PORTUGALIAE MATHEMATICA
Vol. 51 Fasc. 3 – 1994
NON LOCAL SOLUTIONS OF A NONLINEAR
HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
Cı́cero Lopes Frota
Abstract: In this work we prove that the mixed problem for a temporally nonlinear
Kirchhoff-Carrier model, for vibrations of a nonhomogeneous stretched string, has unique
nonlocal solution for small data. The solution is obtained in S.L. Sobolev spaces.
Introduction
The nonlinear model of Kirchhoff-Carrier, cf. Carrier [5], for vibrations of an
elastic string, of lenght L, is given by:
∂2u
−
∂t2
(1)
µ
E
Po
+
ρ.h 2Lρ
Z
0
¯
¯2 ¶ 2
¯
∂ u
¯
¯
¯ ∂s (s, t)¯ ds ∂x2 = 0
L ¯ ∂u
where 0 ≤ x ≤ L and t > 0 represent the string in repose, u(x, t) is the vertical
displacement of the point x at the instant t, ρ is the mass density, h is the area of
the cross section of the string, L is the lenght of the string, Po the initial tension
on the string and E the Young’s modulus of the material.
The natural generalization of the model (1) is given by the following nonlinear
mixed problem
(2)

¯
µX
¶
n Z ¯
¯ ∂u ¯2

∂2u

¯
¯

−M

¯ ∂x ¯ dx ∆u = f on Q = Ω × (0, T )


∂t2
i

i=1 Ω



u = 0 on Σ = Γ × (0, T )


u(x, 0) = φo (x) on Ω






∂u


(x, 0) = φ1 (x) on Ω
∂t
Received : December 15, 1992.
456
C.L. FROTA
where Ω is a bounded open set of Rn with smooth boundary Γ, M : [0, ∞) → R
n ∂2
P
is a positive real function and ∆ =
2 is the Laplace operator.
i=1 ∂xi
Remark 1. In the Kirchhoff-Carrier model (1), M : [0, ∞) → R is M (λ) =
E
Po
+
λ.
ρ.h 2Lρ
Several authors have investigated the nonlinear problem (2). When n = 1
and Ω = (0, L), it was studied by Dickey [8] and Bernstein [3] whom considered
φo and φ1 analytic functions with some growth conditions. Assuming Ω bounded
open set of Rn , φo and φ1 analytic functions, Pohozaev [18] obtained existence
and uniqueness of global solutions for the mixed problem (2). In Lions [12] he
formulated the Pohozaev’s results in an abstract context obtaining better results
and presenting a collection of problems. One of the problems proposed by Lions
[12] was the study of the problem (2) with M : Ω × [0, ∞) → R, i.e., the problem
(3)

¯
µ X
¶
n Z ¯¯

∂u ¯¯2
∂2u

¯

−
M
x,
dx
∆u = f on Q

¯ ∂x ¯


∂t2
i

i=1 Ω



u = 0 on Σ


u(x, 0) = φo (x) on Ω






∂u


(x, 0) = φ1 (x) on Ω
∂t
that is, for nonhomogeneous materials. This case has it’s origin in the model (1)
when the physic elements ρ, h and E are not constants, but depends on the point
x in the string. In Rivera Rodrigues [20] the author proved the existence and
uniqueness of local solutions for the problem (3).
In a more general context it is correct to consider ρ, h and E changing not
only with the point x in the string but with the instant t too, i.e., ρ = ρ(x, t),
h = h(x, t) and E = E(x, t). In this case, we have the problem
(4)

¯
¶
µ
n Z ¯
X
¯ ∂u ¯2

∂2u

¯
¯

dx
∆u = f on Q
−
M
x,
t,

¯
¯


∂t2
Ω ∂xi

i=1



u = 0 on Σ


u(x, 0) = φo (x) on Ω






∂u


(x, 0) = φ1 (x) on Ω
∂t
where M : Ω × [0, T ] × [0, ∞) → R.
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
457
In this work we study the problem (4) and making use of the same technique
∂M
used by Rivera Rodrigues [20], we prove that if φo , φ1 , f and
are small
∂t
in some sense, then exist one, and only one, nonlocal solution for the problem
∂M
(4). It’s important to observe that it’s a good assumption to consider
small,
∂t
because in normal conditions ρ, h and E have a small variation with the time.
For the study of problem (2) with dissipative terms we have, for instance, Brito
[4] and Medeiros-Milla Miranda [14]. The problem (2) in the degenerate case
can be find in Arosio-Spagnolo [1], Ebihara-Medeiros-Milla Miranda [9], ArosioGaravaldi [2], Crippa [6], Yamada [21], Nishihara-Yamada [17] and Nishihara
[16].
The plan of this paper is the following:
1) Notations and preliminary results;
2) Assumptions and statement of the principal result;
3) Galerkin’s approximation and a priori estimates;
4) Proof of the theorem;
5) Uniqueness.
1 – Notation and preliminary results
Let Ω be a bounded open set of Rn with smooth boundary Γ. By L2 (Ω) we
represent the usual space of Lebesgue square integrable functions on Ω whose
inner product and norm will be denoted by ( · , · ) and | · | respectively. In the
Sobolev space Ho1 (Ω) we consider the norm
¯2
n Z ¯
X
¯ ∂u
¯
¯
¯
||u|| =
¯ ∂x (x)¯ dx
2
(5)
i=1 Ω
and inner product
(6)
((u, v)) =
n Z
X
i=1 Ω
i
∂u ∂v
dx .
∂xi ∂xi
Let (−∆) be the operator defined by {Ho1 (Ω), L2 (Ω), (( · , · ))}. Then as we
well known (−∆) is an unbounded selfadjoint operator in L2 (Ω) with domain
(7)
n
o
D(−∆) = u ∈ Ho1 (Ω); ∆u ∈ L2 (Ω) = Ho1 (Ω) ∩ H 2 (Ω)
and it has the following properties:
458
C.L. FROTA
(a) There exist mo > 0 such that
(−∆u, u) ≥ mo |u|2 , ∀ u ∈ D(−∆) ;
(8)
(b)
(−∆u, u) = ||u||2 , ∀ u ∈ D(−∆) ;
(9)
(c) There exist a sequence (λj )j∈N of real numbers and (wj )j∈N a sequence
of L2 (Ω) vectors such that
(10)
m o ≤ λ1 ≤ λ2 ≤ . . .
(11)
−∆wj = λj wj , ∀ j ∈ N
(12)
(13)
lim λj = ∞
j→∞
{wj } is a orthonormal complete set in L2 (Ω) and orthogonal complete set in Ho1 (Ω) and in Ho1 (Ω)∩H 2 (Ω).
Remark 2. We introduce the equivalent norm
(14)
||u||Ho1 (Ω)∩H 2 (Ω) = | − ∆u|,
∀ u ∈ Ho1 (Ω) ∩ H 2 (Ω)
for smooth boundary Γ.
In order to complete this section we introduce a compactness result. It is a
version of Arzela’s theorem and it’s proof follows the same argument as the usual
proof of scalar Arzela’s theorem.
Lemma 1. Let E and F be Banach spaces, E ,→ F with compact injection.
Let (σm )m∈N be a sequence of functions from the interval [a, b] ⊂ R into E.
If (σm )m∈N is uniformly bounded in [a, b] with respect to the norm of E and
equicontinuous with respect to the norm of F , then there exist a subsequence
(σmν )ν∈N of (σm )m∈N and a continuous function σ: [a, b] → F such that
(15)
lim σmν (t) = σ(t) in F uniformly for t ∈ [a, b] .
ν→∞
Moreover, if E is a reflexive Banach space then we find that σ ∈ L∞ (a, b; E).
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
459
2 – Assumptions and principal result
Let Ω be as in section 1, T > 0 a real number. We consider a real function
M : Ω × [0, T ] × [0, ∞) −→
R
(x, t, λ)
7−→ M (x, t, λ)
such that the following assumptions are satisfied:
1,∞ (Ω × (0, T ))), i.e., for each k > 0 we have M ∈
(H.1) M ∈ L∞
loc ([0, ∞); W
∂M
∂M
L∞ (Ω × (0, T ) × (0, k)),
∈ L∞ (Ω × (0, T ) × (0, k)) and
∈
∂t
∂xi
L∞ (Ω × (0, T ) × (0, k)) for i = 1, . . . , n.
∂M
∈ L∞ (Ω × (0, T ) × (0, L)).
(H.2) For each L > 0 we have
∂λ
(H.3) There exist a real number m1 > 0 such that m1 ≤ M (x, t, λ), ∀ x ∈ Ω,
t ∈ [0, T ] and λ ≥ 0.
Now we define
ko = 4(mo m31 )−1/2 ,
k1 =
1
m1
¯
¯
¯ ∂M
¯
¯
θo = ess sup ¯
(x, t, 0)¯¯
→
∂t
x∈Ω
0<t<T
(16)
(17)
1
1 + ||M ||L∞ (Ω×(0,T )×(0,1))
2
·µ
¶
´¸
4
T ³
k2 +
1 + e(1+k1 θo )T
k3 =
m m
2
° o 1°
° ∂M °
°
k4 = °
° ∂λ ° ∞
L (Ω×(0,T )×(0,k3 ))
·
k2 =
½
δ = min 1; m1/2
o ;
·
(18)
¸
ln 2
;
3T [1 + T ko k4 + T ko k4 e(1+k1 θo )T ]
ln 2
6T ko k2 k4 (1 + e(1+k1 θo )T )
k δ = k2 δ 2 +
¸1/2 ¾
T
δ.
2
Theorem. Let M : Ω×[0, T ]×[0, ∞) → R be a real function satisfying (H.1)–
(H.3), φo ∈ Ho1 (Ω) ∩ H 2 (Ω), φ1 ∈ Ho1 (Ω) and f : [0, T ] → Ho1 (Ω) a continuous
460
C.L. FROTA
function. If
(19)
|∆φo |2 + ||φ1 ||2 + 0 ≤ t ≤ T → Máx ||f (t)||2 ≤ δ 2
and
°
°
° ∂M °
°
°
° ∂t °
(20)
L∞ (Ω×(0,T )×(0,k3 ))
≤
ln 2
.
3T k1
Then there exist one, and only one, function u: [0, T ] → Ho1 (Ω) such that
(21)
u ∈ C([0, T ]; Ho1 (Ω)) ∩ C 1 ([0, T ]; L2 (Ω)) ∩ C 2 ([0, T ]lH −1 (Ω)) ,
(22)


u ∈ L∞ (0, T ; Ho1 (Ω) ∩ H 2 (Ω))





(23)
u0 ∈ L∞ (0, T ; Ho1 (Ω))
u00 ∈ L∞ (0, T ; L2 (Ω)) ,

00
2
2


 u (t) − M (t, ||u(t)|| ) ∆u(t) = f (t) in L (Ω), 0 ≤ t ≤ T
u(0) = φ
o


 u0 (0) = φ .
1
Remark 3. In (23)1 we are making use of the following notation: if ψ: Ω ×
(0, T ) → R is a function then ψ(t): Ω → R is defined by ψ(t)(x) = ψ(x, t).
3 – Galerkin’s approximation and a priori estimates
We consider Vo = {0} and Vm = [w1 , . . . , wm ] for m = 1, 2, . . . i.e., Vm is the
vector space spanned by w1 , . . . , wm ; where (wm )m∈N is as in the section 1. The
sequence of Galerkin’s approximation is defined by induction as follows: we put
uo : [0, T ] −→
Vo
t
7−→ uo (t) = 0
and for m = 1, 2, . . ., we consider
um : [0, Tm ] −→
V
Pm
t
7−→ um (t) = m
j=1 gjm (t)wj
the unique solution of the initial value problem, with the coefficient of −∆um (t)
depends on the time t:
(24)

00
2


 um (t) − M (t, ||um−1 (t)|| ) ∆um (t) = fm (t) in Vm , ∀ t ∈ [0, Tm ]
u (0) = ϕ
m
om


 u0 (0) = ϕ
1m
m
461
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
where
n
(25)
o
Tm = sup τ ; 0 < τ ≤ Tm−1 and um : [0, τ ] → Vm is solution of (24) ,
(26)
fm (t) =
m
X
(f (t), wj ) wj ,
j=1
(27)
ϕom =
0≤t≤T ,
m
X
(φo , wj ) wj ,
m
X
(φ1 , wj ) wj .
j=1
(28)
ϕ1m =
j=1
Remark 4. The Galerkin’s approximation is well defined. It’s sufficient we
note that the initial value problem (24) is equivalent to the following system of
ordinary differential equations:

m
´
³
X

00


λk gkm (t) M (t, ||um−1 (t)||2 )wk , wj = (f (t), wj )
g
(t)
+

jm



k=1

(29)




 gjm (0) = (φo , wj )


 0
0 ≤ t ≤ Tm ; j = 1, . . . , m
gjm (0) = (φ1 , wj ) .
Estimate (i) From (24)1 we have the approximate equation
³
´
´
(u00m (t), v) − M (t, ||um−1 (t)||2 ∆um (t), v = (fm (t), v),
(30)
Take v = −∆u0m (t) in (30) we get
1 d 0
||u (t)||2 +
2 dt m
Z
Ω
∀ v ∈ Vm .
M (x, t, ||um−1 ||2 ) ∆um (x, t).∆u0m (x, t) dx = ((fm (t), u0m (t))),
since
Z
Ω
³
´
M x, t, ||um−1 (t)||2 ∆um (x, t) ∆u0m (x, t) dx =
´
1 d³
M (t, ||um−1 (t)||2 )∆um (t), ∆um (t)
2 dt
Z
1
∂M
(x, t, ||um−1 (t)||2 ) (∆um (x, t))2 dx
−
2 Ω ∂t
Z
∂M
(x, t, ||um−1 (t)||2 ) (∆um (x, t))2 dx
− ((um−1 (t), u0m−1 (t)))
Ω ∂λ
=
462
C.L. FROTA
we have
(31)
³
´
d 1
||u0m (t)||2 + M (t, ||um−1 (t)||2 )∆um (t), ∆um (t)
dt 2
½ ·
1
2
¸¾
=
∂M
(x, t, ||um−1 (t)||2 ) (∆um (x, t))2 dx
Ω ∂t
Z
∂M
0
+ ((um−1 (t), um−1 (t)))
(x, t, ||um−1 (t)||2 ) (∆um (x, t))2 dx,
Ω ∂λ
= ((fm (t), u0m (t))) +
Z
∀ t ∈ [0, Tm ], m = 1, 2, . . .
Lemma 2. Let be
(32)

Zo (t) = 0




1h
Zm (t) =




2
³
||um (t)||2 + M (t, ||um−1 (t)||2 )∆um (t), ∆um (t)
α=
0 ≤ t ≤ Tm , m = 1, 2, . . . ,
0
=
sup Zm (t) , αm
0≤t≤Tm
θm
°
°
° ∂M °
°
=°
° ∂t °
,
L∞ (Ω×(0,T )×(0,α0m ))
´i
2
αm ,
mo m1
° ∂M °
°
βm = °
.
° ∂λ ° ∞
L (Ω×(0,T )×(0,α0m ))
°
°
Then, Tm = T , αm is finite ∀ m ∈ N and
(33)
1
Zm (t) ≤ Zm (0) +
2δ
·
Z
t
0
2
¸
||fm (s)|| ds e(δ+k1 θm−1 +ko αm−1 βm−1 )t .
Proof: The proof will be done by induction on m. Clearly the solution of
the problem

00


 g11 (t) + λ1 (M (t, 0)w1 , w1 ) g11 (t) = (f (t), w1 )
g (0) = (φ , w )
11
o
1


 g 0 (0) = (φ , w )
1
1
11
is defined in all [0, T ]. This show us that T1 = T . Moreover if we consider the
assumption (H.3) on M we have
(34)
|∆u1 (t)|2 ≤
2
Z1 (t),
m1
∀ t ∈ [0, T ] .
From (31) and (34) we get
Z10 (t) − (δ + k1 θo ) Z1 (t) ≤
1
||f1 (t)||2 ,
2δ
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
463
where δ is given by (17). By the last inequality we obtain
·
Z1 (t) ≤ Z1 (0) +
1
2δ
Z
t
0
¸
||f1 (s)||2 ds e(δ+k1 θo )t
and it proves that α1 is finite and (33) is true when m = 1. Now we make the
induction assumption, i.e., we assume that for m ≥ 1 we have Tm = T , αm finite
and (34) true for this m. Then (31) for m + 1 implies
1
||fm+1 (t)||2 + δZm+1 (t)
2δ Z ¯
¯
¯ ∂M
¯
1
2 ¯
¯
+
(x, t, ||um (t)|| )¯ (∆um+1 (x, t))2 dx
¯
2 Ω ∂t
¯
Z ¯
¯
¯ ∂M
2 ¯
0
¯
(x, t, ||um (t)|| )¯ (∆um+1 (x, t))2 dx .
+ ||um (t)|| ||um (t)|| ¯
Ω ∂λ
0
Zm+1
(t) ≤
By the other hand, we note that
(35)
||um (t)||2 ≤
2
1
|∆um (t)|2 ≤
Zm (t)
mo
mo m1
2
0
≤
αm = α m
,
mo m1
0 ≤ t ≤ T.
It follows that:
0
Zm+1
(t) − (δ + k1 θm + ko αm βm ) Zm+1 (t) ≤
1
||fm+1 (t)||2 .
2δ
The above inequality shows that (33) is true for (m + 1), αm+1 is finite and
Tm+1 = T , i.e., the proof of Lemma 2 is complete.
We denote,
(36)
τm = Zm (0) +
1
2δ
Z
T
0
||fm (t)||2 dt , m = 1, 2, . . . ,
and then the sequence (τm )m∈N is bounded. In fact, by (26), (27) and (28) we
have that
(37)

2

 ∆ϕom → ∆φo strong in L (Ω)




ϕ1m → φ1 strong in Ho1 (Ω)
fm (t) → f (t) strong in Ho1 (Ω), uniformly on [0, T ]
and from the hypothesis of small data (17) we obtain
(38)
|∆ϕom |2 + ||ϕ1m ||2 + 0 ≤ t ≤ T → Máx ||fm (t)||2 ≤ δ 2 ,
∀m ∈ N .
464
C.L. FROTA
Therefore,
||ϕom ||2 ≤
1
1 2
|∆ϕom |2 ≤
δ ≤ 1,
mo
mo
∀m ∈ N ,
and then,
τm
1
M (x, 0, ||ϕo(m−1) ||2 ) (∆ϕom (x))2 dx
= ||ϕ1m ||2 +
2
Ω
Z
T
1 T
||fm (t)||2 dt ≤ k2 δ 2 + δ = kδ .
+
2δ 0
2
·
Z
¸
We conclude that:
(39)
0 ≤ τ m ≤ kδ , ∀ m ∈ N ,
and
(40)
Zm (t) ≤ τm e(δ+k1 θm−1 +ko αm−1 βm−1 )t ,
∀ t ∈ [0, T ], m ∈ N .
Lemma 3. Exists a constant co (independent of m ∈ N and t ∈ [0, T ]) such
that
(41)
Zm (t) ≤ 2co , ∀ t ∈ [0, T ], ∀ m ∈ N .
Proof: We consider co = kδ [1 + e(1+k1 θo )T ]. Then, we have by (39):
(42)
τ m ≤ co , ∀ m ∈ N ,
and by (40)
Z1 (t) ≤ τ1 e(δ+k1 θo )t ≤ kδ e(1+k1 θo )T ≤ co ≤ 2co ,
it shows that (41) is true for m = 1. Now, we do the follows induction assumption:
given m ≥ 1 we assume that (41) is true for this m. In order to prove that (41)
is true for (m + 1) we have
αm = sup Zm (t) ≤ 2co
0≤t≤T
and
0
αm
2αm
4co
4
=
≤
=
kδ [1 + e(1+k1 θo )T ] =
mo m1
mo m1
mo m1
½µ
¶³
´¾
4
T
2
(1+k1 θo )T
=
k2 δ + δ 1 + e
≤ k3 .
mo m1
2
½
¾
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
465
Therefore, we can see that
° ∂M °
°
= k4
βm ≤ °
° ∂λ ° ∞
L (Ω×(0,T )×(0,k3 ))
°
(43)
and
(44)
θm
°
°
° ∂M °
°
°
≤°
∂t °
°
L∞ (Ω×(0,T )×(0,k3 ))
≤
ln 2
.
3T k1
By (40), (42), (43) and (44) we get
ln 2
Zm+1 (t) ≤ τm+1 e(δ+k1 θm +ko αm βm )t ≤ co e(δ+ 3T +2ko k4 co )t .
We note that, from our choice we have
µ
h
i
ln 2
δ+
+ 2ko k4 co = 1 + T ko k4 + T ko k4 e(1+k1 θo )T δ +
3T
ln 2 ln 2 ln 2
ln 2
ln 2
≤
+
+
=
.
+ 2ko k2 k4 [1 + e(1+k1 θo )T ] δ 2 +
3T
3T
3T
3T
T
¶
Therefore,
µ
(45)
δ+
ln 2
+ 2ko k4 co t ≤ ln 2 , ∀ t ∈ [0, T ] ,
3T
¶
and then
Zm+1 (t) ≤ 2co , ∀ t ∈ [0, T ] .
The above relation complete the proof of lemma 3.
We obtain from (41) the first estimate: There exists a constant c1 such that
(46)
||um (t)||2 + ||u0m (t)||2 + |∆um (t)|2 ≤ c1 , ∀ t ∈ [0, T ], ∀ m ∈ N
Estimate (ii) We start observing that
Z ¯
¯
¯2
¯2
¯
¯
¯
¯
¯M (t, ||um−1 (t)||2 ) ∆um (t)¯ =
¯M (x, t, ||um−1 (t)||2 )¯ |∆um (x, t)|2 dx
Ω
≤ ||M ||L∞ (Ω×(0,T )×(0,c1 )) · c1
and
|fm (t)|2 =
m
X
j=1
|(f (t), wj )|2 ≤ |f (t)|2 ≤
δ2
1
||f (t)||2 ≤
≤1.
mo
mo
466
C.L. FROTA
Thus, using (24)1 we obtain the existence of a constant c2 such that
|u00m (t)|2 ≤ c2 , ∀ t ∈ [0, T ], ∀ m ∈ N .
(47)
By (46), (47) and the fundamental theorem of calculus we choose t, s ∈ [0, T ]
and we have that
||um (t) − um (s)|| ≤
(48)
√
c1 |t − s| ,
|u0m (t) − u0m (s)| ≤ c2 |t − s| .
(49)
In order to obtain an estimate for (u00m ) analogous to (48) and (49) we choose
t, s ∈ [0, T ] and by (24)1 we get
u00m (t) − u00m (s) = M (t, ||um−1 (t)||2 ) ∆(um (t) − um (s)) +
h
i
+ M (t, ||um−1 (t)||2 ) − M (s, ||um−1 (s)||2 ) ∆um (s) + (fm (t) − fm (s)) .
On the other hand, for v ∈ Ho1 (Ω) we note that
=
°
°2
°
°
°M (t, ||um−1 (t)||2 ).v ° =
¯2
m Z ¯
X
¯
¯ ∂M
2
2 ∂v
¯ dx
¯
(x,
t,
||u
(t)||
).v(x)
+
M
(x,
t,
||u
(t)||
)
(x)
m−1
m−1
¯
¯ ∂x
∂x
i=1 Ω
i
i
°
n °
X
° ∂M °2
°
°
≤ 2|v|
° ∂x ° ∞
2
i=1
i
+
L (Ω×(0,T )×(0,c1 ))
2||M ||2L∞ (Ω×(0,T )×(0,c1 )) .
°
n °
X
° ∂M °
°
°
≤ 2 ||M ||L∞ (Ω×(0,T )×(0,c1 )) +
° ∂x °
·
i
i=1
L∞ (Ω×(0,T )×(0,c1 ))
Whence, there exists a constant c3 such that
(50)
¸·
n ¯
X
¯ ∂v
¯
¯ ∂x
i
i=1
¯2
¯
¯
¯
n ¯
X
¯ ∂v
¯
|v| +
¯ ∂x
2
i
i=1
¯2 ¸
¯
¯ .
¯
°
°2
°
°
°M (t, ||um−1 (t)||2 ).v ° ≤ c3 ||v||2 , ∀ t ∈ [0, T ], ∀ m ∈ N .
By the above estimate we have
(M (t, ||um−1 (t)||2 ) ∆(um (t) − um (s)), v) =
³
= ∆(um (t) − um (s)), M (t, ||um−1 (t)||2 ).v
³
= (um (s) − um (t), M (t, ||um−1 (t)||2 )v)
√
≤ c3 ||v|| ||um (s) − um (t)||
´
´
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
467
and using (48) we get
(51)
¯³
´¯ √
¯
¯
¯ M (t, ||um−1 (t)||2 ) ∆(um (t) − um (s)), v ¯ ≤ c1 c3 ||v|| |t − s| .
Now, if we consider g(x, t) = (x, t, ||um−1 (t)||2 ) then we have
M (x, t, ||um−1 (t)||2 ) − M (x, s, ||um−1 (s)||2 ) =
Z t
∂
=
(M ◦ g)(x, ξ) dξ
s ∂ξ
=
Z
t
s
+2
Z
∂M
(x, ξ, ||um−1 (ξ)||2 ) dξ
∂ξ
t
s
∂M
(x, ξ, ||um−1 (ξ)||2 ) ((um−1 (ξ), u0m−1 (ξ))) dξ .
∂λ
Then we can see that there exists a constant c4 such that
¯
¯
¯
¯
¯M (x, t, ||um−1 (t)||2 ) − M (x, s, ||um−1 (s)||2 )¯ ≤ c4 |t − s|
and this estimate shows that there exists a constant c5 such that
(52)
¯ µh
¶¯
i
¯
¯
¯ M (t, ||um−1 (t)|2 ) − M (s, ||um−1 (s)||2 ) ∆um (s), v ¯ ≤ c5 ||v|| |t − s| .
¯
¯
Finally, we note that
(53)
|(fm (t) − fm (s), v)| ≤
1
||f (t) − f (s)|| ||v|| .
mo
From (51), (52) and (53) we obtain that there exists a constant c 6 such that
(54)
³
´
||u00m (t) − u00m (s)||H −1 (Ω) ≤ c6 |t − s| + ||f (t) − f (s)|| .
The estimate (ii) is the relations (47), (48), (49) and (54).
4 – Proof of the theorem
By estimates (i) and (ii) we have:
(um )m∈N uniformly bounded in [0, T ] with respect
H 2 (Ω) and equicontinuous with respect to the norm of
to the norm of Ho1 (Ω) ∩
Ho1 (Ω).
(u0m )m∈N uniformly bounded in [0, T ] with respect to the norm of Ho1 (Ω) and
equicontinuous with respect to the norm of L2 (Ω).
468
C.L. FROTA
(u00m )m∈N uniformly bounded in [0, T ] with respect to the norm of L2 (Ω) and
equicontinuous with respect to the norm of H −1 (Ω).
Then, by lemma 1, there exists a function u: Ω×[0, T ] → R and a subsequence
(umν )ν∈N extracted from (um )m∈N , such that
(55)
(56)
u ∈ C([0, T ]; Ho1 (Ω)) ∩ C 1 ([0, T ]; L2 (Ω)) ∩ C 2 ([0, T ]; H −1 (Ω)) ,

1


 umν (t) → u(t) strongly in Ho (Ω), uniformly in [0, T ]
u0 (t) → u0 (t)
mν


 u00 (t) → u00 (t)
mν
strongly in L2 (Ω), uniformly in [0, T ]
strongly in H −1 (Ω), uniformly in [0, T ] .
Moreover, since Ho1 (Ω)∩H 2 (Ω), Ho1 (Ω) and L2 (Ω) are reflexive Banach spaces,
we still have


u ∈ L∞ (0, T ; Ho1 (Ω) ∩ H 2 (Ω)


(57)



u0 ∈ L∞ (0, T ; Ho1 (Ω))
u00 ∈ L∞ (0, T ; L2 (Ω)) .
The convergences don’t allow us to pass to the limit in the approximate equation. Indeed, the sequence (umν )ν∈N have the properties, but we can’t say the
same for (umν −1 )ν∈N . In order to solve this problem we will prove the following
lemma.
Lemma 4.
lim ||um+1 (t) − um (t)||2 = 0 uniformly on [0, T ].
m→∞
Proof: For each m ∈ N we define wm = um+1 − um . Then
2
||um+1 (t) − um (t)|| =
n Z µ
X
∂wm
i=1 Ω
∂xi
(x, t)
¶2
dx
and making use of the assumption (H.3) we can see that there exists a constant
c7 such that
(58)
||um+1 (t) − um (t)||2 ≤
≤ c7
n
X
∂wm
1
∂wm
0
(t),
(t)
|wm
(t)|2 +
M (t, ||um (t)||2 )
2
∂x
∂xi
i
i=1
½ ·
µ
Hence, we are motivated to put
(59)
n
X
1
∂wm
∂wm
0
ψm (t) = |wm
M (t, ||um (t)||2 )
(t)|2 +
(t),
(t)
2
∂xi
∂xi
i=1
·
µ
¶¸
¶ ¸¾
.
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
469
and then, we will conclude with the proof of lemma showing that ψm (t) → 0
uniformly in [0, T ].
Differentiating ψm (t), we have
0
ψm
(t) =
(60)
1 d 0
|w (t)|2 +
2 dt m
+
n
∂M
1X
∂wm
∂wm
(t, ||um (t)||2 )
(t),
(t) +
2 i=1 ∂t
∂xi
∂xi
+
((um (t), u0m (t)))
+
n µ
X
µ
¶
n µ
X
∂M
i=1
i=1
M (t, ||um (t)||2 )
∂wm ∂wm
(t, ||um (t)|| )
,
(t) +
∂λ
∂xi ∂xi
2
¶
0
∂wm
∂wm
.
(t),
∂xi
∂xi
¶
From the approximation equation we find
h
i
00
wm
(t) + M (t, ||um−1 (t)||2 ) − M (t, ||um−1 (t)||2 ) ∆um (t) −
− M (t, ||um (t)||2 ) ∆wm (t) = fm+1 (t) − fm (t)
and then
³
´
1 d 0
0
|wm (t)|2 = M (t, ||um (t)||2 )∆wm , wm
(t)
2 dt
¶
µh
i
0
(t)
+ M (t, ||um (t)||2 ) − M (t, ||um−1 (t)||2 ) ∆um (t), wm
³
´
0
+ fm+1 (t) − fm (t), wm
(t) .
From the above relation and (60) we obtain
(61)
0
ψm
(t) = Am (t) + Bm (t) + Cm (t) + Dm (t) + Em (t)
where
(62)

¶
n µ
X

∂M

2 ∂wm
0

(t, ||um (t)|| )
(t), wm (t)
Am (t) = −



∂xi
∂xi

i=1


¶
µh

i


2
2
0


Bm (t) = M (t, ||um (t)|| ) − M (t, ||um−1 (t)|| ) ∆um (t), wm (t)





¶
n µ
X
∂M
∂wm
0
2 ∂wm
C
(t)
=
((u
(t),
u
(t)))
(t,
||u
(t)||
)
(t),
(t)
m
m
m

m

∂λ
∂xi
∂xi


i=1



¶
µ
n


∂wm
∂wm
∂M
1X


(t, ||um (t)||2 )
(t),
(t)
D
(t)
=

m


2 i=1 ∂t
∂xi
∂xi





0
Em (t) = (fm+1 (t) − fm (t), wm (t)) .
470
C.L. FROTA
By (59) and the estimates we find constants c8 , c9 , c10 and c11 such that
Am (t) ≤ c8 ψm (t),
Bm (t) ≤ c9 [ψm−1 (t) − ψm (t)]
Cm (t) ≤ c10 ψm (t),
Dm (t) ≤ c11 ψm (t)
1
and Em (t) ≤ |fm+1 (t) − fm (t)|2 + ψm (t).
2
Then we prove that there exists a constant c12 , independent of m and t ∈
[0, T ], such that
0
ψm
(t) − c12 ψm (t) ≤
1
|fm+1 (t) − fm (t)|2 + c12 ψm−1 (t)
2
and then,
ψm (t) ≤ e
c12 T
·
1
ψm (0) +
2
+ c12 ec12 T
Now we denote by
γm
Z
t
Z
T
0
2
|fm+1 (t) − fm (t)| dt
¸
ψm−1 (s) ds .
0
1
= ψm (0) +
2
Z
T
0
|fm+1 (t) − fm (t)|2 dt ,
and choose
n
o
c13 = Máx ec12 T , c12 ec12 T , 0 ≤ t ≤ T → Máx ψ1 (t) .
Then, we can see that
(63)


 ψ1 (t) ≤ c13

 ψm (t) ≤ c13 γm + c13
Z
t
0
ψm−1 (s) ds .
By induction we find
(64)
ψm (t) ≤ c13
m−1
X
j=0
(c13 + t)j
γm−j , ∀ t ∈ [0, T ], m = 2, 3, . . .
j!
If we consider (37) we get
(65)
lim γm = 0
m→∞
and, as we well know,
(66)
∞
X
(c13 T )j
j=1
j!
= ec13 T .
A NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
471
Therefore, from (64), (65) and (66) we conclude that ψm (t) → 0 uniformly in
[0, T ] and the proof of lemma 4 is complete.
The result of lemma 4 implies that
lim ||umν −1 (t)||2 = ||u(t)||2 uniformly in [0, T ] .
(67)
ν→∞
Then, we have the following convergences:
(68)
M (t, ||umν −1 (t)||2 ).v → M (t, ||u(t)||2 ).v
strongly in L2 (Ω), uniformly in [0, T ], ∀ v ∈ L2 (Ω) ,
(69)
∆umν (t) → ∆u(t) weakly in L2 (Ω), 0 ≤ t ≤ T .
The convergences (68) and (69) imply
(70)
M (t, ||umν −1 (t)||2 )∆umν (t) → M (t, ||u(t)||2 )∆u(t)
weakly in L2 (Ω), 0 ≤ t ≤ T .
We have then by passage to the limit in ν that
u00 (t) − M (t, ||u(t)||2 ) ∆u(t) = f (t) in L2 (Ω), 0 ≤ t ≤ T .
Clearly we also have u(0) = φo and u0 (0) = φ2 .
5 – Uniqueness
Let u and v be satisfying (21), (22) and (23). Then, if we define w = u − v
we get
(71)
(
w00 (t) + M (t, ||v(t)||2 ) ∆v(t) − M (t, ||u(t)||2 ) ∆u(t) = 0
w(0) = w 0 (0) = 0 .
Now we put
(72)
n
X
∂w
1
∂w
M (t, ||u(t)||2 )
ψ(t) = |w0 (t)|2 +
(t),
(t)
2
∂xi
∂xi
i=1
·
µ
¶¸
.
Therefore, using again the same analysis used in the proof of lemma 4, we
obtain a constant c14 such that
ψ 0 (t) − c14 ψ(t) ≤ 0
472
C.L. FROTA
and this imply
(73)
ψ(t) ≤ cc14 t ψ(0) , ∀ t ∈ [0, T ] .
But, from (72) there exists a constant c15 such that
h
i
0 ≤ ψ(t) ≤ c15 |w0 (t)|2 + ||w(t)||2 , 0 ≤ t ≤ T .
By (71)2 , if we take t = 0 in the above relation, we have ψ(0) = 0. This fact
with (73) shows that ψ(t) = 0, 0 ≤ t ≤ T ; and then we have uniqueness.
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Cı́cero Lopes Frota,
Departamento de Matemática, Universidade Estadual de Maringá,
Agencia Postal UEM, 87 020-900 Maringá - PR – BRAZIL