Electronic Journal of Differential Equations, Vol. 2002(2002), No. 22, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Existence of solutions for a variational unilateral
system ∗
Marcondes R. Clark & Osmundo A. Lima
Abstract
In this work the authors study the existence of weak solutions of the
nonlinear unilateral mixed problem associated to the inequalities
utt − M (|∇u|2 )∆u + θ ≥ f,
θt − ∆θ + ut ≥ g,
where f , g, M are given real-valued functions with M positive.
1
Introduction
Let Ω be a bounded and open set of Rn , with smooth boundary Γ = ∂Ω, and
let T be a positive real number. Let Q = Ω×]0, T [ be the cylinder with lateral
boundary Σ = Γ×]0, T [.
We study the variational nonlinear system
utt − M (|∇u|2 )∆u + θ ≥ f in Q,
θt − ∆θ + ut ≥ g in Q,
u = θ = 0 in Σ
u(0) = u0 , u0 (0) = u1 , θ(0) = θ0 .
(1.1)
(1.2)
(1.3)
(1.4)
The above system with M (s) = m0 + m1 s (m0 and m1 positive constants) and
θ = 0 is a nonlinear perturbation of the canonical Kirchhof model
Z
utt − m0 + m1
|∇u2 dx ∆u = f .
(1.5)
Ω
This model describes small vibrations of a stretched string when only the transverse component of the tension is considered, see for example, Arosio & Spagnolo
[1], Pohozaev [12].
∗ Mathematics Subject Classifications: 35L85, 49A29.
Key words: weak solutions, variational unilateral nonlinear problem, Galerkin method,
penalization method.
c
2001
Southwest Texas State University.
Submitted November 17, 2000. Published February 21, 2002.
M. R. Clark was a visiting professor of State University of Piauı́
1
2
Existence of solutions of a variational unilateral system
EJDE–2002/22
Several authors have studied (1.5). For Ω bounded, we can cite: D’ancona
& Spagnolo [5], Medeiros & Milla Miranda [9], Hosoya & Yamada [7], Lions [8],
Medeiros [10], and Matos [9]. For Ω unbounded, we can cite Bisiguin [2], Clark
& Lima [4], and Matos [9]. The system (1.1)–(1.4) was studied also in the case
when (1.1) and (1.2) are equations, see for example [3].
In the present work we show the existence of a weak solution for the variational nonlinear system (1.1)–(1.4), under appropriate assumptions on M , f and
g. We employ Galerkin’s approximation method and the penalization method
used by Frota & Lar’kin [6].
2
Notation and main result
We represent the Sobolev space of order m on Ω by
W m,p (Ω) = {u ∈ Lp (Ω); Dα u ∈ Lp (Ω), ∀ |α| ≤ m}
and its associated norm by
X
1/p
kukm,p =
|Dα u|pLp (Ω)
,
u ∈ W m,p (Ω),
1 ≤ p < ∞.
|α|≤m
When p = 2, we have the usual Sobolev space H m (Ω). Let D(Ω) be the space
of the test functions on Ω, and let W0m,p (Ω) be the closure of D(Ω) in W m,p (Ω).
When p = 2, we have W02,p (Ω) = H0m (Ω). The dual space of W0m,p (Ω) is denoted
0
by W −m,p (Ω), with p0 such that p1 + p10 = 1. For the rest of this paper we use
the symbol (·, ·) to indicate the inner product in L2 (Ω), and ((·, ·)) to indicate
the inner product in H01 (Ω).
Let K = {ψ ∈ W02,4 (Ω); |∆ψ| ≤ 1 and ψ ≥ 0 a. e. in Ω }. Then we have the
following proposition whose proof can be found in [6]
Proposition 2.1 The set K is a closed and connected in W02,4 (Ω).
Definition Let V be a Banach space and V 0 its dual. An operator β from V
to V 0 is called hemicontinous if the function
λ → hβ(u + λv), wi
is continuous for all λ ∈ R. The operator β is called monotone if
hβ(u) − β(v), u − vi ≥ 0,
∀u, v ∈ V.
We consider the penalization operator β : W02,4 (Ω) → W −2,4/3 (Ω) such that
β(z) = β1 (z) + β2 (z), z ∈ W02,4 (Ω), where β1 (z) and β2 (z) are defined by
Z
hβ1 (z), vi = −
z − (x)v(x)dx,
Ω
Z
hβ2 (z), vi = − (1 − |∆z(x)|2 )− ∆z(x)∆v(x)dx
Ω
for all v in
W02,4 (Ω).
EJDE–2002/22
M. R. Clark & Osmundo A. Lima
3
Proposition 2.2 The operator β defined above satisfies the following coditions:
i) β is monotone and hemicontinous
ii) β is bounded; this is, β(S) is bounded in W 2,4/3 (Ω) for all bounded set S
in W02,4 (Ω).
iii) β(u) = 0 if only if u belongs to K.
The proof of this proposition can be found in [6].
In this article, we assume the following hypotheses:
R∞
A1) M ∈ C 1 [0, ∞), M (s) ≥ 0 for s ≥ 0, and 0 M (s)ds = ∞
A2) f, g belong to H 1 (0, T ; L2 (Ω).
The main result of this paper is stated as follows.
Theorem 2.3 Assume A1) and A2). For u0 ∈ H01 (Ω) ∩ H 2 (Ω), u1 , θ0 in the
interior of K, there exist functions u, θ : Q → R such that
u ∈ L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω))
u0 ∈ L1 (0, T ;
∞
θ ∈ L (0, T ;
(2.1)
W02,4 (Ω))
00
∞
and u0 (t) ∈ K a.e. in [0, T ]
u ∈ L (0, T ; L (Ω))
(2.3)
H01 (Ω))
and θ(t) ∈ K a.e. in [0, T ] .
(2.4)
2
(2.2)
Also
(u00 (t) − M (ku(t)k2 )∆u(t) + θ(t) − f (t), v − u0 (t) ≥ 0, ∀v ∈ K a.e. in [0, T ]
(2.5)
(θ0 (t) − ∆θ(t) + u0 (t) − g(t), v − θ(t)) ≥ 0 ∀v ∈ K a.e. in [0, T ]
u(0) = u0 , u0 (0) = u1 , θ(0) = θ0 .
(2.6)
(2.7)
To obtain the solution {u, θ} of problem (2.1)–(2.4) in Theorem 2.3, we
consider the following associated penalized problem. For 0 < ε < 1, consider
1
u00ε (t) − M (kuε (t)k2 )∆uε (t) + θε (t) + β(u0ε (t)) = f (t) in Q
ε
1
0
0
θε (t) − ∆θε (t) + uε + β(θε (t)) = g(t) in Q
ε
uε (0) = u0ε , u0ε (0) = u1ε , θε (0) = θ0ε in Ω
(2.8)
(2.9)
(2.10)
Here β is a penalization operator, M , f , and g are as above. The solution
{uε , θε } of the penalized problem (2.8)–(2.10) are guaranteed by the following
theorem.
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Existence of solutions of a variational unilateral system
EJDE–2002/22
Theorem 2.4 Suppose the hypotheses of the Theorem 2.3 hold, and for
0 < ε < 1, then there exist functions {uε , θε } such that
uε , θε ∈ L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω))
(2.11)
u0ε ∈ L4 (0, T ; W02,4 (Ω))
u00ε ∈ L∞ (0, T ; L2 (Ω))
θε ∈ L4 (0, T ; W02,4 (Ω))
(2.12)
(2.13)
(2.14)
1
(u00ε (t), v) + M (kuε (t)k2 )((uε (t), v)) + (θε (t), v) + hβ(u0ε (t)), vi
ε
2,4
= (f (t), v) a.e. in [0, T ] for all v ∈ W0 (Ω),
(2.15)
1
(θε0 (t), v) + ((θε (t), v)) + (u0ε (t), v) + hβ(θε (t)), vi
ε
= (g(t), v)a.e. in [0, T ] for all v ∈ W02,4 (Ω),
(2.16)
uε (0) = u0ε , u0ε (0) = u1ε , θε (0) = θ0ε .
(2.17)
Proof We will use Galerkin’s method and a compactness argument.
First step (Approximated system) Let w1 , . . . , wm , . . . be an orthonormal base
of W02,4 (Ω) consisting of eigenfunctions of the Laplacian operator. Let
Vm = [w1 , . . . , wm ] the subspace of W02,4 (Ω), generated by the first m vectors
wj . We look for a pair of functions
uεm (t) =
m
X
gjm (t)wj ,
j=1
θεm (t) =
m
X
hjm (t)wj
in Vm
j=1
with gjm ∈ C 2 ([0, T ]) and hjm ∈ C 1 ([0, T ]), for all j = 1, . . . , m. Which are
solutions of the following system of ordinary differential equations
(u00εm (t), wj ) + M (kuεm (t)k2 )((uεm (t), wj )) + (θεm (t), wj )+
1
hβ(u0εm (t)), wj i = (f (t), wj ),
ε
0
(θεm
(t), wj ) + ((θεm (t), wj )) + (u0εm (t), wj )+
1
hβ(θεm (t)), wj i = (g(t), wj ),
ε
(2.18)
(2.19)
for j = 1, . . . , m, with the initial conditions: uεm (0) = u0εm , u0εm (0) = u1εm ,
θεm (0) = θ0εm , where
u0εm =
m
X
(u0ε , wj )wj → u0 strongly in H01 (Ω) ∩ H 2 (Ω),
j=1
u1εm =
θ0εm =
m
X
j=1
m
X
(u1ε , wj )wj → u1 strongly in H01 (Ω),
(θ0ε , wj )wj → θ0 strongly in W02,4 (Ω).
j=1
(2.20)
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M. R. Clark & Osmundo A. Lima
5
The system (2.18)–(2.20) contains 2m unknowns functions gjm (t), hjm (t);
j = 1, 2, . . . , m. By Caratheodory’s Theorem it follows that (2.18)–(2.20) has a
local solution {uεm (t), θεm (t)} on [0, tm [. In order to extend these local solution
to the interval [0, T [ and to take the limit in m, we must obtain some a priori
estimates.
Estimate (i) Note that finite linear combinations of the wj are dense in
W02,4 (Ω), then we can take w ∈ W02,4 (Ω) in (2.18) and (2.19) instead of wj .
Taking w = 2u0εm (t) in (2.18) and w = 2θεm (t) in (2.19) we obtain
2
d 0
d c
|uεm (t)|2 + M
(kuεm (t)k2 ) + hβ(u0εm (t)), u0εm (t)i
dt
dt
ε
= 2(f (t), u0εm (t)) − 2(θεm (t), u0εm (t)),
(2.21)
d
2
|θεm (t)|2 + kθεm (t)k2 + hβ(θεm (t)), θεm (t)i
dt
ε
= −2(u0εm (t), θεm (t)) + 2hg(t), θεm (t)i ,
(2.22)
Rλ
c(λ) =
where M
M (s)ds. Adding (2.21) and (2.22), and integrating from 0 to
0
t ≤ tm we have
Z kuεm (t)k2
Z t
0
2
2
|uεm (t)| + |θεm (t)| +
M (s)ds +
kθεm (s)k2 ds+
0
0
Z
Z
2 t
2 t
0
0
hβ(uεm (s)), uεm (s)ids +
hβ(θεm (s)), θεm (s)ids ≤
ε 0
ε 0
(2.23)
Z T
Z t
Z t
2
0
2
2
|f (t)| ds + 3
|uεm (s)| ds + 3
|θεm (s)| ds+
0
0
0
Z
T
|g(t)|2 dt + |θ0εm |2 + |u1εm |2 .
0
From (2.20) and hypothesis (A2) there exists a positive constant C, independently of ε > 0 and m such that
Z kuεm (t)k2
Z t
0
2
2
|uεm (t)| + |θεm (t)| +
M (s)ds +
kθεm (s)k2 ds+
0
0
Z
Z t
i
2h t
0
0
(2.24)
hβ(uεm (s)), uεm (s)ids +
hβ(θεm (s)), θεm (s)ids ≤
ε 0
0
Z t
Z t
0
2
C +3
|uεm (s)| ds + 3
|θεm (s)|2 ds.
0
0
Rt
hβ(u0εm (s)), u0εm (s)ids.
0
Next we analyze the sign of the term
−u0εm (t) ≤ u0εm (t)− . Then, by the definition of β, we have
Note that
hβ(u0εm (t)), u0εm (t)i =hβ1 (u0εm (t)), u0εm (t)i + hβ2 (u0εm (t)), u0εm (t)i
Z
= − (u0εm (x, t))− u0εm (x, t)dx+
Z Ω
(1 − |∆u0εm (t)|2 )− (∆u0εm (t))2 dx ≥ 0.
Ω
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Existence of solutions of a variational unilateral system
EJDE–2002/22
Similarly, we have,
hβ(θεm (t)), θεm (t)i ≥ 0 .
Because M (s) ≥ 0 for all s, from (2.24) and Gronwall’s inequality it follows that
|u0εm (t)|2 + |θεm (t)|2 ≤ C1 ,
∀ε, m, ∀t ∈ [0, tm [.
Returning to (2.24), we obtain
|u0εm (t)|2
2
+ |θεm (t)| +
Z
kuεm (t)k2
M (s)ds +
0
Z t
Z
t
kθεm (s)k2 ds+
0
(2.25)
Z
2 t
0
0
[ hβ(uεm (s)), uεm (s)ids +
hβ(θεm (s)), θεm (s)ids] ≤ C + 3C1 T .
ε 0
0
R∞
Since 0 M (s)ds = ∞, by (2.25) we can find C1 such that
kuεm (t)k2 ≤ C1 ,
∀ε, m, ∀t ∈ [0, tm [.
Thus there exists, other constant C = C(T ) independently of ε, m and t ∈ [0, tm [
such that
Z t
|u0εm (t)|2 + |θεm (t)|2 + kuεm (t)k2 +
kθεm (s)k2 ds+
0
(2.26)
Z
Z t
2
2 t
0
0
hβ(uεm (s)), uεm (s)ids +
hβ(θεm (s)), θεm (s)ids ≤ C
ε 0
ε 0
Estimate (ii) We will obtain a bound for |u00εm (0)|. For this, we note that u1
being in the interior of K and u1εm → u1 imply that u1εm is in the interior of
K, for m large. Therefore, |∆u1εm | ≤ 1 and u1εm ≥ 0 a. e. in Ω. Also we have
(u1εm )− = 0 and (1 − |∆u1εm |2 )− = 0 a. e. in Ω. Thus
hβ(u1εm ), u00εm (0)i = 0
(2.27)
Taking t = 0 and v = u00εm (0) in (2.14), and observing (2.27), we obtain
|u00εm (0)|2 + M (ku0εm k2 )((u0εm , u00εm (0))) + (θεm , u00εm (0)) = (f (0), u00εm (0))
which implies
|u00εm (0)|2 ≤ |f (0)||u00εm (0)| + M (ku0εm k2 )|∆u0εm ||u00εm (0)| + |θ0εm ||u00εm (0)|.
From u0εm → u0 in H01 (Ω) ∩ H 2 (Ω), θ0εm → θ0 in H01 (Ω), M ∈ C 1 [0, ∞), and
f ∈ H 1 (0, T ; L2 (Ω), we obtain
|u00εm (0)| ≤ C,
(2.28)
with C independent of ε, m, and t ∈ [0, T [.
Rt
Estimate (iii) We obtain estimates for |∆u0εm (t)|, |∆θεm (t)|, 0 |u0εm (s)|3 ds,
Rt 0
and 0 |θεm
(s)|3 ds. For this, we need the following lemma whose proof can be
found in [6].
EJDE–2002/22
M. R. Clark & Osmundo A. Lima
7
Lemma 2.5 Let h : Ω → R be an arbitrary function. Then
h4 − 1 ≤ 2(1 − h2 )− h2 .
By this lemma, we have
(∆u0εm )4 − 1 ≤ 2[1 − (∆u0εm )2 ]− (∆u0εm )2 .
Therefore,
k∆u0εm k4L4 (Q) =
Z
T
Z
0
≤2
|∆u0εm (x, t)|4 dx dt
Ω
Z
T
Z
0
=2
Z
≤2
Z
(1 − ∆|u0εm (x, t)|2 )− (∆u0εm (x, t))2 dx dt + meas(Q)
Ω
T
hβ2 (∆u0εm (t)), u0εm (t)idx dt + meas(Q)
0
T
(β(u0εm (t)), u0εm (t))dt + meas(Q) .
0
Using (2.26), we obtain
k∆u0εm k4L4 (Q) ≤ Cε + meas(Q) < C + meas(Q)
(2.29)
with C independent of ε, m and t ∈ [0, T [. Analogously, using the Lemma 2.5
with h = ∆θεm and (2.26), we obtain
k∆θεm k4L4 (Q) ≤ C + meas(Q)
(2.30)
On the other hand, from (2.18) and (2.19), we obtain
1
1
hβ(u0εm (t)), vi + hβ(θεm (t)), vi ≤ C(|f (t)|) + |g(t)|)kvk+
ε
ε
M (kuεm (t)k2 )kuεm (t)k.kvk + C(|θεm (t)|) + |u0εm (t)|) ≤
|f (t)||v| + |g(t)||v| + M (kuεm (t)k2 )kuεm (t)k.kvk|u00εm (t)||v| + |θεm (t)||v|+
0
|θεm
(t)||v| + kθεm (t)k.kvk + |u0εm (t)||v| ≤
00
0
C{|f (t)| + |g(t)| + |uεm (t)| + |θεm (t)| + |θεm
(t)| + |u0εm (t)|}kvk+
(M (kuεm (t)k2 )kuεm (t)k + kθεm (t)k)kvk.
Since f, g ∈ C 0 ([0, T ]; L2 (Ω)), from the inequality above we obtain
1
|hβ(u0εm (t)), vi| ≤ C1 kvk ∀v ∈ W02,4 (Ω) ,
ε
1
|hβ(θεm (t)), vi| ≤ C1 kvk ∀v ∈ W02,4 (Ω),
ε
(2.31)
(2.32)
independent of ε, m and t ∈ [0, T ]; this is,
kβ(u0εm )kL∞ (0,T ;W 2,4/3 (Ω)) ≤ C1 ,
(2.33)
kβ(θεm )kL∞ (0,T ;W 2,4/3 (Ω)) ≤ C1 .
(2.34)
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Existence of solutions of a variational unilateral system
EJDE–2002/22
To estimate |∆uεm (t)|, we note that
Z t
d
|∆uεm (t)|2 =|∆u0εm |2 +
|∆uεm (s)|2 ds
0 ds
Z t
=|∆u0εm |2 + 2C
|∆uεm (s)|k∆u0εm (s)k
0
Z t
2
≤|∆u0εm | + C
(|∆uεm (s)|2 + k∆u0εm (s)k2 )ds ,
0
where C is the constant of the embedding from H01 (Ω) into L2 (Ω). From (2.20),
(2.29) and Gronwall’s inequality, we obtain
|∆uεm (t)|2 < C ,
(2.35)
where C is a constant independent of ε, m and t ∈ [0, T [.
Rt
Next, we obtain an estimate for 0 k∆u0εm (s)k3 ds. Let C represent various
positives constants of the embedding in the sequence
W02,4 (Ω) ,→ H02 (Ω) ,→ H01 (Ω) .
Observing that WH 2 (Ω) ≤ C|∆w| we obtain
Z
t
ku0εm (s)k3 ds
0
≤C
Z
t
ku0εm (s)k3H 2 (Ω) ds
≤C
0
Z
t
|∆u0εm (s)|3 ds,
(2.36)
0
independently of ε and m. It follows from Höder’s inequality that
Z t
Z T
Z t
|∆u0εm (s)|3 ds ≤ (
11 ds)1/4 (
k∆u0εm (s)k4 ds)3/4
0
0
0
and substituting in (2.36) and observing (2.29), we obtain
Z t
ku0εm (s)k3 ds ≤ C,
(2.37)
0
independent of ε, m and t ∈ [0, T [.
Estimate (iv) We will obtain the estimative for |u00εm (t)|. Let us consider the
functions
Ψh (t) = h1 [uεm (t + h) − uεm (t)] ,
Mh (t) = h1 [M (kuεm (t + h)k2 ) − M (kuεm (t)k2 )] ,
fh (t) = h1 [f (t + h) − f (t)].
Setting w = 2Ψ0h (t) in (1.14), we obtain
2(u00εm (t), Ψ0h (t)) + 2M (kuεm (t)k2 )((uεm (t), Ψ0h (t)))+
.
2
hβ(u0εm (t)), Ψ0h (t))i = 2(f (t), Ψ0h (t))
ε
(2.38)
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M. R. Clark & Osmundo A. Lima
9
Substituting t by t + h ∈ [0, T ] in (2.18) and taking w = 2Ψ0h (t), we set
2(u00εm (t + h), Ψ0h (t)) + 2M (kuεm (t + h)k2 )((uεm (t + h), Ψ0h (t)))+
2
hβ(u0εm (t + h)), Ψ0h (t))i = 2(f (t + h), Ψ0h (t)).
ε
(2.39)
Now, from (2.38) and (2.39) it follows, for h 6= 0, that
u00εm (t + h) − u00εm (t) 0
2
, Ψh (t)) + M (kuεm (t + h)k2 )((uεm (t + h), Ψ0h (t)))−
h
h
2
2
M (kuεm (t)k2 )((uεm (t), Ψ0h (t))) +
hβ(u0εm (t + h)) − β(u0εm (t)), Ψ0h (t)i =
h
hε
f (t + h) − f (t) 0
2(
, Ψh (t)),
h
2(
which implies
d 0
2
|Ψh (t)|2 + M (kuεm (t + h)k2 )(uεm (t + h), Ψ0h (t))−
dt
h
2
M (kuεm (t)k2 )((uεm (t), Ψ0h (t)))+
h
2
hβ(u0εm (t + h)) − β(u0εm (t)), Ψ0h (t)i = 2(fh (t), Ψ0h (t)).
hε
(2.40)
Nothing that
2
2
M (kuεm (t + h)k2 )((uεm (t + h), Ψ0h (t))) − M (kuεm (t)k2 )((uεm (t), Ψ0h (t))) =
h
h
2M (kuεm (t + h)k2 )
2
0
2M (kuεm (t + h)k )((Ψh (t), Ψh (t))) +
((uεm (t), Ψ0h (t)))−
h
2M (kuεm (t)k2 )
((uεm (t), Ψ0h (t))) =
h
d
M (kuεm (t + h)k2 ) (kΨh (t)k2 ) + 2Mh (t)((uεm (t), Ψ0h (t))).
dt
From (2.40) it follows that
d 0
d
|Ψh (t)|2 + M (kuεm (t + h)k2 ) (kΨh (t)k2 )+
dt
dt
2
hβ(u0εm (t + h)) − β(u0εm (t)), u0εm (t + h) − u0εm (t)i =
h2 ε
−2Mh (t)((uεm (t), Ψ0h (t))) + 2(fh (t), Ψ0h (t)).
By the monotonicity of the operator β, we obtain
d 0
d
|Ψ (t)|2 + M (kuεm (t + h)k2 ) (kΨh (t)k2 )
dt h
dt
≤ 2|Mh (t)(∆uεm (t), Ψ0h (t))| + 2|(fh (t), Ψ0h (t))|.
(2.41)
10
Existence of solutions of a variational unilateral system
EJDE–2002/22
Integrating (2.41) in t we have
Z
t
d
M (kuεm (s + h)k2 ) (kΨh (s)k2 )ds ≤
ds
0
Z t
Z t
|Ψ0h (0)|2 + 2
|Mh (s)(∆uεm (s), Ψ0h (s))|ds + 2
|(fh (s), Ψ0h (s))|ds .
|Ψ0h (t)|2 +
0
0
Taking the limit as h → 0, it follows
Z t
d
|u00εm (t)|2 +
M (kuεm (s)k2 ) ku0εm (s)k2 ds ≤
ds
0
Z t
00
2
0
2 d
|uεm (0)| + 2
[M (kuεm (s)k ) kuεm (s)k2 ]|∆uεm (s), u00εm (s)|ds+
ds
0
Z t
2
|(f 0 (s), u00εm (s))|ds .
(2.42)
0
Using Assumption (A2) and (2.28), we obtain, from (2.42),
Z t
d
|u00εm (t)|2 +
M (kuεm (s)k2 ) ku0εm (s)k2 ds ≤
ds
0
Z t
C +4
|M 0 (kuεm (s)k2 )|ku0εm (s)kkuεm (s)kk∆uεm (s)kku00εm (s)kds+
0
Z t
|u00εm (s)|2 ds.
(2.43)
0
From (2.26), (2.35) and (2.37) it follows that there exists a positive constant C
such that
Z t
kuεm (t)k2 + |∆uεm (t)|2 +
ku0εm (s)k2 ds ≤ C, ∀ε, m, t.
(2.44)
0
Since M ∈ C 1 ([0, ∞)), we also obtain from (2.44),
|M 0 (kuεm (s)k2 )| ≤ C,
∀ε, m, t.
(2.45)
On the other hand, using integration by parts, we get
Z t
d
M (kuεm (s)k2 ) ku0εm (s)k2 ds =
ds
0
M (kuεm (s)k2 )ku0εm (s)k2 − M (ku0εm (s)k2 )ku1εm (s)k2 −
Z t
d
M 0 (kuεm (s)k2 ) ku0εm (s)k2 ku0εm (s)k2 ds.
ds
0
Estimates (2.37), (2.44), and (2.45) together imply
Z
Z t
0
2 d
2
0
2
− M (kuεm (s)k ) kuεm (s)k kuεm (s)k ds ≥ −C
ku0εm (s)k3 ≥ −C,
ds
0
EJDE–2002/22
M. R. Clark & Osmundo A. Lima
independently of ε, m, and t. Therefore,
Z t
d
M (kuεm (s)k2 ) ku0εm (s)k2 ds ≥ M (kuεm (t)k2 )ku0εm (t)k2 − C,
ds
0
11
(2.46)
independently of ε, m and t. Here, C denote various positive constants. Making
use of inequalities (2.44)–(2.46) in (2.43) we obtain
|u00εm (t)|2 + M (kuεm (t)k2 )ku0εm (s)k2 ≤ C + C
Z
t
|u00εm (s)|2 ds,
(2.47)
0
independently of ε, m, and t. From (2.47) and using Gronwall’s inequality, we
have
|u00εm (t)|2 ≤ C,
(2.48)
independently of ε, m and t.
Passage to the limit
By estimates (2.26) and (2.35) we obtain
(uεm ) is bounded in L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)),
(u0εm ) is bounded in L∞ (0, T ; L2 (Ω)),
(θεm ) is bounded in L∞ (0, T ; L2 (Ω)).
Therefore, we can get subsequences, if necessary, denoted by (uεm ) and (θεm ),
such that
uεm → uε
u0εm
→
weak star in
u0ε
θεm → θε
L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)),
weak star in
weak star in
(2.49)
∞
2
(2.50)
∞
2
L (0, T ; L (Ω)).
(2.51)
L∞ (0, T ; L2 (Ω)).
(2.52)
L (0, T ; L (Ω)),
Similarly by (2.48), we obtain
u00εm → u00ε
weak star in
Also, by (2.33) and (2.34), there exist functions Xε , φε ∈ L4/3 (0, T ; W 2,4/3 (Ω))
such that
β(u0εm ) → Xε
in
β(θεm ) → φε
in
L4/3 (0, T ; W 2,4/3 (Ω)),
4/3
L
(0, T ; W
2,4/3
(Ω)).
(2.53)
(2.54)
It follows from the embeding W02,4 (Ω) into L4 (Ω) and of (2.29) that
|u0εm |4L4 (0,T ;W 2,4 (Ω)) ≤ Ck∆u0εm k4L4 (Ω) ≤ K.
0
Therefore, there exists a subsequence of (uεm ) such that
u0εm → u0ε
weak star in
L4 (0, T ; W02,4 (Ω)).
(2.55)
12
Existence of solutions of a variational unilateral system
EJDE–2002/22
Analogously, by (2.30) we obtain
weak star in L4 (0, T ; W02,4 (Ω)).
θεm → θε
(2.56)
Being the embedding from H01 (Ω) ∩ H 2 (Ω) into H01 (Ω) compact, we can set a
subsequence, again denoted by (uεm ), such that:
uεm → uε
L2 (0, T ; H01 (Ω)).
strong in
(2.57)
By assumption (A1) we obtain
M (kuεm (t)k2 ) → M (kuε (t)k2 ).
(2.58)
From the compactness of the embedding H01 (Ω) ,→ L2 (Ω) we obtain
u0εm → u0ε
L2 (0, T ; L2 (Ω)).
strong in
(2.59)
Then taking limit in the system (2.18)–(2.20), when m → ∞, with w = vϕ(t),
v ∈ W02,4 (Ω), ϕ(t) ∈ D(0, T ) instead of wj , and using the fact that β is monotone
and hemicontinous operator, we obtain that {uε , θε } is a weak solution of the
system (2.18)–(2.20).
The initial conditions (2.19) can be obtained by observing the convergence
above and the definition of weak solution; this is,
u0ε (0) = lim u0εm = lim
m→∞
m→∞
u0ε (0) = lim u1εm = lim
m→∞
φε (0) = lim θ0εm = lim
m→∞
m
X
(u0ε , wj )wj = u0 ,
j=1
m
X
(u1ε , wj )wj = u1 ,
j=1
m
X
m→∞
(θ0ε , wj )wj = θ0 .
j=1
This concludes the proof of Theorem 2.4
3
Main Result
In this section, we will prove the Theorem 2.3. By Theorem 2.4, there exists
functions uε , θε : Q → R such that
uε ∈ L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)),
u0ε , θε ∈ L4 (0, T ; W02,4 (Ω)),
u00ε ∈ L∞ (0, T ; L2 (Ω)),
θε ∈ L∞ (0, T ; L2 (Ω)),
EJDE–2002/22
M. R. Clark & Osmundo A. Lima
13
satisfying the system
1
(u00ε (t), w) + M [kuε (t)k2 ]((uε (t), w)) + hβ(u0ε (t), w)i = (g(t), w),
ε
1
0
(θε (t), w) + ((θε (t), w)) + (uε (t), w) + hβ(θε (t), w)i = (g(t), w),
ε
a.e. in [0, T ], for all w ∈ W02,4 (Ω). uε (0) = u0 ; u0ε (0) = u1 , and θε (0) = θ0 .
Being the estimates (2.26), (2.29), (2.30), (2.33), (2.34), (2.32) and (2.44)
independently of ε, m and t we obtain by Uniform Boundedness Theorem that
there exists a positive constant C such that
Z T
0
2
2
2
|uε (t)| + |θε (t)| + kuε (t)k +
kθε (t)k2 ds+
0
2
ε
T
Z
0
2
hβ(u0ε (s), u0ε (s))ids +
ε
T
Z
hβ(θε (s)), θε (s))ids ≤
0
Ck∆u0ε k4L4 (Q) ≤C ,
and
k∆θε k4L4 (Q) ≤ C ,
kβ(u0ε )k
kβ(θε )kL4/3 (0,T ;W 2,4/3 (Ω)) ≤ C ,
≤C,
4
L 3 (0,T ;W 2,4/3 (Ω))
|∆uε (t)|2 ≤ C , |u00ε (t)|2
≤ C.
Consequently, we can find a subnet, which we still represent by (uε ), (θε ) such
that
uε → u
weak star in L∞ (0, T ; H01 (Ω) ∩ H 2 (Ω)),
u0ε → u0
weak star in
L∞ (0, T ; L2 (Ω)),
u00ε → u00
weak star in
L∞ (0, T ; L2 (Ω)),
β(u0ε ) → β(u0 )
weak in
L4/3 (0, T ; W −2, 3 (Ω)),
β(θε ) → β(θ)
weak in
L4/3 (0, T ; W −2, 3 (Ω)),
4
4
u0ε → u0
weak in
L4 (0, T ; W02,4 (Ω)),
θε → θ
weak in
L4 (0, T ; W02,4 (Ω)).
By the compactness theorem of Aubin-Lions [8], we obtain
uε → u
u0ε
→u
0
strongly
strongly
L2 (0, T ; H01 (Ω)),
2
2
L (0, T ; L (Ω)).
(3.1)
(3.2)
We observe that
(u00ε (t), v(t)) + M (kuε (t)k2 )((uε (t), v(t))) + (θε (t), v(t))+
1
hβ(u0ε (t)), v(t)i = (f (t), v(t)) ,
ε
1
(θε0 (t), v(t)) + ((θε (t), v(t))) + (u0ε (t), v(t)) + hβ(θε (t)), v(t)i = (g(t), v(t)) .
ε
14
Existence of solutions of a variational unilateral system
EJDE–2002/22
is true for all v ∈ L4 (0, T ; W02,4 (Ω)).
On the other hand, being u0ε , θε ∈ L4 (0, T ; W02,4 (Ω)) implies
(u00ε (t), u0ε (t)) + M (kuε (t)k2 )((uε (t), u0ε (t))) + (θε (t), u0ε (t))+
1
hβ(u0ε (t)), u0ε (t)i = (f (t), u0ε (t)) ,
ε
1
(θε0 (t), θε (t)) + ((θε (t), θε (t))) + (u0ε (t), θε (t)) + hβ(θε (t)), θε (t)i = (g(t), θε (t)).
ε
Subtracting the equations of the system above, we obtain
(u00ε (t), v(t) − u0ε (t)) + M (kuε (t)k2 )((uε (t), v(t) − uε (t)))+
(3.3)
1
(θε (t), v − u0ε (t)) + hβ(u0ε (t)), v(t) − u0ε (t)i =(f (t), v(t) − u0ε (t)) ,
ε
(θε0 (t), v(t) − θε (t)) + ((θε (t), v(t) − θε (t)))+
(3.4)
1
(u0ε (t), v(t) − θε (t)) + hβ(θε (t)), v(t) − θε (t)i = (g(t), v(t) − θε (t)),
ε
for all v ∈ W02,4 (Ω).
Let us consider v(t) ∈ K a. e. in [0, T ]. Then we obtain β(v(t)) = 0 and
being β a monotone operator, we have
hβ(u0ε (t)) − β(v(t)), v(t) − u0ε (t)i ≤ 0 ,
hβ(θε (t)) − β(v(t)), v(t) − θε (t)i ≤ 0 .
Therefore,
Z T
(u00ε (t) − M (kuε (t)k2 )∆uε (t) + θε (t) − f (t), v(t) − u0ε (t))dt ≥ 0,
(3.5)
0
Z
T
(θε (t) − ∆θε + u0ε − g(t), v(t) − θε (t))dt ≥ 0,
(3.6)
0
for all v ∈ L4 (0, T ; W02,4 (Ω)) with v(t) ∈ K a.e. in [0, T ]. Now, taking the
limit in (3.6) and (3.7), when ε → 0 and using (3.1)–(3.3) and observing that
∆uε → ∆u weak in L2 (0, T ; L2 (Ω)) it follows that u, θ satisfy (1.5) and (1.6) in
Theorem 2.3.
To conclude the proof of the existence of a solution, we show that
u0 (t), θ(t) ∈ K a.e. in [0, T ]. In fact, by (2.33) and (2.34) we have
kβ(u0ε )k
kβ(θε )k
4
≤ Cε,
4
≤ Cε.
L∞ (0,T ;W 2, 3 (Ω))
L∞ (0,T ;W 2, 3 (Ω))
4
Therefore, as ε → 0, β(u0ε ) → 0 and β(θε ) → 0 strong L∞ (0, T ; W 2, 3 (Ω)).
On the other hand we have β(u0ε ) → β(u0 ) and β(θε ) → β(θ) weak in
4/3
L (0, T ; W 2,4/3 (Ω)). Then, β(u0 (t)) = β(θ(t)) = 0 in L∞ (0, T ; W 2,4/3 (Ω)).
Therefore, u0 (t), θ(t) ∈ K a.e. in [0, T ].
The initial conditions (1.7) can be verified easily. This concludes the proof
of Theorem 2.3.
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4
M. R. Clark & Osmundo A. Lima
15
Uniqueness
For proving uniqueness of solutions in Theorem 2.3, we consider the restriction
u0 ∈ H01 (Ω) ∩ H 2 (Ω),
u0 (x) ≥ 0 a. e. in Ω, and ku0 k > 0.
Consequently ku(t)k > 0, for all t ∈ [0, T ]. In fact, if there exists t0 ∈ [0, T ]
such that ku0 k = 0, then
Z
|u(x, t0 )|2 dx ≤ Cku(t0 )k2 = 0,
Ω
where C is the constant of the embedding H01 (Ω) ,→ L2 (Ω). Therefore,
u(x, t0 ) = 0, a.e. in Ω.
Since u0 (t) ∈ K a.e. in [0, T ], we have u0 (t) ≥ 0 a.e. in Ω. This implies that
u(x, t) ≥ u(x, 0) = u0 (x) in Ω a.e. in [0, T ].
(4.1)
Being ku0 k > 0, there exists Ω0 ⊂ Ω with kΩ0 k > 0 such that that u0 (x) > 0.
By (3.1) it follows that u(x, t0 ) > 0 in Ω. This is a contradiction.
Theorem 4.1 Under the hypotheses of Theorem 2.3, if
i) M (λ) > 0 for all λ > 0, and M (0) = 0.
ii) u0 ∈ H01 (Ω) ∩ H 2 (Ω), u0 (x) ≥ 0 a.e. in Ω, and ku0 k > 0,
Then the solution {u, θ} of Theorem 2.3 is unique.
Proof.
From (i) and (ii) it follows that
m0 = min{M (ku(t)k2 ); t ∈ [0, T ]} > 0.
Suppose we have two pairs of solutions {u, θ} and {w, ϕ} satisfying the conditions of Theorem 2.3. Let Ψ = u − w and φ = θ − ϕ. Thus, Ψ and φ satisfy
(Ψ00 (t) − M (ku(t)k2 )∆Ψ(t) + {M (kw(t)k2 ) − M (ku(t)k2 )}∆w + φ(t), Ψ0 (t)) ≤ 0,
(φ0 (t) − ∆φ(t) + Ψ0 (t), φ(t)) ≤ 0,
which implies
1 d
1 d
{|Ψ0 (t)|2 + |φ(t)|2 + kφ(t)k2 } + M (ku(t)k2 )
kΨ(t)k2 + 2(φ(t), Ψ0 (t)) ≤
2 dt
2 dt
{M (ku(t)k2 ) − M (kw(t)k2 )}(∆w(t), Ψ0 (t)) .
Since
M (ku(t)k2 )
d
d
d
kΨ(t)k2 = {M (ku(t)k2 )kΨ(t)k2 } − [M (ku(t)k2 )]kΨ(t)k2
dt
dt
dt
16
Existence of solutions of a variational unilateral system
EJDE–2002/22
we obtain
1 d
d
{|Ψ0 (t)|2 + |φ(t)|2 + kφ(t)k2 } + {M (ku(t)k2 )kΨ(t)k2 } + 2(φ(t), Ψ0 (t)) ≤
2 dt
dt
{M (ku(t)k2 ) − M (kw(t)k2 )}(∆w(t), Ψ0 (t))+
M 0 (ku(t)k2 )((u0 (t), u(t))kΨ(t)k2 .
Now, integrating this inequality form 0 to t < T , we obtain
1
{|Ψ0 (t)|2 + |φ(t)|2 + kφ(t)k2 } + M (ku(t)k2 )kΨ(t)k2 +
2
Z t
2
(φ(t), Ψ0 (t))ds ≤
0
Z t
{M (ku(s)k2 ) − M (kw(s)k2 )}(∆w(s), Ψ0 (s))ds+
0
Z t
M 0 (ku(s)k2 )((u0 (s), u(s))kΨ(s)k2 ds.
(4.2)
o
Note that ku(t)k and ku0 (t)k ∈ L∞ (0, T ). Then there exists a positive constant
C0 such that
ku(t)k ≤ C0
and ku0 (t)k ≤ C0
a.e. in [0, T ].
Since M ∈ C 1 ([0, ∞)), it follows |M 0 (ξ)| ≤ C1 , for all ξ ∈ [0, C0 ].
Now, by the Mean Value Theorem, for each s ∈ [0, T ], there exists ξs between
ku(s)k2 and kw(s)k2 such that
|M (ku(s)k2 ) − M (kw(s)k2 )| ≤ C1 |ku(s)k2 − kw(s)k2 | ≤
C2 ku(s) − w(s)k = C2 kΨ(s)k.
(4.3)
Observing that |∆w(s)| ≤ C3 , from (4.2) and (4.3) we obtain that
|Ψ0 (t)|2 + kφ(t)k2 + M (ku(t)k2 k)Ψ(t)k2 ≤
Z t
0
C4
|Ψ (s)|2 + kΨ(s)k2 + kφ(s)k2 ds,
o
which implies
|Ψ0 (t)|2 + kφ(t)k2 + kΨ(t)k2 ≤ C5
Z
o
t
|Ψ0 (t)|2 + kΨ(t)k2 + kφ(s)k2 ds.
where C5 = C4 / min{1, m0 }. From the above inequality and Gronwall inequality
if follows that kφ(t)k = kΨ(t)k = 0, i.e., φ and Ψ are zero almost everywhere.
This completes the proof of uniqueness.
EJDE–2002/22
M. R. Clark & Osmundo A. Lima
17
References
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[2] Bisiguin, E. : Perturbation of Kirchhoff-Carrier’s operator by Lipschitz
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[7] Hosoya, M. & Yamada, Y.: On some nonlinear wave equation I - local
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[8] Lions, L.: Qulques mthods des résolutions des problèmes aux limites nonlinears, Dumod, Paris, 1969.
[9] Matos, M. P.: Mathematical analysis of the nonlinear model for the vibrations of a string, Nonlinear Analysis, Theory, Methods & Applications, vol.
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[10] Medeiros, L. A. & Milla Miranda, M.: Local solutions for a nonlinear degenerate hyperbolic equation, Nonlinear Analysis 10, 27–40 (1986).
[11] Medeiros, L. A.: On some nonlinear perturbations of Kirchhoff-Carrier’s
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[12] Pohozaev, S. I.: On a class of quasilinear hyperbolic equations, Mat. Sbornic
96(138)(1)(1975), 152 - 166 (Mat. Sbornic 25(1)(1975), 145–158, English
Translation).
18
Existence of solutions of a variational unilateral system
Marcondes R. Clark
Federal University of Piauı́ - CCN - DM
Av. Ininga S/N - 64.049-550 - Teresina - PI - Brazil
e-mail: mclark@ufpi.br
Osmundo A. Lima
State University of Paraı́ba-DM
CEP 58.109-095 - Campina Grande - PB- Brazil
e-mail: osmundo@openline.com.br
EJDE–2002/22