Electronic Journal of Differential Equations, Vol. 2006(2006), No. 130, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
WEAK SOLUTIONS FOR A STRONGLY-COUPLED NONLINEAR
SYSTEM
OSMUNDO A. LIMA, ALDO T. LOURÊDO, ALEXANDRO O. MARINHO
Abstract. In this paper the authors study the existence of local weak solutions of the strongly nonlinear system
u00 + Au + f (u, v)u = h1
v 00 + Av + g(u, v)v = h2
where A is the pseudo-Laplacian operator and f , g, h1 and h2 are given
functions.
1. Introduction
Let Ω be an open and bounded subset in Rn with smooth boundary Γ and let
T
be
P a positive real number. In the cylinder Q = Ω×]0, T [, with lateral boundary
= Γ×]0, T [, we consider the nonlinear system
u00 + Au + f (u, v)u = h1
v 00 + Av + g(u, v)v = h2
u(0) = u0 ,
u0 (0) = u1 ,
v(0) = v0 ,
v 0 (0) = v1
(1.1)
u = v = 0 on Σ = Γ×]0, T [
where
Au = −
n
X
∂ ∂u p−2 ∂u ,
|
∂xi ∂xi
∂xi
i=1
p > 2,
is the pseudo-Laplacian operator, f is a continuous function in the first variable and
Lipschitz in the second variable and g is a Lipschitz’s function in the first variable
and continuous in the second variable, with f (0, 0) = g(0, 0) = 0 and u0 , v0 , u1 , v1 ,
h1 and h2 are given functions.
When p ≥ 2, many authors studied the system (1.1). For instance, we can
mention: Segal [11], where the physical meaning of (1.1) is presented, Medeiros
and Menzala [9], Medeiros and M. Miranda [10], Castro [3], Biazutti [1] and more
recently, Clark and Lima [6] showed the existence, a local solution and its uniqueness
for the system
u00 − ∆u + f (u, v)u = h1
in Q = Ω × (0, T )
2000 Mathematics Subject Classification. 35L85, 35L05, 35L20, 35L70, 49A29.
Key words and phrases. Weak solutions; coupled system; monotonic operator.
c
2006
Texas State University - San Marcos.
Submitted March 3, 2006. Published October 16, 2006.
O. A. Marinho is partially supported by CNPq-Brazil.
1
2
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
v 00 − ∆u + g(u, v)v = h2
u(0) = u0 ,
v(0) = v0 ,
u = 0,
in Q
0
in Ω
0
in Ω
u (0) = u1
v (0) = v1
EJDE-2006/130
v = 0 on Σ = Γ × (0, T ),
where the functions f and g satisfying the same conditions of the problem (1.1).
Castro [3] showed the existence of solution for the system
u00 + Au − ∆u0 + |v|ρ+2 |u|ρ u = f1
00
0
ρ+2
v + Av − ∆v + |u|
u(0) = u0 ,
v(0) = v0 ,
u = 0,
0
ρ
|v| v = f2
u (0) = u1
in Q
in Q
in Ω
0
v (0) = v1 in Ω
v = 0 on Σ,
where A is the pseudo-Laplacian operator. We can show that the functions f (u, v) =
|u|ρ+2 |v|ρ and g(u, v) = |v|ρ+2 |u|ρ , ρ ≥ −1, satisfy the conditions of the system
(1.1). Consequently the above system, without the dissipations ∆u0 and ∆v 0 , is
a particular case of (∗). Thus, we see that (1.1) generalizes the above mentioned
problems.
To show the existence of a local solution for (1.1), we encounter following technical difficulties:
(i) The choices of the functional spaces;
(ii) In the a priori estimate for u00m , we had that to use the projection operator,
since, to derive the approximated equation we will have much technical
difficulties because of the pseudo-Laplacian operator in the equation;
(iii) In the passage to the limit, we use strongly the fact that A is a monotonic
and hemicontinuous operator.
We remark that these difficulties do not appear in [6].
Notation. We represent the Sobolev space of order m in Ω by
W m,p (Ω) = {u ∈ Lp (Ω) : Dα u ∈ Lp (Ω)∀|α| ≤ m},
with the norm
kukm,p =
X
|Dα u|pLp (Ω)
1/p
, u ∈ W m,p (Ω), 1 ≤ p < ∞.
|α|≤m
Let D(Ω) be the space of test functions in Ω and by W0m,p (Ω) we represent the
0
closure of D(Ω) in W m,p (Ω). The dual space of W0m,p (Ω) is denoted by W −m,p (Ω)
with p0 is such that p1 + p10 = 1. We use the symbols (·, ·) and | · |, to indicate the
inner product and the norm in L2 (Ω). We use h·, ·iW −1,p (Ω),W 1,p (Ω) to indicate the
0
0
duality between W −1,p (Ω) and W01,p (Ω) and k · k0 to indicate the norm W01,p (Ω).
The pseudo-Laplacian operator A is such that
0
A : W01,p (Ω) → W −1,p (Ω)
u
7→
Au
and it satisfies the following properties:
• A is monotonic, that is, hAu − Av, u − vi ≥ 0, ∀u, v ∈ W01,p (Ω);
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WEAK SOLUTIONS
3
• A is hemicontinuous, that is, for each u, v, w ∈ W01,p (Ω) the function λ 7→
hA(u + λv), wi is continuous in R;
• hAu(t), u(t)iW −1,p0 (Ω)×W 1,p (Ω) = kukp0 ;
0
• hAu(t), u0 (t)iW −1,p0 (Ω)×W 1,p (Ω) =
0
p d
1 d
p dt kuk0 , dt
=0 ;
Ckuk0p−1 ,
where C is a constant;
• kAu(t)kW −1,p0 (Ω) ≤
We will use the same notation for the operator P and its restrictions, as well as for
the operator P ∗ .
The next lemma plays a central role in the proof of the Existence Theorem. Its
proof can be found in [6].
Lemma 1.1. Let φ be a positive real function, α, β and γ, positive real constants,
with γ > 1, such that
Z t
φ(t) ≤ α + β
φ(s) + φγ (s) ds.
0
Then, there exists T0 ∈ R, with 0 < T0 < T , such that φ is bounded in [0, T0 [.
Definition. A local weak solution of the problem (1.1) is a pair of functions
u = u(x, t), v = v(x, t) defined for all (x, t) ∈ QT0 = Ω × (0, T0 ), and T0 > 0 fixed,
satisfying
u, v ∈ L∞ (0, T0 ; W01,p (Ω));
u0 , v 0 ∈ L∞ (0, T0 ; L2 (Ω));
d 0
(u , w) + hAu, wi + hf (u, v)u, wi = h1 , w , ∀w ∈ W01,p (Ω)in D0 (0, T0 );
dt
d 0
(v , w) + hAv, wi + hg(u, v)v, wi = h2 , w , ∀w ∈ W01,p (Ω) in D0 (0, T0 );
dt
u(0) = u0 , u0 (0) = u1 , v(0) = v0 , v 0 (0) = v1 .
2. Existence Results
Theorem 2.1. Let f and g be functions of two variables such that f is continuous
in the first variable and Lipschitz in the second variable and g is Lipschitz in the
first and continuous in the second variable, with f (0, 0) = g(0, 0) = 0.
h1 , h2 ∈ L2 (0, T ; L2 (Ω));
u 0 , v0 ∈
W01,p (Ω);
2
(2.1)
(2.2)
u1 , v1 ∈ L (Ω).
(2.3)
Then it exists T0 > 0, T0 ∈ R and functions u : QT0 → R and v : QT0 → R
satisfying
u, v ∈ L∞ (0, T0 ; W01,p (Ω));
0
0
∞
2
u , v ∈ L (0, T0 ; L (Ω));
d 0
(u , w) + hAu, wi + hf (u, v)u, wi = h1 , w ,
dt
(2.4)
(2.5)
∀w ∈ W01,p (Ω), in D0 (0, T0 );
(2.6)
d 0
(v , w) + hAv, wi+ig(u, v)v, wi = h2 , w , ∀w ∈ W01,p (Ω), in D0 (0, T0 ); (2.7)
dt
u(0) = u0 , v(0) = v0 ;
(2.8)
4
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
u0 (0) = u1 ,
EJDE-2006/130
v 0 (0) = v1 .
(2.9)
The main tools in the proof of this theorem are the Faedo-Galerkin
method and
compactness arguments. Let H0s (Ω), with s > m = n 12 − p1 + 1 a separable
Hilbert space such that H0s (Ω) ,→ W01,p (Ω), is a continuous and dense immersion.
In H0s (Ω), there exists a complete orthonormal hilbertian base {wj }j∈N in L2 (Ω).
We consider Vm = [w1 , . . . , wm ] the subspace of H0s (Ω) generated by the m first
vectors of the base {wj }j∈N . Also, we have the following chain of continuous and
dense immersions.
0
H0s (Ω) ,→ W01,p (Ω) ,→ L2 (Ω) ,→ W −1,p (Ω) ,→ H −s (Ω).
(2.10)
We will divide the proof in three steps: (i) Approximated Problem, (ii) A Priori
Estimates I and (iii) A Priori Estimates II.
Approximated Problem. We want to find um (t), vm (t) in Vm satisfying the approximated problem.
(u00m (t), w) + hAum (t), wi + hf (um (t), vm (t))um (t), wi = (h1 (t), w),
(2.11)
00
(vm
(t), w)
(2.12)
+ hAvm (t), wi + hg(um (t), vm (t))vm (t), wi = (h2 (t), w),
for all w ∈ Vm ; and
um (0) = u0m ,
u0m (0) = u1m ,
vm (0) = v0m ,
0
vm
(0) = v1m ;
(2.13)
So that
u0m → u0 ,
u1m → u1 ,
v0m → v0 ,
v1m → v1 ,
in W01,p (Ω);
in L2 (Ω).
It can be shown that the above system satisfies the Caracthodory’s conditions;
therefore there exists solutions um (t), vm (t) in [0, tm ), tm < T satisfying (2.11)–
(2.13).
A priori estimates I. Let us consider w = 2u0m (t) in (2.11). It follows that
2(u00m (t), u0m (t)) + 2hAum (t), u0m (t)i + 2hf (um (t), vm (t))um (t), u0m (t)i
= (h1 (t), u0m (t)).
Thus
2 d
d 0
|u (t)|2 +
kum (t)kp0 = 2(h1 (t), u0m (t)) − 2hf (um (t), vm (t))um (t), u0m (t)i.
dt m
p dt
0
Similarly, setting w = 2vm
(t) in (2.12) it follows that
d 0
2 d
0
0
(t)) − 2hg(um (t), vm (t))um (t), vm
(t)i.
|v (t)|2 +
kvm (t)kp0 = 2(h2 (t), vm
dt m
p dt
Summing the two equalities above, then integrating from 0 to t, t < tm , and using
2
2
the Cauchy-Schwarz’s inequality and ab ≤ a +b
2 , we obtain
2
2
0
|u0m (t)|2 + |vm
(t)|2 + kum (t)kp0 + kvm (t)kp0
p
p
2
2
p
0
≤ |u0m (0)|2 + |vm
(0)|2 + kum (0)k0 + kvm (0)kp0
p
p
Z tZ
+2
|f (um (s), vm (s))||um (s)||u0m (s)|ds
0
Ω
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WEAK SOLUTIONS
Z tZ
0
|g(um (s), vm (s))||vm (s)||vm
(s)|ds
+2
Z
+
0
t
5
Ω
|u0m (s)|2
+
0
|vm
(s)|2
Z
ds +
T
|h1 (t)|2 + |h2 (t)|2 dt.
0
0
From (2.1), (2), and (2), it follows that
2
2
0
|u0m (t)|2 + |vm
(t)|2 + kum (t)kp0 + kvm (t)kp0
p
p
Z t
0
|u0m (s)|2 + |vm
(s)|2 ds
≤C+
0
Z t
|f (um (s), vm (s))||um (s)||u0m (s)|ds
+2
0
Z t
0
+2
|g(um (s), vm (s))||vm (s)||vm
(s)|ds.
(2.14)
0
From the Sobolev immersions it is well known that
W01,p (Ω) ,→ Lq (Ω),
Let α, β > 0, such that
1
α
+
1
β
+
1
2
∀1 ≤ q ≤
np
.
n−p
= 1, with 1 ≤ α, β ≤
np
n−p .
2
2
and the
Now, using Holder and Young inequalities, the inequality ab ≤ a +b
2
hypothesis over f , we have
Z tZ
2
|f (um (s), vm (s))||um (s)||u0m (s)|ds
0
Ω
Z tZ
≤C
|vm (s)||um (s)||u0m (s)|ds
0
Ω
Z tZ
α1 Z
2
β1 Z
≤C
|vm (s)|α
|um (s)|β
|u0m (s)|2
0
Ω
Ω
Ω
Z t
=C
|vm (s)|Lα (Ω) |um (s)|Lβ (Ω) |u0m (s)|L2 (Ω) ds
0
Z tn
o
p
p−1
1
0
|vm (s)|pLα (Ω) +
|um (s)|Lp−1
≤C
β (Ω) |um (s)|L2 (Ω) ds
p
p
0
Z tn
p
(p−1)
1
1
p − 2o 0
≤C
+
|vm (s)|pLα (Ω) + |um (s)|Lp−1
|um (s)|L2 (Ω) ds
β (Ω)
p
p
p−1
0
Z tn
1
1
p − 2o 0
=C
|vm (s)|pLα (Ω) + |um (s)|pLβ (Ω) +
|um (s)|L2 (Ω) ds
p
p
p−1
0
Z tn
1
1
p − 2 o2
≤C
|vm (s)|pLα (Ω) + |um (s)|pLβ (Ω) +
+ |u0m (s)|2L2 (Ω) ds
p
p
p
−
1
0
Z tn
o
1
1
p − 2 2
2p
2p
0
2
|v
(s)|
|u
(s)|
+
+
|u
(s)|
≤C
2 (Ω) ds
α (Ω) +
m
m
β
m
L
L
L
(Ω)
p2
p2
p−1
0
Z tn
o
1
1
2p
2p
0
2
≤C
|v
(s)|
+
|u
(s)|
+
1
+
|u
(s)|
2 (Ω) ds.
α
m
m
β
m
L
L (Ω)
L (Ω)
p2
p2
0
6
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
EJDE-2006/130
Since W01,p (Ω) ,→ Lα (Ω) and W01,p (Ω) ,→ Lβ (Ω), it follows that
Z tZ
2
|f (um (s), vm (s))||um (s)||u0m (s)|ds
0
Ω
Z tn
o
1
1
2p
2p
0
2
≤C
+
+
1
+
|u
(s)|
kv
(s)k
ku
(s)k
2 (Ω) ds.
m
m
m
0
0
L
p2
p2
0
Similarly, we have
Z tZ
0
2
|g(um (s), vm (s))||vm (s)||vm
(s)|ds
0
Ω
Z tn
o
1
1
2p
2p
0
2
≤C
ku
(s)k
+
kv
(s)k
+
1
+
|v
(s)|
2
m
m
m
0
0
L (Ω) ds.
p2
p2
0
(2.15)
(2.16)
Substituting, (2.15) and (2.16) in (2.14),
2
2
0
|u0m (t)|2 + |vm
(t)|2 + kum (t)kp0 + kvm (t)kp0
p
p
Z t
Z t
2p 0
≤C +C
|u0m (s)|2 + |vm
(s)|2 ds + C
kum (s)k2p
0 + kvm (s)k0
0
0
Z t
+C
2 ds
0
Z t
Z t
2p 0
≤C +C
|u0m (s)|2 + |vm
(s)|2 ds + C
kum (s)k2p
0 + kvm (s)k0
0
Z
(2.17)
0
T
+C
2 ds
0
Z
≤C +C
t
|u0m (s)|2
+
0
0
|vm
(s)|2
Z
ds + C
t
2p kum (s)k2p
0 + kvm (s)k0 .
0
Note that
2 0
2
2
2 0
|um (t)|2 + |vm
(t)|2 + kum (t)kp0 + kvm (t)kp0
p
p
p
p
2
2
p
0
2
0
2
≤ |um (t)| + |vm (t)| + kum (t)k0 + kvm (t)kp0 ,
p
p
with p > 2, It follows that
0
|u0m (t)|2 + |vm
(t)|2 + kum (t)kp0 + kvm (t)kp0
Z t
Z t
2p 0
2
0
2
≤C +C
|um (s)| + |vm (s)| ds + C
kum (s)k2p
0 + kvm (s)k0
0
0
Z tn
2
2
0
≤C +C
|u0m (s)|2 + |vm
(s)|2 + kum (s)kp0 + kvm (s)kp0
0
o
0
0
+ 2 |um (s)|2 + |vm
(s)|2 kum (s)kp0 + kvm (s)kp0 ds
Z t
0
0
+C
|um (s)|2 + |vm
(s)|2 + kum (s)kp0 + kvm (s)kp0 ds
0
Z t
0
2
0
=C +C
|um (s)|2 + |vm
(s)|2 + kum (s)kp0 + kvm (s)kp0 ds
0
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WEAK SOLUTIONS
Z
+C
7
t
0
|u0m (s)|2 + |vm
(s)|2 + kum (s)kp0 + kvm (s)kp0 ds.
0
By setting
0
φ(t) = |u0m (t)|2 + |vm
(t)|2 + kum (t)kp0 + kvm (t)kp0 ,
the above inequality can be rewritten as
Z t
φ(s) + φ2 (s) ds.
φ(t) ≤ C + C
(2.18)
0
Then, by Lemma 1.1, there exists T0 ∈ R, with 0 < T0 < T , such that φ is bounded
in [0, T0 ). From this, we have
0
|u0m (t)|2 + |vm
(t)|2 + kum (t)kp0 + kvm (t)kp0 ≤ C
∀t ∈ [0, T0 ),
∀m ∈ N.
(2.19)
Therefore, by prolongation results, we can extend the solutions um (t), vm (t), to the
interval [0, T0 ].
00
We will estimate, now, the second derivatives u00m (t), vm
(t). Since the procedure,
00
00
to estimates um (t) and vm (t) are similar, we will fix our attention only on bounding
u00m (t).
2.1. A priori Estimates II. Let Pm : L2 (Ω) → Vm ⊂ L2 (Ω) be
Pm (h) =
m
X
(h, wj )wj ,
j=1
∗
the projection operator on L2 (Ω). Observe that Pm = Pm
and Pm ∈ L(H0s (Ω)).
Now, by the approximate equation (2.12),
(u00m (t), w) + hAum (t), wi + hf (um (t), vm (t))um (t), wi = (h1 (t), w)
(2.20)
for all w ∈ Vm . By the chain of immersions (2.10) we have
hu00m (t) + Aum (t) + f (um (t), vm (t))um (t) − h1 (t), wiH −s (Ω),H0s (Ω) = 0,
for all w ∈ Vm . From this equality and the fact that Pm w = w, ∀w ∈ Vm , we have
∗
Pm
(u00m (t) + Aum (t) + f (um (t), vm (t))um (t) − h1 (t)) = 0
∗
in Vm . From this, by the linearity of Pm
, the fact that u00m ∈ Vm , and by the
continuous and dense immersions, we have
∗
∗
∗
u00m (t) = −Pm
(Aum (t)) − Pm
(f (um (t), vm (t))um (t)) + Pm
(h1 (t))
in H −s (Ω). Thus
∗
ku00m (t)kH −s (Ω) ≤ kPm
(f (um (t), vm (t))um (t))kH −s (Ω)
∗
∗
+ kPm
(Aum (t))kH −s (Ω) + kPm
(h1 (t))kH −s (Ω)
0
∗
With Pm ∈ L(H0s (Ω)) which implies Pm
∈ L(H −s (Ω)). Since W −1,p (Ω) ,→
−s
∗
−1,p0
H (Ω), it follows that Pm ∈ L(W
(Ω), H −s (Ω)), Then
∗
kPm
(Aum (t))kH −s (Ω) ≤ Ck(Aum (t))kW −1,p0 (Ω) ≤ Ckum (t)k0p−1 .
2
Since, L (Ω) ,→ H
−s
(Ω), we have
∗
Pm
2
∈ L(L (Ω), H
∗
kPm
(h1 (t))kH −s (Ω)
−s
(2.21)
(Ω). Furthermore,
≤ C|h1 (t)|L2 (Ω) .
(2.22)
∗
Now, to bound the term kPm
(f (um (t), vm (t))um (t))kH −s (Ω) , it is necessary to place
np
], such
f (um (t), vm (t))um (t) in some space contained in H −s (Ω). Let γ, θ ∈ [1, n−p
8
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
that
1
γ
+
W01,p (Ω)
1
θ
= 1. Since W01,p (Ω) ,→ Lq (Ω) for 1 ≤ q ≤
EJDE-2006/130
np
n−p ,
we have, in particular
γ
,→ L (Ω). Therefore,
0
0
Lγ (Ω) ,→ W −1,p (Ω).
0
From the chain of immersions (2.10), we have W −1,p (Ω) ,→ H −s (Ω), from where
0
(2.23)
Lθ (Ω) = Lγ (Ω) ,→ H −s (Ω)
Now, it is sufficient to show that f (um (t), vm (t))um (t) ∈ Lθ (Ω). From the Hölder
inequality and the hypothesis on f we have
Z
Z
θ
|f (um (s), vm (s))um (s)| dx =
|f (um (s), vm (s))|θ |um (s)|θ dx
Ω
Ω
Z
|vm (s))|θ |um (s)|θ dx
≤ Cfθ
Ω
Z
1/α0 Z
10
0
0
β
θ
,
≤ Cf
|vm (s))|α θ
|um (s))|β θ
Ω
Ω
(2.24)
where Cf is the Lipschitz constant, associated f and
np
np
If θα0 ≤ n−p
and θβ 0 ≤ n−p
, then
θ≤
1 np
,
α0 (n − p)
and θ ≤
1
α0
+
1
β0
= 1.
1 np
,
β 0 (n − p)
from which,
2θ ≤
Then, we have
1≤θ≤
1
1 np
+ 0
.
α0
β n−p
np
np
<
.
2(n − p)
n−p
0
0
Noticing that W01,p (Ω) ,→ Lθα (Ω) and W01,p (Ω) ,→ Lθβ (Ω), we have
Z
|f (um (s), vm (s))um (s)|θ dx ≤ Cfθ |vm (t)|θLα0 θ |um (t)|θLβ0 θ ≤ Ckum (t)kθ0 kvm (t)kθ0 .
Ω
From this estimate and (2.19), it follows
Z
|f (um (s), vm (s))um (s)|θ dx < ∞;
(2.25)
Ω
that is,
0
f (um (t), vm (t))um (t) ∈ Lθ (Ω) = Lγ (Ω) ,
for 1 ≤ θ ≤
np
,
2(n − p)
(2.26)
and
kf (um (t), vm (t))um (t)kLθ (Ω) ≤ C,
Similarly, we have
kg(um (t), vm (t))vm (t)kLθ (Ω) ≤ C,
f (um (t), vm (t))u2m (t)
We will also need that
ity,
Z
|f (um (s), vm (s))u2m (s)|θ dx
Ω
θ
∀m, t ∈ [0, T0 ]
(2.27)
∀m, t ∈ [0, T0 ]
(2.28)
∈ L (Ω). In fact, by Hölder inequal-
EJDE-2006/130
WEAK SOLUTIONS
9
Z
|f (um (s), vm (s))|θ |u2m (s)|θ dx
Z
θ
≤ Cf
|vm (s))|θ |um (s)|θ |um (s)|θ dx
Ω
Z
1 Z
1/δ Z
1/ω
θ
ξθ ξ
δθ
,
≤ Cf
|vm (s))|
|um (s))|
|um (s))|ωθ
=
Ω
Ω
Ω
Ω
1
δ
where Cf is the Lipschitz constant, associated to f and
np
np
θδ ≤ n−p
and θω ≤ n−p
then
θ≤
1 np
,
ξ n−p
1 np
,
δn−p
θ≤
θ≤
+ ω1 + 1ξ = 1. If θξ ≤
np
n−p ,
1 np
ωn−p
which implies
3θ ≤
1 1
1 np
+ +
.
ξ
δ
ω n−p
Then
np
np
<
.
3(n − p)
n−p
1≤θ≤
Observing that W01,p (Ω) ,→ Lθξ (Ω) , W01,p (Ω) ,→ Lθδ (Ω) and W01,p (Ω) ,→ Lθω (Ω),
it follows that
Z
|f (um (s), vm (s))u2m (s)|θ dx ≤ Cfθ |vm (t)|θLξθ |um (t)|θLωθ |um (t)|θLδθ
(2.29)
Ω
θ
≤ Ckum (t)k2θ
kv
(t)k
.
m
0
0
This estimate and (2.19) lead us to
Z
|f (um (s), vm (s))u2m (s)|θ dx < ∞;
Ω
that is,
0
f (um (t), vm (t))u2m (t) ∈ Lθ (Ω) = Lγ (Ω) ,
for 1 ≤ θ ≤
kf (um (t), vm (t))u2m (t)kLθ (Ω) ≤ C,
np
,
3(n − p)
(2.30)
∀m, t ∈ [0, T0 ]
(2.31)
∀m, t ∈ [0, T0 ]
(2.32)
Similarly, we have
2
kg(um (t), vm (t))vm
(t)kLθ (Ω) ≤ C,
np
3(n−p) ,
−s
Note that if θ ≤
Thus, as Lθ (Ω) ,→ H
np
3(n−p)
np
we still have (2.26) and (2.30), because
< 2(n−p)
.
∗
θ
−s
(Ω), we have that Pm ∈ L(L (Ω), H (Ω)). Therefore
∗
kPm
(f (um (t), vm (t))um (t))kH −s (Ω) ≤ Ckf (um (t), vm (t))um (t)kLθ (Ω) .
(2.33)
Hence, from the estimates (2.21), (2.22) and (2.33). we have
ku00m (t)kH −s (Ω) ≤ C kum (t)kp−1
+ kf (um (t), vm (t))um (t)kLθ (Ω) + |h1 (t)| .
0
From this inequality, it results
Z T0
Z
ku00m (t)k2H −s (Ω) dt ≤ C
0
Z
+
0
T0
0
T0
2(p−1)
kum (t)k0
Z
dt +
T0
|h1 (t)|2 dt
0
kf (um (t), vm (t))um (t)k2Lθ (Ω) dt .
10
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
EJDE-2006/130
Therefore, from (2.17), (2.25) and (2.1), we conclude that
ku00m (t)kL2 (0,T0 ;H −s (Ω) ≤ C,
∀m ∈ N.
(2.34)
Arguing in a similar way, one can deduce that
00
kvm
(t)kL2 (0,T0 ;H −s (Ω) ≤ C, ∀m ∈ N.
(2.35)
From (2.19), we have
kum (t)k0 ≤ C
|u0m (t)|
and kvm (t)k0 ≤ C,
≤C
0
|vm
(t)|
and
≤ C,
∀m, t ∈ [0, T0 ].
∀m, t ∈ [0, T0 ].
From where, it follows that ess supt∈[0,T0 ] kum (t)k0 ≤ C; that is
kum kL∞ (0,T0 ;W 1,p (Ω)) ≤ C,
∀m ∈ N.
(2.36)
kvm kL∞ (0,T0 ;W 1,p (Ω)) ≤ C,
∀m ∈ N;
(2.37)
0
Similarly, we have
0
ku0m kL∞ (0,T0 ;L2 (Ω))
0
kvm
kL∞ (0,T0 ;L2 (Ω))
≤ C,
∀m ∈ N;
(2.38)
≤ C,
∀m ∈ N;
(2.39)
Therefore, from (2.27), (2.28), (2.31), (2.32), (2.34), (2.35), (2.36), (2.37), (2.38),
(2.39), we have
are bounded in L∞ (0, T0 ; W01,p (Ω));
(um )m , (vm )m
0
(u0m )m , (vm
)m
00
00
(um )m , (vm )m
∞
2
are bounded in L (0, T0 ; L (Ω));
2
are bounded in L (0, T0 ; H
−s
(2.41)
(Ω));
∞
(2.40)
(2.42)
θ
(f (um , vm )um )m , (g(um , vm )vm )m
are bounded in L (0, T0 ; L (Ω));
(2.43)
2
(f (um , vm )u2m )m , (g(um , vm )vm
)m
are bounded in L∞ (0, T0 ; Lθ (Ω));
(2.44)
Furthermore, since A is bounded, we have
(Aum )m , (Avm )m
0
are bounded in L∞ (0, T0 ; W −1,p (Ω)).
Taking Limits. From the estimates and Banach-Alaoglu-Boubarki theorem guarantee the existence of subsequences (uν )ν , (vν )ν of (um )m , (vm )m , respectively, such
that
∗
uν * u,
∗
u0ν * u0 ,
∗
u00ν * u00 ,
∗
Auν * χ,
∗
vν * v
∗
vν0 * v 0
∗
vν00 *
∗
Avν *
v
η
00
in L∞ (0, T0 ; W01,p (Ω)).
(2.45)
in L∞ (0, T0 ; L2 (Ω)).
(2.46)
2
in L (0, T0 ; H
−s
(Ω)).
0
in L∞ (0, T0 ; W −1,p (Ω)).
(2.47)
(2.48)
As L2 (0, T0 ; H −s (Ω)) is reflexive, the convergence (2.47) becomes
u00ν * u00 , vν00 * v 00
in L2 (0, T0 ; H −s (Ω)).
(2.49)
Let us consider the approximate equation (2.11) in the form
(u00ν (t), w) + hAuν (t), wi + hf (uν, (t), vν, (t))uν (t), wi = (h1 (t), w) ∀w ∈ Vm , ν ≥ m
EJDE-2006/130
WEAK SOLUTIONS
11
Now, multiplying the above equality by ϕ ∈ D(0, T0 ) and integrating from 0 for T0
we obtain
Z T0
Z T0
Z T0
(u00ν (t), w)ϕdt +
hAuν (t), wiϕdt +
hf (uν, (t), vν, (t))uν (t), wiϕdt
0
0
Z
0
T0
(h1 (t), w)ϕdt ∀w ∈ Vm , ν ≥ m.
=
0
Integrating by parts, we obtain
Z T0
Z T0
Z
−
(u0ν (t), w)ϕ0 dt +
hAuν (t), wiϕdt +
0
0
T0
hf (uν, (t), vν, (t))uν (t), wiϕdt
0
T0
Z
(h1 (t), w)ϕdt ∀w ∈ Vm , ν ≥ m.
=
0
(2.50)
0
∗
With u0ν * u0 in L∞ (0, T0 ; L2 (Ω)) = L1 (0, T0 ; L2 (Ω)) then
hu0ν , φi → hu0 , φi, ∀φ ∈ L1 (0, T0 ; L2 (Ω)).
(2.51)
0 R T0 0
Convergence (2.51) with uν , φ = 0 (uν (t), φ(t))dt, and assuming φ(x, t) =
w(x)ψ(t) imply hat
Z T0
Z T0
(u0ν (t), φ(t))dt =
(u0ν (t), w(x))ψ(t)dt, ∀w ∈ L2 (Ω), ∀ψ ∈ L1 (0, T0 ).
0
0
Consequently, for all w ∈ L2 (Ω) and all ψ ∈ L1 (0, T0 ),
Z T0
Z T0
(u0ν (t), w(x))ψ(t)dt →
(u0 (t), w(x))ψ(t)dt .
0
0
In fact,
T0
Z
(u0ν (t), w(x))ϕ0 (t)dt
0
Z
T0
→
(u0 (t), w(x))ϕ0 (t)dt,
0
for all w ∈ Vm ⊂ W01,p (Ω) ⊂ L2 (Ω) and all ψ = ϕ0 , ϕ ∈ D(0, T0 ) ⊂ L1 (0, T0 ). In a
similar way,
Z T0
Z T0
< Auν (t), w(x) > ψ(t)dt →
< χ(t), w(x) > ψ(t)dt,
0
for all w ∈
0
W01,p (Ω)
T0
Z
1
and all ψ ∈ L (0, T0 ). In fact,
Z T0
(Auν (t), w(x))ϕ(t)dt →
(χ(t), w(x))ϕ(t)dt,
0
0
for all w ∈ Vm ⊂ W01,p (Ω) and all ϕ ∈ D(0, T0 ) ⊂ L1 (0, T0 ).
From (2.24), we have the existence of a subsequence (f (uν, , vν, )uν )ν such that
∗
f (uν, , vν, )uν * λ,
∞
θ
θ
in L∞ (0, T0 ; Lθ (Ω)).
(2.52)
θ
Since L (0, T0 ; L (Ω)) ,→ L (0, T0 ; L (Ω)), we have from (2.29) that
(f (um (t), vm (t))um (t))m , (g(um (t), vm (t))vm (t))m
are bounded in Lθ (0, T0 ; Lθ (Ω)); Thus we guarantee the existence of a subsequence,
denoted as above, such that
f (uν, , vν, )uν * λ,
in Lθ (0, T0 ; Lθ (Ω)).
(2.53)
12
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
EJDE-2006/130
Since
(u0m )m ,
is bounded in L∞ (0, T0 ; L2 (Ω)),
c
is bounded in L∞ (0, T0 ; W01,p (Ω))W01,p (Ω) ,→ L2 (Ω),
(um )m ,
we have by Aubin-Lions theorem, the existence of a subsequence (uν )ν such that
uν → u,
inL2 (0, T0 ; L2 (Ω)) ≡ L2 (QT0 )
uν → u,
a.e. in QT0
(2.54)
(2.55)
0
Since, the sequences (vm )m , (vm
)m satisfy the same conditions, it follows that, there
exists a subsequence (vν )ν such that
vν → v,
inL2 (0, T0 ; L2 (Ω)) ≡ L2 (QT0 )
vν → v,
a.e, inQT0
(2.56)
(2.57)
From (2.55), (2.57), and of the hypothesis on f, g, we have
f (uν, , vν, )uν → f (u, v)u,
a.e. in QT0 .
(2.58)
g(uν, , vν, )vν → g(u, v)v,
a.e. in QT0 .
(2.59)
From (2.27), we have
kf (um , vm )um kLθ (QT0 ) ≤ C,
∀m,
where Lθ (QT0 ) ≡ Lθ (0, T0 ; Lθ (Ω)). From this and (2.58), by means of Lion’s
Lemma, it follows that
f (uν, , vν, )uν * f (u, v)u, in Lθ (QT0 ),
np
. Therefore, from (2.53), we have λ = f (u, v)u and from (2.52).
for 1 ≤ θ ≤ 3(n−p)
This implies
∗
in L∞ (0, T0 ; Lθ (Ω)).
∗
in L∞ (0, T0 ; Lθ (Ω)).
f (uν, , vν, )uν * f (u, v)u,
(2.60)
Similarly,
g(uν, , vν, )vν * g(u, v)v,
The convergence in (2.60) implies
Z T0
Z
f (uν (t), vν (t))uν (t), w(x) ψ(t)dt →
0
T0
f (u(t), v(t))u(t), w(x) ψ(t)dt,
0
W01,p (Ω)
γ
1
⊂ L (Ω), for all ψ ∈ L (0, T0 ). In fact,
for all w ∈
Z T0
Z T0
f (uν (t), vν (t))uν (t), w(x) ϕ(t)dt →
f (u(t), v(t))u(t), w(x) ϕ(t)dt,
0
0
W01,p (Ω)
γ
for all w ∈ Vm ⊂
⊂ L (Ω), for all ϕ ∈ D(0, T0 ) ⊂ L1 (0, T0 ). Taking the
limit, as ν → ∞, in (2.50) and using the convergences obtained above, we have
Z T0
Z T0
Z T0
−
(u0 (t), w)ϕ0 dt +
hχ(t), wiϕdt +
hf (u(t), v(t))u(t), wiϕdt
0
0
0
(2.61)
Z T
0
=
(h1 (t), w)ϕdt,
0
∀w ∈ Vm , ϕ ∈ D(0, T0 ).
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13
Note that, with a similar reasoning for the approximate equation (2.12) we obtain
Z T0
Z T0
Z T0
hg(u(t), v(t))v(t), wiϕdt
hη(t), wiϕdt +
(v 0 (t), w)ϕ0 dt +
−
0
0
0
(2.62)
Z T
0
(h2 (t), w)ϕdt,
=
∀w ∈ Vm , ϕ ∈ D(0, T0 ).
0
Now, using the basis definition and the fact that Vm is dense in W01,p (Ω), expressions
(2.61) and (2.62) take the form
Z T0
Z T0
Z T0
hf (u(t), v(t))u(t), wiϕdt
< χ(t), w > ϕdt +
(u0 (t), w)ϕ0 dt +
−
0
0
0
T0
Z
(h1 (t), w)ϕdt,
=
∀w ∈ W01,p (Ω), ϕ ∈ D(0, T0 ),
0
(2.63)
and
T0
Z
−
(v 0 (t), w)ϕ0 dt +
0
Z
Z
T0
Z
0
T0
=
(h2 (t), w)ϕdt,
T0
hη(t), wiϕdt +
hg(u(t), v(t))v(t), whϕdt
0
(2.64)
∀w ∈ W01,p (Ω), ϕ ∈ D(0, T0 ).
0
Note that, the mappings t 7→ (u0 (t), w), t 7→ (v 0 (t), w) being functions in L∞ (0, T0 ),
they define distributions on (0, T0 ). Therefore, the first integrals of (2.63), (2.64)
are the derivative of these distributions. Thus, from (2.63) we have
Z T0
d 0
(u (t), w) + hχ(t), wi + hf (u(t), v(t))u(t), wi − (h1 (t), w) ϕdt = 0
dt
0
for all w ∈ W01,p (Ω) and all ϕ ∈ D(0, T0 ). Thus,
d 0
(u (t), w) + hχ(t), wi + hf (u(t), v(t))u(t), wi = (h1 (t), w),
dt
for all w ∈ W01,p (Ω), in D0 (0, T0 ). Similarly,
d 0
(v (t), w) + hη(t), wi + hg(u(t), v(t))v(t), wi = (h2 (t), w),
dt
for all w ∈ W01,p (Ω), in D0 (0, T0 ).
If one shows that Au(t) = χ(t) and Av(t) = η(t), the proof of the theorem will be
complete; since the verification of the initial conditions can be done in a standard
way.
The monotonocity of A implies that
Z T0
hAuν (t) − Aw, uν − widt ≥ 0, ∀w ∈ W01,p (Ω);
0
that is,
Z
0≤
T0
Z
hAuν (t), uν idt −
0
T0
Z
0
hAw, uν (t) − widt,
0
for all w ∈ W01,p (Ω).
Z T0
Z
hAuν (t), uν idt −
0 ≤ lim sup
0
T0
hAuν (t), widt −
0
T0
Z
hχ(t), widt −
T0
hAw, u(t) − widt,
0
14
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
EJDE-2006/130
for all w ∈ W01,p (Ω). Considering the approximate equation (2.11) with m = ν and
w = uν (t) we have
(u00ν (t), uν (t)) + hAuν (t), uν (t)i + hf (uν , vν )uν , uν i = (h1 (t), uν (t)).
Therefore,
d 0
(u (t), uν (t)) − |u0ν (t)|2 + hAuν (t), uν (t)i + hf (uν , vν )uν , uν i = (h1 (t), uν )
dt ν
Integrating from 0 the T0 we have
Z T0
Z T0
hAuν (t), uν (t)idt = (u0ν (0), uν (0)) − (u0ν (T0 ), uν (T0 )) +
|u0ν (t)|2 dt
0
0
(2.65)
Z T
Z T
0
−
0
hf (uν , vν )uν , uν idt +
0
(h1 (t), uν )dt
0
Recall that W01,p (Ω) ,→ L2 (Ω). Since uν (0) * u(0) in W01,p (Ω) it implies
uν (0) → u(0)inL2 (Ω). Since u0ν (0) * u0 (0) in L2 (Ω), it implies
(u0ν (0), uν (0)) → (u0 (0), u(0))
in R
(2.66)
W01,p (Ω)
Recall that (um (T0 ))m is bounded in
and (u0m (T0 ))m is
2
L (Ω). Thus, there exists subsequences (uν (T0 ))ν and (u0ν (T0 ))ν such
uν (T0 ) * u(T0 )
bounded in
that
c
in W01,p (Ω) ,→ L2 (Ω),
which implies
uν (T0 ) → u(T0 ), inL2 (Ω),
u0ν (T0 ) * u0 (T0 )inL2 (Ω)
Consequently,
(u0ν (0), uν (T0 )) → (u0 (T0 ), u(T0 ))
We have that
(u0m )
∞
in R.
(2.67)
2
bounded in L (0, T0 ; L (Ω)). Since
L∞ (0, T0 ; L2 (Ω)) ,→ L2 (0, T0 ; L2 (Ω)),
it follows that (u0m ) is bounded in L2 (0, T0 ; L2 (Ω)). We also have that (u00m ) is
bounded in L2 (0, T0 ; H −s (Ω)). Therefore, by the Aubin-Lions Theorem, there exists a subsequence (u0ν ) such that
u0ν → u0
in L2 (0, T0 ; L2 (Ω)) ≡ L2 (QT0 ).
Hence
T0
Z
0
|u0ν (t)|2 dt
Z
→
T0
|u0 (t)|2 dt
(2.68)
0
Note that
hf (um (t), vm (t))um (t), um (t)iLθ ,Lγ = hf (um (t), vm (t))u2m (t), 1iLθ ,Lγ .
From (2.68) we have u2ν → u2 a.e. in QT0 . Similarly
Z T0
Z T0
0
2
|vν (t)| dt →
|v 0 (t)|2 dt
0
hence, we have
vν2
0
2
→ v a.e. in QT0 , From (2.31), we have
kf (uν , vν )u2ν kLθ (0,T0 ;Lθ (Ω))≡Lθ (QT0 ) ≤ C,
∀m.
(2.69)
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WEAK SOLUTIONS
15
From this inequality and (2.44), we guarantee the existence of a subsequence such
that
∗
in L∞ (0, T0 ; Lθ (Ω))
f (uν , vν )u2ν * σ
f (uν , vν )u2ν *σ
θ
(2.70)
θ
in L (0, T0 ; L (Ω))
(2.71)
Thus, from (2.55), (2.57) and the hypotheses on f, g, we have that
f (uν , vν )u2ν → f (u, v)u2
g(uν , vν )u2ν
→ g(u, v)u
2
a.e. in QT0 ,
(2.72)
a.e in QT0
(2.73)
From (2.69), (2.72) and the Lions’ Lemma it follows that
f (uν , vν )u2ν * f (u, v)u2 inLθ (QT0 ) ≡ Lθ (0, T0 ; Lθ (Ω)),
for 1 ≤ θ ≤
np
3(n − p)
From this convergence and (2.71), we have σ = f (u, v)u2 and from (2.70),
∗
in L∞ (0, T0 ; Lθ (Ω)).
f (uν , vν )u2ν * f (u, v)u2
(2.74)
Similarly,
∗
g(uν , vν )vν2 * g(u, v)u2 inL∞ (0, T0 ; Lθ (Ω)).
The convergence (2.74) implies
hf (uν , vν )u2ν , ψi → hf (u, v)u2 , ψi,
∀ψ ∈ L1 (0, T0 ; Lγ (Ω))
or better
Z
T0
hf (uν , vν )u2ν , w(x)iϕ(t)dt
T0
Z
hf (u, v)u2 , w(x)iϕ(t)dt,
→
0
γ
0
for all w ∈ L (Ω) and all ϕ ∈ L1 (0, T0 ). When fixing w ≡ 1 and ϕ ≡ 1, we have
Z T0
Z T0
hf (uν (t), vν (t))uν (t), uν (t)idt =
hf (uν (t), vν (t))u2ν (t), 1idt
0
0
which approaches
Z T0
Z
2
hf (u(t), v(t))u (t), 1idt =
0
hence
Z
T0
hf (u(t), v(t))u(t), u(t)idt.
0
T0
Z
hf (uν (t), vν (t))uν (t), uν (t)idt →
0
T0
hf (u(t), v(t))u(t), u(t)idt,
(2.75)
0
as ν → ∞. Therefore, taking the limit in (2.65), using the convergence (2.66),
(2.67), (2.68) and (2.75), as ν → +∞, we have
Z T0
Z T0
0
0
lim sup
hAuν (t), uν (t)idt = (u (0), u(0)) − (u (T0 ), u(T0 )) +
|u0 (t)|2 dt
0
0
Z
−
T0
Z
T0
hf (u(t), v(t))u(t), u(t)idt +
0
(h1 (t), u(t))dt
0
From this equality and (2.75), we have
0 ≤ (u0 (0), u(0)) − (u0 (T0 ) − u(T0 )) +
Z
T0
|u0 (t)2 |dt −
0
Z
−
T0
Z
hχ(t), widt −
0
T0
T0
hf (u, v)u, uidt
0
Z
hAw, u(t) − widt +
0
Z
T0
(h1 (t), u(t))dt,
0
(2.76)
16
O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
EJDE-2006/130
for all w ∈ W01,p (Ω). From the approximate equation (2.11), we have
(u00ν (t), w) + hAuν (t), wi + hf (uν (t), vν (t))uν (t), wi = (h1 (t), w),
∀w ∈ Vm , ν ≥ m.
1
Now, let ϕ ∈ C ([0, T0 ]). Then
Z T0
Z T0
Z
(u00ν (t), w)ϕ +
hAuν (t), wiϕ +
0
0
T0
hf (uν (t), vν (t))uν (t), wiϕ
0
T0
Z
=
(h1 (t), w),
0
for all w ∈ Vm and all ν ≥ m. Setting
(u0ν (t), w)ϕ(T0 ) − (u0ν (0), w)ϕ(0) −
Z
T0
(u0ν (t), w)ϕ0 dt
0
T0
Z
T0
Z
hAuν (t), wiϕdt +
+
hf (uν (t), vν (t))uν (t), wiϕ(t)dt
0
0
T0
Z
=
∀w ∈ Vm , ϕ ∈ C 1 ([0, T0 ]), ν ≥ m.
(h1 (t), w)ϕ(t)dt,
0
Taking into account the previous convergence statements, it follows that
Z T0
(u0 (T0 ), w)ϕ(T0 ) − (u0 (0), w)ϕ(0) −
(u0 (t), w)ϕ0 dt
0
T0
Z
Z
hχ(t), wiϕdt +
+
0
T0
hf (u(t), v(t))u(t), wiϕ(t)dt
0
T0
Z
=
(h1 (t), w)ϕ(t)dt,
∀w ∈ Vm , ϕ ∈ C 1 ([0, T0 ])
0
Using a basis argument and the fact that Vm is dense in W01,p (Ω), it follows that
Z T0
(u0 (T0 ), w)ϕ(T0 ) − (u0 (0), w)ϕ(0) −
(u0 (t), w)ϕ0 dt
0
T0
Z
Z
hχ(t), wiϕdt +
+
0
Z
=
T0
hf (u(t), v(t))u(t), wiϕ(t)dt
(2.77)
0
T0
(h1 (t), w)ϕ(t)dt,
∀w ∈ W01,p (Ω), ϕ ∈ C 1 ([0, T0 ]).
0
Observing that the set of the linear combinations of the type wϕ, with w ∈ W01,p (Ω)
and ϕ ∈ C 1 ([0, T0 ]), is dense in the space
V = {v ∈ L2 (0, T0 ; W01,p (Ω)), v 0 ∈ L2 (0, T0 ; L2 (Ω))}.
It follows that (2.77) is true in the space V .
Using the fact that,
u ∈ L∞ (0, T0 ; W01,p (Ω)) ,→ L2 (0, T0 ; W01,p (Ω)),
u0 ∈ L∞ (0, T0 ; L2 (Ω)) ,→ L2 (0, T0 ; L2 (Ω)),
we obtain that u ∈ V . So (2.77) takes the form
(u0 (T0 ), w)ϕ(T0 ) − (u0 (0), w)ϕ(0)
Z T0
Z T0
Z
0
0
−
(u (t), u (t))dt +
hχ(t), u(t)idt +
0
0
0
T0
hf (u, v)u, uidt
EJDE-2006/130
WEAK SOLUTIONS
Z
17
T0
(h1 (t), u(t)dt
=
0
Substituting this expression in (2.76), it follows that
Z T0
Z T0
0≤
hχ(t), u(t) − widt −
hAw, u(t) − widt,
0
∀w ∈ W01,p (Ω).
0
Let us take w = u(t) + λv(t), λ > 0. Thus
Z T0
Z T0
0≤−
hχ(t), λv(t)idt +
hAu(t) + λv(t), λv(t)idt, ∀w ∈ W01,p (Ω)
0
0
which implies
T0
Z
Z
T0
hA(u(t) + λv(t)), λv(t)idt.
hχ(t), λv(t)idt +
0≤−
0
0
Dividing the previous inequality by λ and letting λ → 0+ , by the hemicontinuity
of A, we have
Z T0
Z T0
0≤−
hχ(t), v(t)idt +
hA(u(t)), v(t)idt, ∀v ∈ W01,p (Ω).
0
0
Hence
Z
T0
hAu(t) − χ(t), v(t)idt,
∀v ∈ W01,p (Ω).
Now, for λ < 0 it follows that
Z T0
hAu(t) − χ(t), v(t)idt ≤ 0,
∀v ∈ W01,p (Ω).
0≤
0
0
Therefore,
Z
0≤
T0
hAu(t) − χ(t), v(t)idt ≤ 0,
∀v ∈ W01,p (Ω).
0
Thus Au(t) = χ(t). Similarly, Av(t) = η(t). This completes the proof of the
theorem.
References
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Comportamento Assintótico. Phd Thesis, IM-UFRJ, Rio-Brazil (1988).
[2] Brezis, H; Analyse Fonctionelle-theorie et Applications. Masson, Paris (1983).
[3] Castro, N. N; Existence and asymptotic behavior of solutions for a nonlinear evolution problem. Appl. of Mathematics, 42(1997), No. 6, 411-420.
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Seminário Brasileiro de Análise (2000), 445-451.
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2(2000), 207-219.
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of Pure and Applied Mathematics. Vol. 20 No. 1 (2005), 81-95.
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Dunod, Paris (1969).
[8] Medeiros, L.A., Menzala, G.P; On a mixed problem for a class of nonlinear Klein-Gordon
equation. Acta Math Hung., 52, N o.1 − 2(1988), 61 − 69.
[9] Medeiros, L.A., Menzala, G.P; On the existence of global solutions for a coupled nonlinear
Klein-Gordon equations. Funkcialaj Ekvacioj, 30 (1987), 147-161.
[10] Medeiros, L.A., Miranda, M.M; Weak Solutions for the system of nonlinear Klein-Gordon
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O. A. LIMA, A. T. LOURÊDO, A. O. MARINHO
EJDE-2006/130
[11] Segal, I; Nonlinear partial differential equations in quantum field theory. Proc. Symb. Appl.
Math. A.M.S., (1965), 210-226.
Osmundo A. Lima
Universidade Estadual da Paraı́ba, DME, Campina Grande - PB, Brazil
E-mail address: osmundo@hs24.com.br
Aldo T. Lourêdo
Universidade Estadual da Paraı́ba, DME, Campina Grande - PB, Brazil
E-mail address: aldotl@bol.com.br
Alexandro O. Marinho
Universidade Federal da Paraı́ba, DM, João Pessoa - PB, Brazil
E-mail address: nagasak@ig.com.br