PORTUGALIAE MATHEMATICA
Vol. 55 Fasc. 3 – 1998
ON A VARIATIONAL INEQUALITY FOR
THE NONHOMOGENEOUS DEGENERATED
KIRCHHOFF EQUATION WITH A FRICTIONAL DAMPING
C.L. Frota and N.A. Lar’kin *
Presented by Hugo Beirão da Veiga
Abstract: We study a unilateral problem for the nonhomogeneous degenerated
Kirchhoff equation with a frictional damping. Making use of the penalty method and
Galerkin’s approximations, we establish global existence and uniqueness theorems.
1 – Introduction
The one-dimensional nonlinear equation of motion of an elastic string
(1)
∂2u
ρ 2 −
∂t
µ
E
P0
+
h
2L
Z
0
¯
¯2 ¶ 2
¯
∂ u
¯
¯
¯ ∂s (s, t)¯ ds ∂x2 = 0
L ¯ ∂u
was proposed by Kirchhoff [8]. Here h is the area of the cross section, L is the
length of the string, ρ is the mass density, P0 is the initial tension and E is
Young’s modulus of a material. Another model for the motion of an elastic string
was given in Carrier [2]. The mixed problem for (1) was studied by a number
of authors, see Bernstein [1], Pohozaev [16], Lions [11], Ebihara–Medeiros–Milla
Miranda [4], Menzala [14], D’ancona–Spagnolo [3] and their references.
Most authors considered homogeneous case, when ρ0 , P0 , h and E are constants. These restrictions allowed to obtain an a priori estimate independent of
t. When the initial data are analytic functions, it is possible to prove a global
in t existence theorem, see Bernstein [1], Pohozaev [16], and D’ancona–Spagnolo
[3]. If the initial data are only from some Sobolev space, then the presence of a
Received : December 10, 1996.
* Supported by CNPq-Brasil as Visiting Professor at the State University of Maringá.
278
C.L. FROTA and N.A. LAR’KIN
frictional damping allows to prove the global existence theorem for small initial
data, see Nishihara–Yamada [15].
In [11], Lions proposed to study the problem which modelled a nonhomogeneous material. In this case one does not have such a global estimate and, until
now, we know only local existence results, see Frota [5]. Our goal in this paper
is to consider a unilateral problem for equation (1), having a frictional damping,
in a case of a nonhomogeneous material; that is, ρ, P0 , h and E are functions
of x, t; and the initial tension P0 can be zero. We study the case when ρ(x, t)
is only nonnegative. It means that the material is composite and in some places
the density of the material is much times smaller then in others, or the string has
holes inside.
We formulate our problem as follows: Let Ω be a bounded open set of Rn with
a sufficiently smooth boundary Γ, Q = Ω × (0, T ), and Σ = Γ × (0, T ), where T
is a finite positive number. We denote by K the closed convex set of W02,2p+2 (Ω),
1 ≤ p < ∞, defined by
½
K= σ∈
(2)
W02,2p+2 (Ω) ;
|∆σ(x)| ≤ 1 and σ(x) ≥ 0 a.e. on Ω
¾
.
Given M : Ω × [0, T ] × [0, ∞) → R, ρ : Ω × [0, T ] → R, f : Ω × (0, T ) → R,
φ0 : Ω → R and φ1 : Ω → R, find a function u = u(x, t) satisfying
(3)
Z µ
Q
∀ v ∈ K,
(4)
µ
ρ utt − M x, t,
¯2 ¶
¶
n Z ¯¯
X
¯
¯ ∂u (s, t)¯ ds ∆u + α ut − f (v − ut ) dx dt ≥ 0 ,
¯
¯ ∂s
i=1 Ω
i
∂u
(x, t) ∈ K
∂t
a.e. on [0, T ]
and taking the following initial and boundary values
(5)
(6)
u=0
on Σ = Γ × (0, T ) ,
u(x, 0) = φ0 (x) ,
∂u
(x, 0) = φ1 (x) in Ω .
∂t
We observe that if ut ∈ int(K) = interior of K, then u is a solution of the
equation
¯2 ¶
µ
n Z ¯
X
¯ ∂u
¯
∂2u
∂u
¯
ρ(x, t) 2 − M x, t,
(s, t)¯¯ ds ∆u + α
= f,
¯
∂t
∂si
∂t
i=1 Ω
in Q = Ω × (0, T )
and from (4) the velocity ∂u
∂t is positive. Moreover, one can note that the restriction |∆ut | ≤ 1, does not have an explicit physical meaning. Nevertheless, it
THE NONHOMOGENEOUS DEGENERATED KIRCHHOFF EQUATION
279
implies boundedness of |ut | and |∇u| in Q. This, in turn, gives boundedness of
the internal energy of an elastic string whose motion is simulated by the above
equation.
When a function M is strictly positive and does not depend on x and t, α = 0
and ρ = 1, the unilateral problem (3)–(6) for n = 1, was studied by Lar’kin [9],
where global existence and uniqueness theorems were proved. Later this results
were generalized in Lar’kin–Medeiros [10], where Ω was a square in R2 , and in
Frota [6] for dimensions n ≥ 2. Local existence and uniqueness theorems for
(3)–(6) were proved in Medeiros–Milla Miranda [13].
In this paper we prove the existence and uniqueness of a global solution to
(3)–(6) without smallness conditions for the initial data and without geometrical
restrictions for Ω. We need a positive coefficient α in order to have a sufficient
regularity of solutions when ρ is degenerated. It is known that if α = 0 and ρ ≥ 0,
then the Cauchy problem even in a linear case is ill-posed. If ρ is strictly positive,
α can be equal to zero, see Frota–Lar’kin [7]. To prove a global existence theorem
without smallness restrictions for the initial data and f , we use properties of a
convex set to which belongs our solution. A choice of this set depends on growth
properties of a function M (x, t, ku(t)k2 ).
Another problem for the degenerate Kirchhoff equation is to prove a global in
t uniqueness theorem. In Ebihara–Medeiros–Milla Miranda [4], the local uniqueness theorem was proved. In order to establish a global uniqueness result, we
again use properties of a convex set to which belongs our solution.
2 – Penalized problem
Let β be an operator from W02,2p+2 (Ω) into the dual space W
by
hβ(u), vi = −
Z
u− v dx +
Ω
Z ³
1 − |∆u|2p
Ω
´−
−2, 2p+2
2p+1
(Ω) defined
∆u ∆v dx ,
where h− (x) = max{−h(x), 0}. We can verify that β is a monotone and hemicontinuous operator,
(7)
β(S) is bounded for each S bounded in W02,2p+2 (Ω)
and
(8)
β(u) = 0 ⇐⇒ u ∈ K .
Under this conditions, we have the following existence result for the penalized
problem.
280
C.L. FROTA and N.A. LAR’KIN
Theorem 1. Let ρ ∈ C 1 (Q), M ∈ C 1 (Q × [0, ∞)) be real functions and α
be a positive real number such that:
(9)
0 ≤ ρ(x, t), ∀ (x, t) ∈ Q
(10)
0 ≤ M (x, t, λ) ,
∀ (x, t, λ) ∈ Q × [0, ∞) ,
M (x, t, λ) ≤ C(1 + λp ) ,
(11)
(12)
with 0 < ρ0 ≤ ρ(x, 0), ∀ x ∈ Ω ,
¯
¯
¯ ∂M
¯
p
¯
¯
¯ ∂x (x, t, λ)¯ ≤ C(1 + λ ) ,
∀ (x, t, λ) ∈ Q × (0, ∞) ,
∀ (x, t, λ) ∈ Q × (0, ∞), i = 1, ..., n ;
i
(13)
0 < a0 ≤ 2 α − |ρt (x, t)| ,
∀ (x, t) ∈ Q .
If f, f 0 ∈ L2 (0, T ; L2 (Ω)), φ0 ∈ H01 (Ω) ∩ H 2 (Ω), φ1 ∈ int(K), then there exists
²0 ∈ (0, 1) such that for each ² ∈ (0, ²0 ) and ν ∈ N there is a function uν² :


u ∈ L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)),

 ν²
u0ν² ∈ L∞ (0, T ; H01 (Ω)) ∩ L2p+2 (0, T ; W02,2p+2 (Ω)),
(14)



u00ν² ∈ L∞ (0, T ; L2 (Ω)) ,
satisfying the following initial boundary value problem
(15)
µ
¶

00
2
0

ρν (t) uν² (t) − M (t, kuν² (t)k ) ∆uν² (t) + α uν² (t), v +





+







1
hβ(u0ν² (t)), vi = (f (t), v)
²
uν² (0) = φ0 (x) ,
a.e. on [0, T ], ∀ v ∈ W02,2p+2 (Ω) ,
u0ν² (0) = φ1 (x) .
Here | · |,(·, ·) and k · k,((·, ·)) denote the norm and the inner product in L 2 (Ω) and
H01 (Ω) respectively, ρν (x, t) = ν1 + ρ(x, t) and ρ0 , C, a0 are positive constants.
Proof: Let (wj )j∈N be a basis in W02,2p+2 (Ω), orthonormal in L2 (Ω). For
each m ∈ N we define approximate solutions
uν²m (x, t) =
m
X
j=1
gν²jm (t) wj (x) ,
281
THE NONHOMOGENEOUS DEGENERATED KIRCHHOFF EQUATION
where gν²jm are solutions to the Cauchy problem for the following system:
´
´ ³
³
2
00

−
M
(t,
ku
(t)k
)
∆u
(t),
w
ρ
(t)
u
(t),
w
ν²m
ν²m
j +
j

ν²m
 ν



1



+ α(u0ν²m (t), wj ) + hβ(u0ν²m (t)), wj i = (f (t), wj ) ,


²




m
X
(16)
u
(0)
=
(φ0 , wj ) wj = φ0m ,
ν²m




j=1





m

X
 0


(φ1 , wj ) wj = φ1m .

 uν²m (0) =
1≤j ≤m ,
j=1
For ν > 0, (16) can be reduced to a normal system of m ordinary differential
equations, which has solutions at some interval [0, Tm ]. This ensures the existence
of approximate solutions uν²m .
Let Vm = [w1 , ..., wm ] be the m-dimensional subspace of W02,2p+2 (Ω) spanned
by w1 , ..., wm . Then we have the approximate equation
³
(17)
µ
´
³
¶
´
ρν (t) u00ν²m (t), v − M t, kuν²m (t)k2 ∆uν²m (t), v +
1
hβ(u0ν²m (t)), vi = (f (t), v) ,
²
+ α(u0ν²m (t), v) +
∀ v ∈ Vm .
The next step is devoted to deriving a priori estimates of uν²m . From now on
Ci , for i ∈ N, denotes a positive constant independent of ν, ², m and t.
A priori estimate 1. Taking v = 2 u0ν²m (t) in (17), we obtain
(18)
E
´
d³
2D
ρν (t), (u0ν²m (t))2 + 2 α |u0ν²m (t)|2 +
β(u0ν²m (t)), u0ν²m (t) =
dt
²
=
³
ρ0ν (t), (u0ν²m (t))2
³
´
+2
Z
Ω
³
´
M x, t, kuν²m (t)k2 ∆uν²m (x, t) u0ν²m (x, t) dx
´
+ 2 f (t), u0ν²m (t) .
Using integration by parts, (11) and (12), we get
¯ Z
¯
³
´
¯
¯
2
0
¯2
M x, t, kuν²m (t)k ∆uν²m (x, t) uν²m (x, t) dx¯¯ ≤
¯
Ω
·
≤ C1 kuν²m (t)k
2p+2
+
ku0ν²m (t)k2
+ kuν²m (t)k
2p
ku0ν²m (t)k2
¸
.
282
C.L. FROTA and N.A. LAR’KIN
On the other hand, by Young’s inequality we have
1
p
kuν²m (t)k2p+2 +
ku0 (t)k2p+2 ,
kuν²m (t)k2p ku0ν²m (t)k2 ≤
p+1
p + 1 ν²m
kuν²m (t)k2 ≤
p
1
+
kuν²m (t)k2p+2 ,
p+1 p+1
ku0ν²m (t)k2 ≤
p
1
+
kuν²m (t)k2p+2 .
p+1 p+1
Therefore,
¯ Z
¯
³
´
¯
¯
2
0
¯2
M x, t, kuν²m (t)k ∆uν²m (x, t) uν²m (x, t) dx¯¯ ≤
¯
(19)
Ω
≤ C2 + C3 kuν²m (t)k2p+2 + C4 ku0ν²m (t)k2p+2 .
Since ρ ∈ C 1 (Q), we conclude
(20)
and
³
¯³
´¯
¯
¯
¯ ρν (t), (u0ν²m (t))2 ¯ ≤ C5 + C6 ku0ν²m (t)k2p+2
From (18)–(21) we have
d³
dt
´
2 f (t), u0ν²m (t) ≤ C7 + |f (t)|2 + C8 ku0ν²m (t)k2p+2 .
(21)
´
E
2D
β(u0ν²m (t)), u0ν²m (t) ≤
²
ρν (t), (uν²m (t))2 + α |u0ν²m (t)|2 +
≤ C9 + |f (t)|2 + C10 |∆uν²m (t)|2p+2 + C11 |∆u0ν²m (t)|2p+2
≤ C12 + |f (t)|2 + C13 |∆u0ν²m (t)|2p+2 .
After integration over t, we obtain
(22)
³
´
ρν (t), (u0ν²m (t))2 +2 α
Z
t
|u0ν²m (s)|2 ds+
0
≤ C14 + C13
Z
t
0
2
²
Z tD
0
E
β(u0ν²m (s)), u0ν²m (s) ds ≤
|∆u0ν²m (s)|2p+2 ds .
Using the identity h = h+ − h− , we can see that
³
´
³
´
³
|∆u0 |2p+2 − 1 = |∆u0 |2 − 1 − |∆u0 |2 1 − |∆u0 |2p
³
´
= |∆u0 |2 − 1 − |∆u0 |2
³
´
³
h³
1 − |∆u0 |2p
≤ |∆u0 |2 − 1 + 1 − |∆u0 |2p
³
≤ 1 − |∆u0 |2p
³
´−
≤ 2 1 − |∆u0 |2p
³
´−
+ 1 − |∆u0 |2p
´−
|∆u0 |2 .
´
´+
|∆u0 |2
´−
³
− 1 − |∆u0 |2p
|∆u0 |2
´− i
283
THE NONHOMOGENEOUS DEGENERATED KIRCHHOFF EQUATION
Then
D
We can rewrite this as follows
·
E
2p+2
0
0
k∆u0ν²m (t)kL
.
2p+2 (Ω) − µ(Ω) ≤ 2 β(uν²m (t)), uν²m (t)
(23)
D
E
1
1
2p+2
0
0
|∆u0ν²m (t)|2p+2 + k∆u0ν²m (t)kL
−
µ(Ω)
≤
2
β(u
(t)),
u
(t)
,
2p+2 (Ω)
ν²m
ν²m
2C
2
¸
hence, after integration in time from 0 to t, we have
(
1
C15
²
"Z
t
0
|∆u0ν²m (s)|2p+2 ds
≤
2
²
Z
+
Z tD
0
t
0
2p+2
k∆u0ν²m (s)kL
2p+2 (Ω) ds
#
− µ(Ω)
Z
t
ds
0
)
≤
E
β(u0ν²m (s)), u0ν²m (s) ds .
From this inequality and (22), one can see
Z
(24)
t
0
|∆u0ν²m (s)|2p+2 ds
+
Z
t
0
2p+2
k∆u0ν²m (s)kL
2p+2 (Ω) ds ≤
≤ C16 + ² C17
Z
t
0
|∆u0ν²m (s)|2p+2 ds ,
so it can be rewritten as follows
(1 − ² C17 )
Z
t
0
|∆u0ν²m (s)|2p+2 ds
If we choose ²0 =
consequently
(25)
Z
t
0
1
C17 ,
+
Z
t
0
2p+2
k∆u0ν²m (s)kL
2p+2 (Ω) ds ≤ C16 .
then, for 0 < ² < ²0 , we have (1 − ² C17 ) > 0, and
|∆u0ν²m (s)|2p+2 ds
≤ C18 and
Z
t
0
2p+2
k∆u0ν²m (s)kL
2p+2 (Ω) ds ≤ C19 .
We note that
|∆u(t)|2p+2 = |∆u(0)|2p+2 +
·
Z
≤ |∆u(0)|2p+2 +
= C20 + (2p + 1)
t
0
Z
Z
ip+1
dh
|∆u(s)|2
ds
ds
T
0
t
¸
|∆u0 (t)|2p+2 dt + (2p + 1)
Z
t
|∆u(s)|2p+2 ds
0
|∆u(s)|2p+2 ds .
0
By Gronwall’s Lemma
|∆u(t)|2p+2 ≤ |∆u(0)|2p+2 e(2p+2)T + e(2p+2)T
Z
t
0
|∆u0 (t)|2p+2 dt ,
284
C.L. FROTA and N.A. LAR’KIN
therefore, we have from (25)
(26)
(27)
|∆uν²m (t)| ≤ C21 ,
³
ρν (t), (u0ν²m (t))2
´
+α
Z
t
0
2
|u0ν²m (s)|2 ds+
²
2
(28)
kuν²m (t)k ≤ C23
(29)
kβ(u0ν²m )k
and
2p+2
L 2p+1 (0,T ;W
Z tD
0
E
β(u0ν²m (s)), u0ν²m (s) ds ≤ C22 ,
ku0ν²m (t)k2 ≤ C24 ,
2p+2
−2, 2p+1
(Ω))
≤ C25 .
A priori estimate 2. Differentiating (17) with respect to t, taking v =
2 u00ν²m (t) and using monotonicity of β, we come to the inequality
³
(30)
≤
³
ρν (t), (u00ν²m (t))2
ρν (0), (u00ν²m (0))2
+2
+2
Z
t
(f
0
0
Z t³
0
´
´
+2
+
Z tZ
0
Z t³
(s), u00ν²m (s)) ds
0
Ω
(2α + ρ0 (x, s)) (u00ν²m (x, s))2 dx ds ≤
´
M 0 (s, kuν²m (s)k2 ) ∆uν²m (s), u00ν²m (s) ds
+2
Z tµ
0
∂M
(s, kuν²m (s)k2 ) ∆uν²m (s), u00ν²m (s) ds
∂λ
¶
´
M (s, kuν²m (s)k2 ) ∆u0ν²m (s), u00ν²m (s) ds .
Now we choose a real number γ such that
(31)
0<γ<
a0
,
4
where a0 is given by (13). Taking into account the first estimate, we can see
(32)
2
Z t³
0
≤
Z tZ
0
Ω
1h
γ
≤ C26 + γ
0
2
M (x, s, kuν²m (s)k ) ∆uν²m (x, s)
Z
t
0
´
M 0 (s, kuν²m (s)k2 ) ∆uν²m (s), u00ν²m (s) ds ≤
i2
dx ds + γ
Z
t
0
|u00ν²m (s)|2 ds
|u00ν²m (s)|2 ds ,
and similarly,
(33)
2
Z tµ
0
∂M
(s, kuν²m (s)k2 ) ∆uν²m (s), u00ν²m (s) ds ≤
∂λ
¶
≤ C27 + γ
Z
t
0
|u00ν²m (s)|2 ds ,
THE NONHOMOGENEOUS DEGENERATED KIRCHHOFF EQUATION
(34)
2
Z t³
0
´
M (s, kuν²m (s)k2 ) ∆u0ν²m (s), u00ν²m (s) ds ≤
≤ C28 + γ
(35)
2
Z t³
0
285
´
f 0 (s), u00ν²m (s) ds ≤ C29 + γ
Z
t
0
Z
t
0
|u00ν²m (s)|2 ds ,
|u00ν²m (s)|2 ds .
Since φ1 ∈ int(K), we have hβ(φ1m ), ϕi = 0, ∀ ϕ ∈ W02,2p+2 (Ω) and for m
sufficiently large. Then taking t = 0 in (17), we get
³
´
ρν (0), (u00ν²m (0))2 ≤ C30 |u00ν²m (0)| .
From this and (9), we have
ρ0 |u00ν²m (0)|2 ≤ C31 |u0ν²m (0)| ,
and, consequently,
|u00ν²m (0)| ≤ C32 .
(36)
From (30), (32)–(36), we obtain
³
´
ρν (t), (u00ν²m (t))2 +
Z tZ ³
0
Ω
´
2 α + ρ0 (x, s) (u00ν²m (x, s))2 dx ds ≤
≤ C33 + 4 γ
Z
t
0
|u00ν²m (s)|2 ds .
This inequality and (13) give
(a0 − 4 γ)
Z
t
0
|u00ν²m (s)|2 ds ≤ C34 ,
hence, from (38), we conclude
Z
(37)
t
0
|u00ν²m (s)|2 ds ≤ C35 .
The a priori estimates obtained imply the existence of a subsequence of
(uν²m )m∈N , which we still denote in the same way, and functions uν² and χν²
such that

∗

in L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)) ,

 uν²m * uν²




u0
* u0ν²


 ν²m
u0
∗
* u0
ν²m
ν²



00
00

u
*
u

ν²
ν²m





0
β(uν²m ) * χν²
in L2p+2 (0, T ; W02,2p+2 (Ω)) ,
in L∞ (0, T ; H01 (Ω)) ,
in L2 (0, T ; L2 (Ω)) ,
2p+2
in L 2p+1 (0, T ; W
−2, 2p+2
2p+1
(Ω)) .
286
C.L. FROTA and N.A. LAR’KIN
Moreover,
in L2 (0, T ; H01 (Ω)) ,
uν²m → uν²
in L2 (0, T ; L2 (Ω)) .
u0ν²m → u0ν²m
Letting m tend to ∞, we obtain
³
´
ρν (t) u00ν² (t) − M (t, kuν² (t)k2 ) ∆uν² (t) + α u0ν² (t), v + hχν² (t), vi = (f (t), v),
a.e. on [0, T ], ∀ v ∈ W02,2p+2 (Ω) .
Obviously, uν² (t) satisfies the initial data (15). Using monotonicity and hemicontinuity of β, we can prove (see Lions [12]) that
χν² (t) = β(u0ν² (t))
2p+2
in L 2p+1 (0, T ; W
−2, 2p+2
2p+1
(Ω)) .
This completes the proof of Theorem 1.
3 – The main result
Theorem 2. Under conditions of Theorem 1, there exists a function u:
(38)

∞
1
2

 u ∈ L (0, T ; H0 (Ω) ∩ H (Ω)) ,



 u0 ∈ L∞ (0, T ; H 1 (Ω)) ∩ L2p+2 (0, T ; W 2,2p+2 (Ω)) ,
0
0
0


u (t) ∈ K a.e. on [0, T ] ,



 00
2
2
u ∈ L (0, T ; L (Ω)) ,
which is a solution to the following problem
(39)
´
³
00 (t) − M (t, ku(t)k2 ) ∆u(t) + α u0 (t) − f (t), v − u0 (t) ≥ 0 ,

ρ(t)
u


∀ v ∈ K,



u(0) = φ0 ,
a.e. on [0, T ] ,
u0 (0) = φ1 .
Moreover, if
(40)
(41)
φ0 (x) ≥ 0 a.e. on Ω
with kφ0 k > 0 ,
M (x, t, 0) = 0 and M (x, t, λ) > 0 ,
then this solution is uniquely defined.
(x, t, λ) ∈ Ω × [0, T ] × (0, ∞) ,
THE NONHOMOGENEOUS DEGENERATED KIRCHHOFF EQUATION
287
Proof: For each ν ∈ N and ² ∈ (0, ²0 ), we have a function uν² given by
Theorem 1. Therefore, uν² satisfies (14)–(15), which imply the following estimates
(42)
|∆uν² (t)| ≤ κ1 ,
(43)
kuν² (t)k ≤ κ2 ,
(44)
ku0ν² (t)k ≤ κ3 ,
Z
(45)
t
0
2p+2
k∆u0ν² (s)kL
2p+2 (Ω) ds ≤ κ4 ,
Z
(46)
t
0
|u00ν² |2 ds ≤ κ5 ,
where κi , for i = 1, ..., 5, are positive constants independent of ν, ² and t.
If we fix ν < ∞, letting ² tend to zero, taking into account (42)–(46) and
the monotonicity of β, then we find a function uν , as a limit of a subsequence
(uν² )²∈(0,²0 ) , such that


u ∈ L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)) ,

 ν
u0ν ∈ L∞ (0, T ; H01 (Ω))∩L2p+2 (0, T ; W02,2p+2 (Ω)) and u0ν (t) ∈ K a.e. on [0, T ] ,



u00ν ∈ L2 (0, T ; L2 (Ω)) ,
and
´
³
00 (t) − M (t, ku (t)k2 ) ∆u (t) + α u0 (t) − f (t), v − u0 (t) ≥ 0

ρ
(t)
u
 ν
ν
ν
ν
ν
ν

∀ v ∈ K,



a.e. on [0, T ] .
u0 (0) = φ1 .
u(0) = φ0 ,
The sequence (uν )ν∈N has the estimates (42)–(46). Thus, taking the limit
in a subsequence as ν tend to ∞, we find a function u which satisfies (38) and
(39). This completes the proof of the existence. To prove uniqueness, let u and
z be two distinct functions satisfying (38) and (39). Then ψ = u − z satisfies the
following inequality
³
´
ρ(t) (ψ 0 (t))2 − 2
≤
Z t³
0
0
Z t³
0
0
ρ (s), (ψ (s))
2
´
´
M (s, ku(s)k2 ) ∆ψ(s), ψ 0 (s) ds + 2 α
ds+2
Z t µh
0
2
Z
t
|ψ 0 (s)|2 ds ≤
0
2
i
0
¶
M (t, ku(s)k )−M (t, kz(s)k ) ∆z(s), ψ (s) ds .
288
C.L. FROTA and N.A. LAR’KIN
On the other hand, using integration by parts, we have
−2
Z t³
´
M (s, ku(s)k2 ) ∆ψ(s), ψ 0 (s) ds =
0
=2
Z tX
n Z
∂ψ 0
∂ψ
(x, s)
(x, s) dx ds
∂xi
∂xi
M (x, s, ku(s)k2 )
0 i=1 Ω
+2
Z tX
n Z
∂M
0 i=1 Ω
∂xi
(x, s, ku(s)k2 )
∂ψ
(x, s) ψ(x, s) dx ds ,
∂xi
and
2
Z tX
n Z
∂ψ
∂ψ 0
(x, s)
(x, s) dx ds =
∂xi
∂xi
M (x, s, ku(s)k2 )
0 i=1 Ω
=
n Z Z
X
t
M (x, s, ku(s)k2 )
i=1 Ω 0
=
n µ
X
i=1
−
∂ψ
(t)
M (t, ku(t)k ),
∂xi
2
µ
n Z tµ
X
∂M
µ
¶2
¶2 ¶
n Z tµ
X
∂ ∂ψ
(x, s)
∂s ∂xi
−
i=1 0
∂ψ
(s, ku(s)k ),
(s)
∂λ
∂xi
i=1 0
µ
2
ds dx
∂ψ
(s)
M (s, ku(s)k ),
∂xi
¶2 ¶ ³³
2
2
0
u0 (s), u(s)
´´
µ
¶2 ¶
ds
ds .
Hence,
³
0
ρ(t), (ψ (t))
2
´
+
n µ
X
i=1
≤ −2
∂ψ
(t)
M (t, ku(t)k ),
∂xi
2
Z tX
n Z
∂M
0 i=1 Ω
+2
Z t µh
∂xi
µ
¶2 ¶
(x, s, ku(s)k2 )
+ 2α
n Z tµ
X
0
2
M (s, ku(s)k ),
n Z tµ
X
∂M
i=1 0
|ψ 0 (s)|2 ds ≤
0
i
¶
M (s, ku(s)k2 ) − M (s, kz(s)k2 ) ∆z(s), ψ 0 (s) ds
i=1 0
+
t
∂ψ
(x, s) ψ 0 (x, s) dx ds
∂xi
0
+
Z
∂λ
µ
(s, ku(s)k2 ),
∂ψ
(s)
∂xi
µ
¶2 ¶
∂ψ
(s)
∂xi
ds +
Z t³
´
ρ0 (s), (ψ 0 (s))2 ds
0
¶2 ¶ ³³
2
u0 (s), u(s)
´´
ds .
THE NONHOMOGENEOUS DEGENERATED KIRCHHOFF EQUATION
289
Assumption (40) implies that all the solutions u of (38) and (39) have the
property
(47)
ku(t)k > 0 ,
∀ t ∈ [0, T ] ,
see Frota–Lar’kin [7]. By virtue of (47) and (41), we find
n
o
0 < m0 = min M (x, t, ku(t)k2 ); (x, t) ∈ Q ,
from which it follows
n µ
X
2
m0 kψ(t)k ≤
2
M (t, ku(t)k ),
i=1
µ
∂ψ
(t)
∂xi
¶2 ¶
∀ t ∈ [0, T ] .
,
Then
³
´
ρ(t), (ψ 0 (t))2 + m0 kψ(t)k2 +
≤2
Z t µh
Z tZ h
0
i
2 α − ρ0 (x, s) (ψ 0 (x, s))2 dx ds ≤
Ω
0
−2
Z tX
n Z
∂M
0 i=1 Ω
+
n Z tµ
X
∂xi
(x, s, ku(s)k2 )
2
0
M (s, ku(s)k ),
i=1 0
+
¶
i
M (s, ku(s)k2 ) − M (s, kz(s)k2 ) ∆z(s), ψ 0 (s) ds
n Z
X
t
2
i=1 0
³³
u0 (s), u(s)
µ
∂ψ
(x, s) ψ 0 (x, s) ds
∂xi
∂ψ
(s)
∂xi
´´ µ ∂M
∂λ
¶2 ¶
ds
(s, ku(s)k2 ),
µ
∂ψ
(s)
∂xi
¶2 ¶
ds .
Let us analize the right-hand side of this inequality. We choose θ such that
(48)
0<θ<
a0
.
2
Therefore,
(49)
2
Z t µh
2
2
i
0
¶
M (s, ku(s)k ) − M (s, kz(s)k ) ∆z(s), ψ (s) ds ≤
0
≤
C
θ
Z
t
0
kψ(s)k2 ds + θ
Z
t
0
|ψ 0 (s)|2 ds ,
290
C.L. FROTA and N.A. LAR’KIN
(50)
2
Z tX
n Z
∂M
∂xi
0 i=1 Ω
(x, s, ku(s)k2 )
C
≤
θ
(51)
n Z tµ
X
Z
t
2
2
kψ(s)k ds + θ
0
2
0
M (s, ku(s)k ),
i=1 0
(52)
n Z t ³³
X
u0 (s), u(s)
i=1 0
≤C
Z
∂ψ
(x, s) ψ 0 (x, s) dx ds ≤
∂xi
t
µ
∂ψ
(s)
∂xi
´´ µ ∂M
∂λ
Z
¶2 ¶
t
|ψ 0 (s)|2 ds ,
0
ds ≤ C
(s, ku(s)k2 ),
µ
Z
t
kψ(s)k2 ds ,
0
∂ψ
(s)
∂xi
¶2 ¶
ds ≤
kψ(s)k2 ds ,
0
where C is a positive constant.
From (13) and (49)–(52), we obtain
³
´
ρ(t), (ψ 0 (t))2 + m0 kψ(t)k2 + (a0 − 2 θ)
Z
t
|ψ 0 (s)|2 ds ≤ C
0
Z
t
kψ(s)k2 ds .
0
This inequality, (48) and Gronwall’s Lemma give kψ(t)k ≤ 0 that implies
u = z. This completes the proof of Theorem 2.
REFERENCES
[1] Bernstein, S. – Sur une classe d’équations fonctionelles aux dérivées partielles,
Isv. Acad. Nauk. SSSR Ser. Math., 4 (1940), 17–26.
[2] Carrier, G.F. – On the nonlinear vibration problem of the elastic string, Q. J.
Appl. Math., 3 (1945), 157–165.
[3] D’ancona, P. and Spagnolo, S. – Global solvability for the degenerate Kirchhoff
equation with real analytic data, Inventiones Mathematicae, 108 (1992), 247–262.
[4] Ebihara, Y., Medeiros, L.A. and Milla Miranda, M. – Local solutions for a
nonlinear degenerated hyperbolic equation, Nonlinear Analysis, 10 (1986), 27–40.
[5] Frota, C.L. – Nonlocal solutions of a nonlinear hyperbolic partial differential
equation, Portugaliae Mathematica, 51 (1994), 455–473.
[6] Frota, C.L. – Global solutions for a nonlinear unilateral problem, An. Acad.
Bras. Ci., 66 (1994), 1–11.
[7] Frota, C.L. and Lar’kin, N.A. – On global existence and uniqueness for the
unilateral problem associated to the degenerated Kirchhoff equation, Nonlinear
Analysis, Theory, Methods & Applications, 28(3) (1997), 443–452.
[8] Kirchhoff, G. – Vorlesungen über Mechanik, Tauber leipzig, 1883.
[9] Lar’kin, N.A. – The unilateral problem for the nonlocal, quasilinear, hyperbolic
equation of the theory of elasticity, Sov. Phys. Dokl., 29(2) (1984), 110–112.
THE NONHOMOGENEOUS DEGENERATED KIRCHHOFF EQUATION
291
[10] Lar’kin, N.A. and Medeiros, L.A. – On a variational inequality for the theory
of the elasticity, in “Well posed boundary value problem for non classical equation”,
Novosibirsk, USSR (in Russian), 1991, pp. 6–11.
[11] Lions, J.L. – On some questions in boundary value problem of Mathematical
Physics, in “Contemporary developments in Continuum Mechanics and Partial
Differential Equations”, North Holland, Math. Studies (G.M. de la Penha and L.A.
Medeiros, Eds.), 1977.
[12] Lions, J.L. – Quelques mèthodes de résolution des problèmes aux limites non
linéaires, Dunod, Gauthier-Villars, 1968.
[13] Medeiros, L.A. and Milla Miranda, M. – Local solutions for a nonlinear unilateral problem, Revue Roumaine de Mathématiques Pures et Appliqueés, XXXI(5)
(1986), 371–382.
[14] Menzala, G.P. – On global classical solutions of a nonlinear wave equation,
Applicable Analysis, 10 (1980), 179–195.
[15] Nishihara, K. and Yamada, Y. – On global solutions of some degenerated quasilinear hyperbolic equations with dissipative terms, Funkcialaj Ekvacioj, 33 (1990),
151–159.
[16] Pohozaev, S. – On a class of quasilinear hyperbolic equations, Math. Sbornick,
96 (1975), 152–166.
Cı́cero Lopes Frota,
Departamento de Matemática – UEM,
Av. Colombo, 5790, 87.020-900, Maringá, PR. – BRASIL
and
Nickolai Andreevitch Lar’kin,
The Institute of Theoretical and Applied Mechanics,
Novosibirsk – 90, 630090 – RUSSIA