation, and then show the existence and uniqueness of solutions.
1. Introduction
The one-dimensional nonlinear equation of motion of an elastic string of the
length L (1.1) was proposed by Kirchhoff [12], in connection with some problems
in nonlinear elasticity, and later rediscovered by Carrier [6],
Z L
∂ 2 u τ0
k
∂u 2 ∂ 2 u
−
+
dx
= 0,
(1.1)
∂t2
m 2mL 0 ∂x
∂x2
where τ0 is the initial tension, m the mass of the string and k the Young’s modulus
of the material of the string. This model describes small vibrations of a stretched
string when only the transverse component of the tension is considered, and for
mathematical aspects of (1.1) see Bernstein [5] and Dickey [8].
Model (1.1) is a generalization of the linearized problem
∂ 2 u τ0 ∂ 2 u
−
= 0,
∂t2
m ∂x2
obtained by d’Alembert and Euler. A particular case of (1.1) can be written, in
general, as
Z
∂2u
2
−
M
|∇u(x,
t)|
dx
∆u = 0,
(1.2)
∂t2
Ω
or
∂2u
+ M ku(t)k2 Au = 0,
(1.3)
∂t2
2010 Mathematics Subject Classification. 35K60, 35F30.
Key words and phrases. Unilateral problem; Kirchhoff-Carrier; Klein-Gordon equation;
weak solution; uniqueness of solutions.
c
2015
Texas State University - San Marcos.
Submitted July 29, 2104. Published May 20, 2015.
1
2
C. RAPOSO, D. PEREIRA, G. ARAUJO, A. BAENA
EJDE-2015/137
in operator notation, where we consider the Hilbert spaces V ,→ H ,→ V 0 , where
V 0 is the dual of V with the immersions continuous and dense. By k · k we denote
the norm in V and A : V → V 0 a bounded linear operator.
Problem (1.3) is called nonlocal because of the presence of the term
Z
M (ku(t)k2 ) = M
|∇u(x, t)|2 dx ,
Ω
which implies that the equation is no longer a pointwise identity. The nonlocal term
provokes some mathematical difficulties which makes the study of such a problem
particulary interesting. On this subject, see a interesting work of Arosio-Panizzi [2],
where was proved that (1.3) is well-posedness in the Hadamard sense (existence,
uniqueness and continuous dependence of the local solution upon the initial data)
in Sobolev spaces.
Nonlocal initial boundary value problems are important from the point of view
of their practical application to modeling and investigation of various phenomena.
For the last several decades, various types of equations have been employed as
some mathematical model describing physical, chemical, biological and ecological
systems. See for example the nonlocal reaction-diffusion system given in Raposo et
all [19].
When we assume that M : [0, ∞) → R real function, M (λ) ≥ m0 > 0,
M ∈ C 1 (0, ∞), Pohozhaev [18] proved that the mixed problem for (1.2) has global
solution in t when the initial data u(x, 0), ut (x, 0) are restricted the class of functions
called Pohozhaev’s Class. This result can also be found in Lions [15] to the operator
given in (1.3), that was also analyzed by Arosio-Spagnolo [3] and Hazoya-Yamada
[10] when M (λ) ≥ 0 and many other authors, for example, Arosio-Espagnolo [3],
Dickey [8], Hazoya-Yamada [10] and Medeiros-Lı́maco-Menezes [16].
Let Ω be a bounded and open set of Rn , with smooth boundary Γ, and let T be
a positive real number. Let Q = Ω×]0, T [ be the cylinder with lateral boundary
Σ = Γ×]0, T [. A unilateral mixed problem associated with a nonlinear perturbation
∂2u
− M (|∇u|2 )∆u + θ ≥ f, in Q,
∂t2
θt − ∆θ + u0 ≥ g, in Q,
u=θ=0
u(0) = u0 ;
in Σ,
ut (0) = u1 ;
θ(0) = θ0 ,
where f, g, M are given real-valued functions with M positive, was studied by ClarkLima in [7], where was proved existence and uniqueness of solution.
In this subject, we consider Ω a bounded open set of Rn . A nonlinear perturbation of the problem (1.3), is given by
ρ
∂2u
+ M (ku(t)k2 )Au ≥ f,
∂t2
where ρ : Ω × (0, T ) → R and f : Ω × (0, T ) → R are real functions. The unilateral
problem associated with this nonlinear perturbation was studied in Frota-Lar’kin
[9] without geometrical restrictions and ρ a positive function.
In the case where ρ is a constant function equal to one, Medeiros-Milla [17] proved
the local existence and uniqueness theorem in non-degenerated case. Lar’kinMedeiros [13] under condition M (λ) ≥ m0 > 0 for all λ ≥ 0 under Ω being a
EJDE-2015/137
UNILATERAL PROBLEMS FOR THE KLEIN-GORDON OPERATOR
3
square (0, 1) × (0, 1) ⊂ R2 , showed the existence and uniqueness of a global solution
theorem.
Unilateral problem is very interesting too, because in general, dynamic contact
problems are characterized by nonlinear hyperbolic variational inequalities. For
contact problem in elasticity and finite element method see Kikuchi-Oden [11] and
reference there in. For Contact Problem Viscoelastic Materials see Rivera-Oquendo
[20]. For dynamic contact problems with friction, for example problems involving
unilateral contact with dry friction of Coulomb, see Ballard-Basseville [4].
For Ω be a bounded and open set of Rn , with smooth boundary Γ, consider the
Cauchy problem associated with the Klein-Gordon operator
∂2u
− M (|∇u|2 )∆u + M1 (|u|2 )u = f,
∂t2
with initial data
u(t) = u0 ∈ H01 (Ω) ∩ H 2 (Ω)
u0 (t) = u1 ∈ H01 (Ω),
u=0
2
and f ∈ L
(0, T ; H01 (Ω)),
(1.4)
on Γ,
where
M, M1 ∈ C 1 ([0, ∞); R),
M (s) ≥ m0 > 0,
∀s ∈ [0, ∞),
M1 (s) ≥ m1 > 0,
∀s ∈ [0, ∞).
(1.5)
For the problem above, the existence and uniqueness of solution was proved in [14]
where the abstract model was considered.
Motivated by the problem (1.4)-(1.5) this work deals with a unilateral problem
associated with the perturbed operator type Klein-Gordon
∂2u
− M (|∇u|2 )∆u + M1 (|u|2 )u ≥ f.
∂t2
More precisely, here we study a unilateral problem, i.e. a variational inequality,
see Lions [15], for the operator L under standard hypothesis on f, f 0 , u0 and u1 .
Making use of the penalty method and Galerkin’s approximations, we prove the
existence and uniqueness of solution.
This work is organized as follows. In the section 2 we introduce the notation and
functional spaces, we use the classical theory of Sobolev spaces as in Adams [1]. In
the section 3 we present the main Theorem. In the section 4 prove the theorem of
existence of solution and finally in the section 5 we prove the uniqueness of solution.
2. Notation and functional spaces
Let T > 0 be a real number, Ω a bounded open set of Rn with boundary Γ
regular and the cylinder Q = Ω×]0, T [ with lateral boundary Σ = Γ×]0, T [.
We propose the variational inequality
u00 − M (|∇u|2 )∆u + M1 (|u|2 )u ≥ f
in Q.
(2.1)
This inequality is satisfied by the unknown; that is, we formulate our problem as
follows. Let K = {v ∈ L2 (Ω); v ≥ 0 a. e. in Ω} be a closed and convex subset of
4
C. RAPOSO, D. PEREIRA, G. ARAUJO, A. BAENA
EJDE-2015/137
H01 (Ω) ∩ L2 (Ω), the variational problem consists in find the solution u = u(x, t)
satisfying
Z
(u00 − M (|∇u|2 )∆u + M1 (|u|2 )u − f ) (v − u0 ) dx dt ≥ 0, ∀v ∈ K,
Q
with u0 (x, t) ∈ K, a. e. on [0, T ] and taking the initial and boundary values
u = 0 on Σ,
u0 = 0 on Σ,
(2.2)
u(x, 0) = u0 (x), u0 (x, 0) = u1 (x)
in Ω.
To study the existence an uniqueness of solutions for the Problem (2.1)-(2.2), we
consider the following hypothesis
(H1) M, M1 ∈ C 1 ([0, ∞); R),
(H2) M (s) ≥ m0 > 0 for all s ∈ [0, ∞),
(H3) M1 (s) ≥ m1 > 0 for all s ∈ [0, ∞).
The notation for the functional spaces are contained in Lions [15]. We denote
the inner product and norm in H01 (Ω) and L2 (Ω) , respectively, by
n Z
n Z
X
X
2
∂u
∂u
∂v
2
((u, v)) =
(x)
(x) dx, kuk =
(x) dx,
∂x
∂x
∂x
i
i
i
i=1 Ω
i=1 Ω
Z
Z
2
(u, v)=
u(x)v(x) dx, |u|2 =
|u(x)| dx.
Ω
Ω
We introduce the bilinear form
n Z
X
∂u
∂v
a(u, v) =
(x)
(x) dx = ((u, v)) ∀v ∈ H01 (Ω).
∂x
∂x
i
i
Ω
i=1
(2.3)
By h·, ·i we denote the duality V and V 0 , where V 0 is the topological dual of the
space V .
3. Existence and uniqueness of weak solution
For the rest of this article, C denotes various positive constants. Next, we present
the main results of this paper.
Theorem 3.1. If
f ∈ L2 (0, T ; H01 (Ω)),
f 0 ∈ L2 (0, T ; L2 (Ω)),
(3.1)
H01 (Ω)
H01 (Ω)
(3.2)
u0 ∈
2
∩ H (Ω),
u1 ∈
∩ K,
and hypothesis (H1)–(H3) hold, then there exists T0 < T and a unique function u
such that
u ∈ L∞ (0, T0 ; H01 (Ω) ∩ H 2 (Ω)),
0
∞
u ∈L
(0, T0 ; H01 (Ω)),
00
∞
0
u (t) ∈ K ∀t ∈ [0, T ],
2
u ∈ L (0, T0 ; L (Ω)),
(3.3)
(3.4)
(3.5)
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UNILATERAL PROBLEMS FOR THE KLEIN-GORDON OPERATOR
5
satisfying
T
Z
(u00 , v − u0 )dt +
Z
T
a M (|∇u|2 )u, v − u0 dt
0
0
T
Z
(M1 (|u|2 )u, v − u0 )dt
+
(3.6)
0
Z
≥
T
(f, v − u0 )dt
∀v ∈ K, a.e. in t,
0
u0 (0) = u1 ,
u(0) = u0 ,
2
0
(3.7)
0
2
2
0
where a(M (|∇u| )u, v − u ) = ((M (|∇u| )u, v − u )) = −(M (|∇u| )∆u, v − u ).
The proof of Theorem 3.1 is made by the penalty method. The method consists
in to consider a perturbation of the operator L with adding singular term, called
penalization, depending on a parameter > 0. We solve the mixed problem in Q
for the penalized operator and the estimates obtained for the local solution of the
penalized equation that allow to pass to the limit, when > 0, in order to obtain
a function u which is the solution of our problem.
First of all, let us consider the penalty operators β : L2 (Ω) → L2 (Ω) associated
to the closed convex sets K, cf. Lions [15, p. 370]. The operator β is monotonous, hemicontinuous, takes bounded sets of L2 (Ω) into bounded sets of L2 (Ω),
its kernel is K and β : L2 (0, T ; L2 (Ω)) → L2 (0, T ; L2 (Ω)) is equally monotone and
hemicontinous.
The penalized problem associated with the variational inequality (2.1) and (2.2)
consists in given 0 < < 1, find u solution in Q of the mixed problem:
1
u00 − M (|∇u |2 )∆u + M1 (|u |2 )u + β(u0 ) = f in Q,
u = 0 on Σ,
(3.8)
u0 = 0 on Σ,
u (x, 0) = u0 (x) u0 (x, 0) = u1 (x)
in Ω.
Definition 3.2. We suppose that f ∈ L2 (0, T ; H01 (Ω)), f 0 ∈ L2 (0, T ; L2 (Ω)), u0 ∈
H01 (Ω) ∩ H 2 (Ω), u1 ∈ H01 (Ω) and hypothesis (H1 ) − (H3 ) hold. A weak solution
to the boundary value problem(3.8) is a functions u , such that for each 0 < < 1,
u ∈ L∞ (0, T0 ; H01 (Ω) ∩ H 2 (Ω)), u0 ∈ L∞ (0, T0 ; H01 (Ω)), u00 ∈ L∞ (0, T0 ; L2 (Ω))),
for T0 > 0, satisfying
1
(u00 , ϕ) + a(M (|∇u |2 u , ϕ) + (M1 |u |2 u , ϕ) + (β(u0 ), ϕ)
(3.9)
= (f, ϕ), ∀ϕ ∈ L2 (0, T0 ; L2 (Ω)),
u (0) = u0 , u0 (0) = u1 .
The solution of (3.8) is given by the following theorem.
Theorem 3.3. Assume that
f ∈ L2 (0, T ; H01 (Ω)),
u0 ∈
f 0 ∈ L2 (0, T ; L2 (Ω)),
H01 (Ω)
u1 ∈
(3.10)
2
(3.11)
H01 (Ω),
(3.12)
∩ H (Ω),
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C. RAPOSO, D. PEREIRA, G. ARAUJO, A. BAENA
EJDE-2015/137
and hypothesis (H1)–(H3) hold. Then for each 0 < < 1 there exists a unique weak
solution of (3.8).
In the next section, we prove the Theorem 3.1. Our goal is first prove the
penalized Theorem 3.3, applying Faedo-Galerkin method.
4. Proof of main results
Proof of Theorem 3.3. To prove Theorem 3.1, we first prove the penalized Theorem 3.3. We apply the Faedo-Galerkin method, noting that the immersions
H01 (Ω) ∩ H 2 (Ω) ,→ H01 (Ω) ,→ L2 (Ω) ,→ H −1 (Ω)
are continuous and dense and that H01 (Ω) is compact L2 (Ω). Let {wν , λν } eigenvectors and eigenfunctions of −∆. We consider (wν ) ⊂ H01 (Ω) ∩ H 2 (Ω) a Hilbertian
basis for Faedo-Galerkin method and Vm = [w1 , w2 , . . . , wm ] the subspace generated
by the vectors w1 , w2 , . . . , wm . Let us consider
um =
m
X
gjm (t)wj
j=1
solution of the approximate problem
(u00m , wj ) + M (|∇um |2 ((um , wj )) + M1 (|um |2 )(um , wj )
1
β(u0m ), wj = (f, wj ), j = 1, 2, . . . m,
+
um (x, 0) → u (x, 0) strongly in H01 (Ω) ∩ H 2 (Ω),
u0m (x, 0) → u0 (x, 0)
(4.1)
strongly in H01 (Ω).
The system of ordinary differential equation (4.1) has a solution um (t) defined in
[0, tm [, 0 < tm ≤ T . The next estimate implies that um (t) is defined in the whole
[0, T ].
To obtain a shorter notation, in the calculation of the following three estimates,
we omit the parameter in the approximate problem .
First estimate. We consider wj = 2u0m in (4.1) to obtain
d
d 0
2
|um (t)| + M (kum (t)k2 ) kum (t)k2
dt
dt
2
2 d
2
+ M1 (|um (t)| ) |um (t)| + (β(u0m ), u0m ) = 2(f (t), u0m (t)).
dt
Now, if we consider
Z λ
c
M (λ) =
M (s) ds and
(4.2)
(4.3)
0
Z
c1 (λ) =
M
λ
M1 (s) ds,
(4.4)
0
from (4.2), (4.3) and (4.4) it follows that
1 d 0
c(kum (t)k2 ) + M
c1 (|um (t)|2 ) ≤ (f (t), u0 (t)),
|u (t)|2 + M
m
2 dt m
because (βu0m (t), u0m (t)) ≥ 0.
(4.5)
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UNILATERAL PROBLEMS FOR THE KLEIN-GORDON OPERATOR
7
Integrating (4.5) from 0 to t, we obtain
1 0
c(kum (t)k2 ) + M
c1 (|um (t)|2 )]
[|u (t)|2 + M
2 m
Z t
1
c(kum (0)k2 ) + M
c1 (|um (0)|2 )].
(f (t), u0m (t)) + [|u0m (0)|2 + M
≤
2
0
From (4.6), (4.1), (3.10) and Cauchy-Schwarz’s inequality it follows that
Z t
|u0m (t)|2 + kum (t)k2 + |um (t)|2 ≤ C + C
|u0m (t)|2 ds.
(4.6)
(4.7)
0
Then Gronwall’s inequality implies
(um ) is bounded in L∞ (0, T ; H01 (Ω)),
(u0m )
∞
2
is bounded L (0, T ; L (Ω)).
(4.8)
(4.9)
Second estimate. Considering wj = −∆u0m in (4.1), we obtain
d 0
d
ku (t)k2 + M (kum (t)k2 ) |∆um (t)|2
dt m
dt
d
2
+ M1 (|um (t)|2 ) kum (t)k2 + (βu0m , −∆u0m )
dt
= 2((f (t), u0m (t))).
(4.10)
Observe that
M (kum (t)k2 )
d
d
|∆um (t)|2 + M1 (|um (t)|2 ) kum (t)k2
dt
dt
d
M (kum (t)k2 )|∆um (t)|2 + M1 (|um (t)|2 )kum (t)k2
dt
d
d
− M (kum (t)k2 )|∆um (t)|2 − M1 (|um (t)|2 )kum (t)k2 .
dt
dt
On the other hand
d
d
M (kum (t)k2 )|∆um (t)|2 + M1 (|um (t)|2 )kum (t)k2
dt
dt
= 2M 0 (kum (t)k2 )((um (t), u0m (t)))|∆um (t)|2
=
(4.11)
(4.12)
+ 2M10 (|um (t)|2 )(um (t), u0m (t))kum (t)k2 .
Using (βu0m , −∆u0m ) ≥ 0 and H01 (Ω) ,→ L2 (Ω), follows from (4.10), (4.11) and
(4.12) that
d 0
kum (t)k2 + M (kum (t)k2 )|∆um (t)|2 + M1 (|um (t)|2 )kum (t)k2
dt
≤ 2|M 0 (kum (t)k2 )|kum (t)kku0m (t)k|∆um (t)|2
(4.13)
+ 2|M10 (|um (t)|2 )| |um (t)|Cku0m (t)kkum (t)k2
+ kf (t)k2 + Cku0m (t)k2 .
Note that (4.8) implies kum (t)k ≤ C, therefore kum (t)k ∈ [0, C], for each m and
t ∈ [0, tm [. Since M ∈ C 1 ([0, ∞); R), this implies that
|M 0 (kum (t)k2 )| ≤ C,
∀m, ∀t ∈ [0, tm [,
(4.14)
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C. RAPOSO, D. PEREIRA, G. ARAUJO, A. BAENA
EJDE-2015/137
and analogously for M1 . Therefore, using (4.14), (4.8), (4.9) and (3.10) we can
write
2|M 0 (kum (t)k2 )| kum (t)kku0m (t)k|∆um (t)|2
+ 2|M10 (|um (t)|2 )| |um (t)|Cku0m (t)kkum (t)k2 + kf (t)k2 + Cku0m (t)k2
≤ C + C|∆um (t)|2 + 2Cku0m (t)k2 |∆um (t)|2 + Cku0m (t)k2
h
2 i
.
≤ C + C ku0m (t)k2 + |∆um (t)|2 + ku0m (t)k2 + |∆um (t)|2
(4.15)
Making
ϕ(t) = ku0m (t)k2 + |∆um (t)|2
(4.16)
and using (4.12), (H2), (H3), (4.1), (4.15) we can write, after integration from 0 to
t,
Z t
ϕ(t) ≤ C + C
(ϕ(s) + ϕ(s)2 )ds.
(4.17)
0
Observe that ϕ(t) is continuous in [0, T0 ], therefore there exists T0 < T such that
ϕ(t) ≤ C for all m and all t ∈ [0, T0 ].
From (4.17) it follows that
ku0m (t)k ≤ C,
∀m, ∀t ∈ [0, T0 ],
(4.18)
|∆um (t)| ≤ C,
∀m, ∀t ∈ [0, T0 ].
(4.19)
That is,
(u0m ) is bounded in L∞ (0, T0 ; H01 (Ω)),
∞
2
(∆um ) is bounded in L (0, T0 ; L (Ω)).
(4.20)
(4.21)
Statements (4.8) and (4.21) imply that
um is bounded in L∞ (0, T0 ; H01 (Ω) ∩ H 2 (Ω)).
(4.22)
Third estimate. Taking derivatives in the distribution sense in (4.1), we obtain
d
M (kum (t)k2 )a(um (t), wj ) + M (kum (t)k2 )a(u0m (t), wj )
dt
d
1
+ M1 (|um (t)|2 )(um (t), wj ) + M1 (|um (t)|2 )(u0m (t), wj ) + ((βu0m (t))0 , wj )
dt
= (f 0 (t), wj ).
(u000
m (t), wj ) +
Considering wj = 2u00m (t) in the above equation, we have
d
d
d 00
|u (t)|2 + 2 M (kum (t)k2 )a(um (t), u00m (t)) + M (kum (t)k2 ) ku0m (t)k2
dt m
dt
dt
d
d
+ 2 M1 (|um (t)|2 )(um (t), u00m (t)) + M1 (|um (t)|2 ) |u0m (t)|2
dt
dt
(4.23)
2
0
0
00
+ ((βum (t)) , um (t))
= 2(f 0 (t), u00m (t)).
Since
β(u0 (t + h)) − β(u0 (t)) u0 (t + h) − u0 (t) m
m
m
, m
≥ 0,
h→0
h
h
((βu0m (t))0 , u00m (t)) = lim
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UNILATERAL PROBLEMS FOR THE KLEIN-GORDON OPERATOR
9
and because β is monotone, we have
d 00
d
d
|u (t)|2 + M (kum (t)k2 ) ku0m (t)k2 + M1 (|um (t)|2 ) |u0m (t)|2
dt m
dt
dt
d
≤ 2|(f 0 (t), u00m (t))| + 2 M (kum (t)k2 )a(um (t), u00m (t))
dt
d
+ 2 M1 (|um (t)|2 )(um (t), u00m (t)).
dt
Using (4.14), (4.8), (4.9) and (4.21), we conclude that
d
M (kum (t)k2 )a(um (t), u00m (t))
dt = 2 M 0 (kum (t)k2 )(um (t), u0m (t))(−∆um (t), u00m (t))
≤ M 0 (kum (t)k2 )(um (t), u0m (t)) |∆um (t)|2 + |u00m (t)|2
(4.24)
(4.25)
2
≤ C + C |u00m (t)| .
Analogously,
d
M1 (|um (t)|2 )(um (t), u00m (t))
dt = 2 M10 |um (t)|2 )(um (t), u0m (t) (um (t), u00m (t))
≤ M10 |um (t)|2 )(um (t), u0m (t) |um (t)|2 + |u00m (t)|2
(4.26)
2
≤ C + C |u00m (t)| .
By Young’s inequality,
2|(f 0 (t), u00m (t))| ≤ |f 0 (t)|2 + |u00m (t)|2 .
(4.27)
We observe that (4.3), (4.4), (4.24), (4.25), (4.26) and (4.27) lead to
d 00
c(ku0m (t)k2 ) + M
c1 (|u0m (t)|2 ) ≤ C + |f 0 (t)|2 + C|u00m (t)|2 . (4.28)
|um (t)|2 + M
dt
Integrating (4.28) from 0 to t < T0 , using (H2), (H3), (3.1), (4.1) we have
Z t
|u00m (t)|2 + ku0m (t)k2 + |u0m (t)|2 ≤ C + C
|u00m (s)|2 ds + |u00m (0)|2 .
(4.29)
0
u00m (0)
We observe that, making t = 0 and wj =
in approximate problem (4.1) we
obtain
1
|u00m (0)|2 ≤ M (ku0m k2 )|∆u0m | + M1 (|u0m |2 )|u0m | + |β(u1m )| + |f (0)| |u00m (0)|;
that is,
|u00m (0)| ≤ C,
(4.30)
because u1m → u1 in H01 (Ω) ,→ L2 (Ω) and β : L2 (Ω) → L2 (Ω) is continuous.
From (4.29) and (4.30) it follows that
Z T
00
2
|um (t)| ≤ C + C
|u00m (s)|2 ds.
(4.31)
0
Using Gronwall’s inequality we conclude that
(u00m ) is bounded in L∞ (0, T0 ; L2 (Ω)).
(4.32)
10
C. RAPOSO, D. PEREIRA, G. ARAUJO, A. BAENA
EJDE-2015/137
Now we return to the notation um . The estimates above and Aubin-Lions
compactness Theorem implies that the existence of a subsequence of (um ), still
denoted by (um ), such that
um → u weak star in L∞ (0, T0 ; H01 (Ω) ∩ H 2 (Ω)),
u0m
u00m
um →
→ u0 weak star in L∞ (0, T0 ; H01 (Ω)),
→ u00 weak star in L∞ (0, T0 ; L2 (Ω)),
u strong in L2 (0, T0 ; L2 (Ω)) and a.e in
u0m
u0
→
(4.33)
(4.34)
(4.35)
Q,
(4.36)
strong in L (0, T0 ; L (Ω)) and a.e in Q.
(4.37)
2
2
Statements (4.36) and (4.37), the continuity of de norm and of β imply
kum k → ku k a.e in Q,
β(u0m )
→
β(u0 )
(4.38)
a.e in Q.
(4.39)
Using (H1) in (4.38) it follows that
M (kum k2 ) → M (ku k)2
a.e in Q,
(4.40)
analogously,
M1 (|um |2 ) → M1 (|u |)2 a.e in Q.
(4.41)
The convergences above are sufficient to pass to the limit with m → ∞ and then
we prove the Theorem 3.3.
Proof of Theorem 3.1. Finally, we prove the main Theorem of this work. Let
v ∈ L2 (0, T0 ; H01 (Ω)) be v(t) ∈ K a. e. for t ∈ (0, T0 ). From (3.8)1 it follows that
Z T0
Z T0
00
0
(u , v − u )dt +
M (|∇u |)2 a(u , v − u0 )dt
0
0
T0
Z
2
M1 (|u | )(u , v −
+
u0 )dt
Z
0
(f, v − u0 )dt
0
1
=
Z
1
Z
=
T0
−
T0
(β(u0 ), u0
(4.42)
− v) dt
0
T0
(β(u0 ) − βv, u0 − v) dt ≥ 0,
0
because v ∈ K (β(v) = 0) and β is monotone.
From (4.33)-(4.37) and the Banach-Steinhauss Theorem, it follows that there
exists a subnet (u )0<<1 , such that it converges to u as → 0, in the sense of
(4.33)-(4.37); that is,
u → u weak star in L∞ (0, T0 ; H01 (Ω) ∩ H 2 (Ω)),
u0
u00
→
→
0
∞
u weak star in L (0, T0 ; H01 (Ω)),
u00 weak star in L∞ (0, T0 ; L2 (Ω)),
(4.43)
(4.44)
(4.45)
2
2
u → u strong in L (0, T0 ; L (Ω)) and a.e in Q,
(4.46)
u0
2
2
(4.47)
0
→ u strong in L (0, T0 ; L (Ω)) and a.e in Q.
The convergences above are sufficient to pass to the limit in (4.42) with → 0
to conclude that (3.6) is valid. To complete the proof of Theorem 3.1, it remains
to show that u0 (t) ∈ K. a. e..
EJDE-2015/137
UNILATERAL PROBLEMS FOR THE KLEIN-GORDON OPERATOR
11
In this position we observe that using convergences (4.33)-(4.37), leting m → ∞
in (4.1), we can find u such that
1
u00 − M (|∇u |2 )∆u + M1 (|u |2 )u + β(u0 ) = f
in L2 (0, T ; L2 (Ω)).
(4.48)
On the other hand, from (4.48) we have
β(u0 ) = [f − u00 + M (|∇u |2 )∆u − M1 (|u |2 )u ].
(4.49)
Then
β(u0 ) → 0
in D0 (0, T ; H −1 (Ω)).
(4.50)
From (4.48) It follows that
β(u0 ) is bounded in L2 (0, T ; L2 (Ω)).
(4.51)
β(u0 ) → 0
(4.52)
Therefore,
weak in L2 (0, T ; L2 (Ω)).
On the other hand we deduce from (4.49) that
T
Z
(β(u0 ), u0 ) dt ≤ C.
0≤
(4.53)
0
Thus
Z
T
(β(u0 ), u0 )dt → 0.
(4.54)
0
We have
Z
T
(β(u0 ) − β(ϕ), u0 − ϕ) dt ≥ 0,
∀ϕ in L2 (0, T ; L2 (Ω)),
0
because β is a monotonous operator. Thus,
Z
0
T
(β(u0 ), u0 ) dt −
T
Z
(β(u0 ), ϕ) dt −
0
Z
T
(β(ϕ), u0 − ϕ) dt ≥ 0.
(4.55)
0
From (4.52) (4.54) and (4.55) we have
Z
T
(β(ϕ), u0 (t) − ϕ) dt ≤ 0.
(4.56)
0
Taking ϕ = u0 − λv, with v ∈ L2 (0, T ; L2 (Ω)) and λ > 0, using the hemicontinuity
of β we deduce that
β(u0 (t)) = 0,
(4.57)
and this implies that u0 (t) ∈ K a. e. and the proof of the existence of solution is
complete.
In the next section we prove the uniqueness of solution to achieve our goal.
12
C. RAPOSO, D. PEREIRA, G. ARAUJO, A. BAENA
EJDE-2015/137
5. Uniqueness of solution
Suppose that u1 , u2 are two solutions of (3.6) and set w = u2 − u1 and t ∈ [0, T0 ].
Taking v = u01 (resp. u02 ) in (3.6) relative to v2 (resp. v1 ) and adding up the results
we obtain
Z t
Z t
Z t
00
0
2
0
−
(w , w )dt +
M (ku2 k ∆u2 , w )dt −
M (ku1 k2 ∆u1 , w0 )dt
0
0
0
(5.1)
Z t
Z t
2
0
2
0
−
(M1 (|u2 | )u2 , w )dt +
(M1 (|u1 | )u1 , w )dt ≥ 0,
0
0
or equivalently
Z t
Z t
Z t
−
(w00 , w0 )ds +
(M (ku2 k2 ∆u2 , w0 )ds −
(M (ku2 k2 ∆u1 , w0 )ds
0
0
0
Z t
Z t
2
0
2
+
(M (ku2 k ∆u1 , w )ds −
(M (ku1 k ∆u1 , w0 )ds
0
0
Z t
Z t
−
(M1 (|u2 |2 )u2 , w0 )ds +
(M1 (|u1 |2 )u1 , w0 )ds
0
0
Z t
Z t
2
0
−
(M1 (|u2 | )u1 , w )ds +
(M1 (|u2 |2 )u1 , w0 )ds
0
0
Z t
Z t
Z t
00
0
2
0
=−
(w , w )ds +
(M (ku2 k ∆w, w )ds −
(M1 (|u2 |2 )w, w0 )ds
0
0
0
Z t
+
([M (ku2 k2 − M (ku1 k2 )]∆u1 , w0 )ds
0
Z t
−
([M1 (|u2 |2 ) − M1 (|u1 |2 )]u1 , w0 )ds ≥ 0,
(5.2)
0
By hypothesis (H1), we can use the Mean Value Theorem to write
Z t
Z t
Z t
(w00 , w0 )ds −
(M (ku2 k2 ∆w, w0 )ds +
(M1 (|u2 |2 )w, w0 )ds
0
0
0
Z t
−
(M 0 ()[ku2 k2 − ku1 k2 ]∆u1 , w0 )ds
0
Z t
+
(M10 (1 )[|u2 |2 − |u1 |2 ]u1 , w0 )ds ≤ 0,
(5.3)
0
where
ku1 k2 ≤ ≤ ku2 k2 ,
|u1 |2 ≤ ≤ |u2 |2 .
From (5.3) It follows that
Z t
Z t
Z t
d
d
d 02
|w | ds +
M (ku2 k2 ) kwk2 ds +
M1 (|u2 |2 ) |w|2 ds
dt
dt
dt
0
0
0
Z t
≤2
|M 0 ()| [(ku2 k − ku1 k) (ku2 k + ku1 k)] |∆u1 ||w0 |ds
0
Z t
+2
|M10 (1 )| [(|u2 | − |u1 |) (|u2 | + |u1 |)] |u1 ||w0 |ds,
0
(5.4)
(5.5)
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UNILATERAL PROBLEMS FOR THE KLEIN-GORDON OPERATOR
13
using an argument similar to that used in (4.11) and (4.12), from (5.5) it follows
that
Z t
d 02
|w | + M (ku2 k2 )kwk2 + M1 (|u2 |2 )|w|2 ds
dt
0
Z t
≤2
|M 0 ()| [(ku2 k − ku1 k) (ku2 k + ku1 k)] |∆u1 ||w0 |ds
0
(5.6)
Z t
0
0
+2
|M1 (1 )| [(|u2 | − |u1 |) (|u2 | + |u1 |)] |u1 ||w |ds
0
Z t
+2
(|M 0 (ku2 k2 )|ku2 kku02 kkwk2 + 2|M10 (|u2 |2 )||u2 ||u02 ||w|2 )ds.
0
Using (H2), (H3), (4.18) (4.19) and (5.6), we conclude that
Z t
Z t
Z t
0 2
2
2
0
0
|w | + kwk + |w| ≤ C
kwk|w |ds + C
|w||w |ds + C
|w|2 ds.
0
0
(5.7)
0
This implies
0 2
2
2
Z
|w | + kwk + |w| ≤ C
t
(|w0 |2 + Ckwk2 + |w|2 )ds.
(5.8)
0
Using Gronwall’s inequality in (5.8), we conclude that w(t) = 0; therefore u1 = u2 .
Acknowledgements. We would like to express our gratitude to the anonymous
referees for a number of valuable comments that led to the clarification of various
points.
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Carlos Raposo
Department of Mathematics, Federal University of São João Del-Rei, São João Del-Rei
- MG 36307-352, Brazil
E-mail address: raposo@ufsj.edu.br
Ducival Pereira
Department of Mathematics, State University of Pará, Belém - PA 66113-200, Brazil
E-mail address: ducival@oi.com.br
Geraldo Araujo
Department of Mathematics, Federal University of Pará, Belém - PA 66075-110, Brazil
E-mail address: gera@ufpa.br
Antonio Baena
Department of Mathematics, Federal University of Pará, Belém - PA 66075-110, Brazil
E-mail address: baena@ufpa.br