ARCHIVUM MATHEMATICUM (BRNO)
Tomus 43 (2007), 289 – 303
ON THE FINITE DIMENSION OF ATTRACTORS OF DOUBLY
NONLINEAR PARABOLIC SYSTEMS WITH L-TRAJECTORIES
Hamid El Ouardi
Abstract. This paper is concerned with the asymptotic behaviour of a class
of doubly nonlinear parabolic systems. In particular, we prove the existence
of the global attractor which has, in one and two space dimensions, finite
fractal dimension.
Introduction
We consider the following doubly nonlinear system (P) of the form
∂H
∂b1 (u1 )
(u1 , u2 )
− div a1 (∇u1 ) + f1 (u1 ) = δ1
∂t
∂u1
∂b2 (u2 )
∂H
(u1 , u2 )
− div a2 (∇u2 ) + f2 (u2 ) = δ2
∂t
∂u2
u1 = u2 = 0
b1 (u1 (x, 0)),b2 (u2 (x, 0)) = b1 (ϕ0 (x)), b2 (ψ0 (x))
in Ω × R+ ,
in Ω × R+ ,
on ∂Ω × R+ ,
in Ω ,
where Ω is a bounded and open subset in Rd , with a smooth boundary ∂Ω.
Problems of form (P) arise in flow in porous media, nonlinear heat equation
with absorption, chemical isothermic reactions and unidirectional Non-Newtonian
fluids, see [4] . Therefore, it is important to obtain information about the existence
of the global attractor and its regularity.
In recent years, numbers of works have contributed to the study of the single
equation of the type (P). Where ai = Id, the elegant work [6, 7] plays a critical
role in this area. The method is extensively used in many other works; we refer the
readers to [1, 2, 3, 4, 5, 8, 9, 19] and the references therein. The basic tools that have
been used are a priori estimates, degree theory and super-subsolution method.
In [21], Miranville has considered the following single equation
2000 Mathematics Subject Classification : 35K55, 35K57, 35K65, 35B40.
Key words and phrases : doubly nonlinear parabolic systems, existence of solutions, global
and exponential attractor, fractal dimension and l-trajectories.
Received March 9, 2007, revised October 2007.
290
H. EL OUARDI
∂α(u)
− div β(u) + f (u) = g ,
∂t
with Dirichlet boundary condition and has obtained the existence of the global
attractor which has, in one and two space dimensions, finite fractal dimension.
Finally, we remind here that doubly nonlinear parabolic systems of the type
∂βi (ui )
− ∆ui = fi (, x, t, u1 , u2 ) , (i = 1, 2)
∂t
have been studied by El Ouardi and El Hachimi [12] who obtained the global
attractor, the regularity and the estimates of Hausdorf and fractal dimensions.
Our aim in this paper is to extend the results of [12] and [21] to the more general
systems (P).
The outline of the paper is as follows: in section 1 we make some assumptions
and we prove the existence of the global attractor. In section 2 we show the
results on the regularity of the global attractor. Section 3 is devoted to the fractal
dimension using the method of l-trajectories.
1. Existence and uniqueness
1.1. Notations and assumptions.
Let Ω be a smooth and bounded domain in Rd , d ∈ {1, 2, 3}. Set for t > 0, Qt :=
Ω × (0, t), St := ∂Ω × (0, t) and also ((., .)) , k.k, (., .) , |.| be respectively the scalar
product and the norms in V = H01 (Ω) and H = L2 (Ω).
Let bi , (i = 1, 2) be continuous functions with bi (0) = 0.
We define for s ∈ R
Z s
τ b′i (τ )dτ ,
Ψi (s) =
0
ds Ai (s) · w = ai (s) · w ,
for all s, w ∈ Rd
(ds denoting the differential).
In the sequel, the same symbol c will be used to indicate some positive constants, possibly different from each other, appearing in the various hypotheses and
computations and depending only on data. When we need to fix the precise value
of one constant, we shall use a notation like Mi , i = 1, 2, . . . , instead.
In the sequel, we shall present the following assumptions:
(A1)
(A2)
(
bi ∈ C3 (R), bi (0) = 0,
γi ≤ b′i (s) ≤ αi ,
(i = 1, 2) ,
for all s ∈ R, (i = 1, 2).

1
d d

ai ∈ C (R ) , ai (0) = 0, ds ai ,

c |s|2 − β ≤ A (s) ≤ d |s|2 + e ,
i1
i
i
i
i
ai (s) · s ≥ ci2 |s|2



ds ai (s) · w · w ≥ ci3 |w|2 ,
is bounded (i = 1, 2),
for all s ∈ Rd (i = 1, 2),
(i = 1, 2),
for all s, w ∈ Rd (i = 1, 2) .
FINITE DIMENSIONAL ATTRACTOR
(A3)
(A4)

fi ∈ C2 (R) ,



sign(s)f (s) ≥ c |s|pi +1 ,
i
i4
pi +1

|f
(s)|
≤
c
|s|
,
i
i5


 ′
fi (s) ≥ −Ki ,
291
for all s ∈ R, pi > 0, (i = 1, 2) ,
for all s ∈ R, (i = 1, 2) ,
for all s ∈ R , (i = 1, 2) .

H ∈ C 2 (R × R ,



 ∂H (0, 0) = 0 ,

(i = 1, 2) ,
∂ui
there
is
positive
constant
M
>
0
such that:
i




∂H
max (s) ≤ Mi ,
(i = 1, 2) .
2 ∂ui
s∈R
(A5)
d = 1, 2 or 3 when bi = ci Id, ci > 0, and that d = 1, or 2 otherwise.
2
(A6)
(ϕ0 , ψ0 ) ∈ (L+∞ (Ω)) .
The following lemma are useful and used.
Lemma 1.1 (Ghidaghia lemma, cf.[19]). Let y be a positive absolutely continuous
function on (0, ∞) which satisfies
y ′ + µy q+1 ≤ λ ,
with q > 0, µ > 0, λ ≥ 0. Then for t > 0
1
λ q+1
−1
+ [µ(q)t] q .
y(t) ≤
µ
Lemma 1.2 (Uniform Gronwall’s lemma, cf. [27]). Let y and h be locally integrable
functions such that: ∃r > 0, a1 > 0, a2 > 0, τ > 0, ∀t ≥ τ
Z t+r
Z t+r
y(s) ds ≤ a1 ,
|h(s)| ds ≤ a2 , y ′ ≤ h .
t
t
Then
y(t + r) ≤
a1
+ a2 ,
r
∀t ≥ τ .
Lemma 1.3 (Theorem 2.4., cf. [21]). For every ϕ ∈ L2 (Ω), the problem
− div(a(∇u)) = ϕ ,
u=0
on
∂Ω ,
possesses a unique solution u such that u ∈ H01 (Ω) ∩ H 2 (Ω).
1.2. Existence theorem.
Theorem 1. Let (A1) to (A6) be satisfied. Then there exists a solution (u1 , u2 )
of problem (P) such that for i = (1, 2), we have
ui ∈ Lpi +1 0, T ; Lpi+1 (Ω) ∩ L2 0, T ; H01 (Ω) ∩ L∞ t0 , T ; L∞ (Ω) , ∀t0 > 0 .
292
H. EL OUARDI
0
pi +1
2
1
Proof. By Theorem
2
in
[6]
,
we
can
choose
u
∈
L
(Q
)
∩
L
0,
T
;
H
(Ω)
∩
T
0
i
∞
∞
L τ, T ; L (Ω) for any τ > 0 such that:
∂b1 (u01 )
− div a1 (∇u01 ) + f1 (u01 ) = 0
∂t
u01 = 0
on ST ,
b1 (u01 )t=0
in Ω ,
in QT ,
= b1 (ϕ0 )
and
∂b2 (u02 )
− div a2 (∇u02 ) + f2 (u02 ) = 0
∂t
u02 = 0
b1 (u02 )t=0
in QT ,
on ST ,
= b1 (ψ0 )
in Ω .
We construct two sequences of functions
(un1 )
and
(un2 ),
such that:
(1.2)
∂b1 (un1 )
∂H n−1 n−1
(u
− div a1 (∇un1 ) + f1 (un1 ) = δ1
, u2 )
∂t
∂u1 1
un1 = 0
on ST ,
(1.3)
b1 (un1 )t=0
in Ω ,
(1.1)
= b1 (ψ0 )
in QT ,
(1.5)
∂H n−1 n−1
∂b2 (un2 )
(u
, u2 )
− div a2 (∇un2 ) + f2 (un2 ) = δ2
∂t
∂u2 1
n
u2 = 0
on ST ,
(1.6)
b2 (un2 )t=0
in Ω .
(1.4)
= b1 (ϕ0
in QT ,
We need Lemma 1.4 and Lemma 1.5 below to complete the proof of Theorem 1.
Lemma 1.4.
(1.7)
∀τ > 0, ∃ cτ > 0
such that
kuni kL∞ (τ,T ;L∞ (Ω)) ≤ cτ .
Proof. For n = 0, (1.7) is proved in [21], so suppose (1.7) for (n − 1).
k
Multiplying (1.1) by |b1 (un1 )| b1 (un1 ), k integer, and integrating over Ω to obtain:
Z
Z
1
n k+2
b′1 (un1 ) |b1 (un1 )|k a1 (∇u1 ) · ∇u1 dx
|b1 (u1 )|
dx + (k + 1)
k+2 Ω
Ω
Z
Z
∂H n−1 n−1
k
n
n
n k
(u1 , u2 )b1 (un1 ) |b1 (un1 )| dx .
δ1
f1 (u1 )b1 (u1 ) |b1 (u1 )| dx =
+
∂u
1
Ω
Ω
We note that
Z
k
b′1 (un1 ) |b1 (un1 )| a1 (∇u1 ) · ∇u1 dx ≥ 0 ,
Z
ZΩ
∂H n−1 n−1
k
k+1
(u1 , u2 )b1 (un1 ) |b1 (un1 )| dx ≤ c
|b1 (un1 )|
dx .
δ1
∂u1
Ω
Ω
FINITE DIMENSIONAL ATTRACTOR
293
We thus have
Z
Z
Z
1
k+2
k+p +2
k+1
|b1 (un1 )|
dx + c
|b1 (un1 )| 1 dx ≤ c′
|b1 (un1 )|
dx .
k+2 Ω
Ω
Ω
Setting yk,n (t) = kb1 (un1 )kLk+2 (Ω) and using Holder inequality on both sides, we
have the existence of two constants λ > 0 and δ > 0 such that
dyk,n (t)
p1 +1
+ λyk,n
(t) ≤ δ,
dt
which implies from Lemma 1.1 that ∀ t ≥ τ > 0
1
δ 1
yk,n (t) ≤ ( ) p1 +
1 .
λ
[λ(p1 )t] p1
As k → ∞, we obtain
|un1 (t)| ≤ cτ
The same holds also for un2 .
∀ t ≥ τ > 0.
Lemma 1.5. ∀τ > 0, ∃ ci = ci (τ, ϕ0 , ψ0 ) > 0:
kuni kL2 (0,T ;H 1 (Ω)) ≤ c ,
0
kuni kL∞ (τ,T ;H 1 (Ω)∩L∞ (Ω)) ≤ c′ ,
0
Z TZ
2 hZ T Z
i
X
2
p
|∇uni | dx + c′
|uni | i dx ≤ c′′ .
0
i=1
Ω
0
Ω
un1
Proof. Multiplying (1.1) by
and (1.4) by un2 , and adding, we get:
Z
2 Z
2 Z
2
X
i X
d Xh
2
p +2
|∇uni | dx + c
|uni | i dx ≤ c′ .
Ψ∗i bi (uni ) dx +
(1.8)
dt i=1 Ω
Ω
Ω
i=1
i=1
But
|ϕ0 |L2 (Ω) + |ψ0 |L2 (Ω) ≤ c ⇒
so we deduce that:
2 Z
X
i=1
0
T
Z
Ω
2
|∇uni |
Z
Ω
Ψ∗1 b1 (ϕ0 ) dx +
dx + c
2 Z
X
i=1
0
T
Z
Ω
Z
Ω
Ψ∗2 b2 (ψ0 ) dx ≤ c ,
pi +2
|uni |
dx ≤ c′ .
Whence Lemma 1.5.
From Lemma 1.4 and Lemma 1.5, there is a subsequence uni (i = 1, 2) with the
following properties:
weakly in
L2 0, T ; H01(Ω) ∩ Lpi +1 0, T ; Lpi +1 (Ω) ,
uni → ui
weakly in
L2 0, T ; L2(Ω) ,
bi (uni ) → χi
bi (uni ) → χi
strongly in
L2 τ, T ; H −1(Ω)
(by the compactness result of Aubin (see [24]). By Lemma 7 in [6], we have
χi = bi (ui ). Moreover,
294
H. EL OUARDI
∂H n−1 n−1
∂H
(u
, u2 ) − fi (·, uni ) → δi
(u1 , u2 ) − fi (·, ui )
∂ui 1
∂ui
in Lr τ, T ; Lr (Ω) , ∀ r ≥ 1, ∀ τ ≥ 1. Taking the limit as n goes to +∞, we deduce
that (u1 , u2 ) is a weak solution of (P).
δi
1.3. Uniqueness.
Let (A1) to (A6) be satisfied. Then (P) has a unique solution (u, v) in QT .
Proof. Let u = (u1 , u2 ) and v = (v1 , v2 ) be solutions of (P), we have:
(1.9)
(1.10)
∂(b1 (u1 ) − b1 (v1 ))
− div(a1 (∇u1 ) − a1 (∇v1 )) + f1 (u1 ) − f1 (v1 )
∂t
∂H
∂H
(u) − δ1
(v) ,
= δ1
∂u1
∂u1
∂(b2 (u1 ) − b2 (u2 ))
− div(a2 (∇u2 ) − a2 (∇v2 )) + f2 (u2 ) − f2 (v2 )
∂t
∂H
∂H
(u) − δ2
(v) .
= δ2
∂u2
∂u2
The difference wi = ui − vi satisfies
b′i (ui )wi′ − div(ai (∇ui ) − ai (∇vi )) + fi (ui ) − fi (vi )
= [b′i (vi ) − b′i (ui )] vi′ + δi
(1.11)
∂H
∂H
(u) − δi
(v)
∂ui
∂ui
(1.11) becomes
b′i (ui )wi′ − div(ai (∇ui ) − ai (∇vi )) + fi (ui ) − fi (vi )
= [b′i (vi ) − b′i (ui )] vi′ + δi
(1.12)
We multiply (1.12) by
∂H
∂H
(u) − δi
(v) .
∂ui
∂ui
wi
b′i (ui )
2 2
X
wi 1 dX
2
(ai (∇ui ) − ai (∇vi ), ∇( ′
|wi |L2 (Ω) +
)
2 dt i=1
bi (ui ) L2 (Ω)
i=1
+
2 X
i=1
(1.13)
2
wi X b′i (vi ) − b′i (ui ) ′
=
fi (ui ) − fi (vi ), ′
vi , wi
′
bi (ui )
bi (ui )
L2 (Ω)
i=1
2 X
∂H
∂H
wi δi
.
(v), ′
(u) − δi
+
∂u
∂ui
bi (ui ) L2 (Ω)
i=1
FINITE DIMENSIONAL ATTRACTOR
295
With the assumptions and standard arguments, we get
2
X
(1.14)
i=1
(ai ∇ui ) − ai (∇vi ),
2
X
1
2
|∇wi | ,
≥
c
∇w
i
b′i (ui )
i=1
′′
2 X
b (ui )
wi ∇ui ai (∇ui ) − ai (∇vi ), i′
(u
)
b
i
i
i=1
(1.15)
≤c
2
X
|∇ui |L4 (Ω) |wi |L4 (Ω) kwi kH 1 (Ω) ,
0
i=1
2 2
X
X
wi 2
|wi | ,
fi (ui ) − fi (vi ), ′
≤c
(u
)
b
i i
i=1
i=1
(1.16)
(1.17)
2 ′
X
b (vi ) − b′ (ui )
i
i=1
i
b′i (ui )
vi′ ,
wi
)L2 (Ω)
b′i (ui )
≤c
2
X
|b′i (vi ) − b′i (ui )|L4 (Ω) |vi′ |L2 (Ω) kwi kL4 (Ω)
i=1
(1.18)
2
2 X
X
∂H
wi ∂H
2
|wi | .
(u) − δi
(v), ′
≤c
δi
(u
)
∂u
∂u
b
i
i
i
i
i=1
i=1
Since (u, v) ∈ (H 2 (Ω))2 and H 1 (Ω) ֒→ L4 (Ω)
(1.19)
2
2
2
2
i 1X
X
1 d hX
1 X
2
2
2
2
|wi | +
|wi | +
kwi kH 1 (Ω) ≤ c
kwi kH 1 (Ω) .
0
0
2 dt i=1
k i=1
2k
i=1
i=1
We finally deduce from Gronwall’s lemma,
2
X
i=1
2
|wi | ≤
2
X
2
|wi (0)| exp(2cT ) , ∀t ∈ (0, T ) .
i=1
Thus, we deduce that u1 = v1 and u2 = v2 .
2. Global attractor
Proposition 1. Assuming that (A1)–(A6) hold, then the solution (u1 , u2 ) of system (P) satisfies
(2.0)
(2.1)
|u1 (t)|L∞ (Ω) + |u2 (t)|L∞ (Ω) ≤ c(r) ,
2
X
i=1
2
|∇ui | ≤ c(r) ,
∀t ≥ r ,
∀t ≥ t0 + r .
296
H. EL OUARDI
Proof. Reasoning as in the proof of Lemma 1.4, we also have (2.0).
Multiplying the first equation of (P) by u1 and the second by u2 , we get
2
2
2 Z
X
X
dX
fi (ui ), ui
ai (∇ui ), ∇ui +
Ψi (ui ) dx +
dt i=1 Ω
i=1
i=1
=−
(2.2)
2
X
∂H
δi
(u)ui dx .
∂ui
i=1
We note that it follows from (A2) that
2
(2.3)
ai (∇ui ), ∇ui ≥ ci |∇ui | .
We deduce from (A3) that
p +2
fi (ui ), ui ≥ c |ui | i − c′ .
(2.4)
For fixed r > 0 and τ > 0, integrate (2.2) on ]t, t + r[
2 Z t+r
X
2
|∇ui | ds ≤ c(τ ), ∀t ≥ τ > 0 ,
i=1 t
2
X Z t+r
(2.5)
i=1
pi +2
|ui |
ds ≤ c(τ ) .
t
1
δ1 (u1 )t
and the second by δ12 (u2 )t , we get
2
2 ∂u 2 1 d Z
X
∂ui X 1 ′
1
i
+
+
ai (∇ui ), ∇
bi (ui ),
Fi (ui ) dx
δ
∂t
δ
∂t
δi dt Ω
i
i
i=1
i=1
Z
(2.6)
=
[H(u1 (T ), u2 (T )) − H(ϕ0 , ψ0 )] dx .
Multiplying the first equation of (P) by
Ω
We note that
(2.7)
(2.8)
(2.9)
1
∂u 2
i
≥ c
,
δi
∂t
∂t
Z
1
1 d
∂ui =
ai (∇ui ), ∇
Ai (∇ui ) dx ,
δi
∂t
δi dt Ω
Z
2
2
αi |∇ui | − c ≤
Ai (∇ui ) dx ≤ di |∇ui | + c′ ,
b′i (ui ),
∂u 2 i
Ω
(2.10)
c |ui |pi +2 − c ≤
Z
Fi (ui ) dx ≤ c |ui |pi +2 + c .
Ω
We finally deduce from (2.5)–(2.10) and the uniform Gronwall’s lemma that, for
r > 0,
(2.11)
2
X
i=1
|∇ui |2 ≤ c(τ ) ,
∀t ≥ t0 + r .
FINITE DIMENSIONAL ATTRACTOR
297
Remark 1. By Proposition 1 we deduce that there exist absorbing sets in Lσ1 (Ω)×
2
Lσ1 (Ω) for any σi : 1 ≤ σi ≤ +∞ and absorbing sets in H01 (Ω) ; then assumptions (1.1),(1.4) and (1.12) in Theorem 1.1 [27, p. 23], are satisfied with
2
U = L2 (Ω) , so we have the following
Theorem 2. Assuming that (A1)–(A6) are satisfied, then the semi-group S(t)
associated with the boundary value problem (P) possesses a maximal attractorA,
2
2
which is bounded in [H01 (Ω) ∩ L∞ (Ω)] , compact and connected in L2 (Ω) . Its
2
2
domain of attraction is the whole space L (Ω) .
1 ∂u2
Proposition 2. We assume that (ϕ0 , ψ0 ) ∈ A. Then, for every t > 0, ∂u
∈
∂t , ∂t
2
2
H01 (Ω) and A ∈ H 2 (Ω) .
Proof. Differentiating equation
∂H
∂bi (ui )
(u1 , u2 ) .
− div [ai (∇ui )] + fi (ui ) = δi
∂t
∂ui
Setting θi =
(2.12)
∂ui
∂t ,
b′i (ui )θi′
+
we get
b′′i (ui )(θi )2
− div (ds ai (∇ui ).∇θi ) +
fi′ (ui )θi
2
X
∂ 2 H(u)
θj .
δi
=
∂ui ∂uj
j=1
Now multiplying (2.12) by θi , and integrating over Ω, we obtain, thanks to (A2),
2
2 Z
2 Z
X
X
X
2
|∇θi |
b′′i (ui )(θi )3 dx + c
b′i (ui )θi′ θi dx +
i=1
Ω
+
(2.13)
i=1 Ω
2 Z
X
i=1
the L
∞
(2.14)
(2.15)
(2.16)
i=1
Z X
2 X
2
∂ 2 H(u) 2
′
δi
fi (ui ) |θi | dx ≤
θi θi dx ,
Ω i=1 j=1 ∂ui ∂uj
Ω
estimate and hypothesis imply successively
Z
Z
Z
1
d
′
′
′
′ 2
bi (ui )θi θi dx +
b (ui ) |θi | dx =
b′′ (ui )(θi )3 dx ,
dt Ω i
2 Ω i
Ω
Z X
2 X
2
2
X
∂ 2 H(u) 2
δi
|θi | ,
θj θi dx ≤ M
∂u
∂u
i
j
Ω i=1 j=1
i=1
2 Z
X
i=1
Ω
2
fi′ (ui ) |θi | dx ≥ −c
2
X
2
|θi | .
i=1
We deduce from (A1) that
Z
b′′ (ui )(u′i )3 dx ≤ c |θi |3L3 (Ω) ≤ c kθk3
Ω
(2.17)
1
H 3 (Ω)
≤ c |θi |2 |∇θi | ≤ M |∇θi |2 + M ′ |θi |4 .
298
H. EL OUARDI
By (2.13)–(2.17) becomes
2
d hX
dt i=1
(2.18)
Setting y =
2 R
P
i=1
Z
Ω
2
2
i
X
X
2
2
4
|θi | .
|θi | + c′
b′i (ui ) |θi′ | dx ≤ c
i=1
i=1
b′ (u ) |θi′ |2
Ω i i
dx, we obtain
dy
≤ cy 2 + c′ .
dt
(2.19)
We have, owing to estimates (2.9) and (2.10)
Z τ +r
(2.20)
ydt ≤ cτ , for any τ ≥ t0 .
τ
The uniform Gronwall’s lemma gives
2 Z
X
2
(2.21)
y(t) =
b′i (ui ) |θi′ | dx ≤ c(r) ,
i=1
for any t ≥ r .
Ω
Now, by (2.21) and hypothesis (A2), we get
2 Z
X
i=1
Then, for every t > 0,
But we have
(E)
(
Ω
θi2 dx ≤ c(t0 ) ,
∂u1 ∂u2
∂t , ∂t
for any
t ≥ t0 .
2
∈ L2 (Ω) .
− div ai (∇ui ) = φi
ui = 0
in Ω ,
on ∂Ω ,
∂H
(u1 , u2 ) ∈ L2 (Ω), for every t > 0.
where φi (x, t) = −fi (ui ) − b′i (ui )θi + δi ∂u
i
It follows from Lemma 1.3 that the problem (E) possesses a unique solution
2
(u, v) such that (u, v) ∈ H01 (Ω) ∩ H 2 (Ω) .
3. Dimension of the global attractor A
Proposition 3. Let (ϕ0 , ψ0 ) ∈ A; u = (u1 , u2 ) and v = (v1 , v2 ) be two solutions
of (P). Then,
2
d hX
dt i=1
(3.1)
Z
Ω
b′i (ui ) |ui
2
i
X
2
− vi | dx + M1 ∇ |ui − vi |
2
i=1
≤ M2
2 Z
X
i=1
Ω
2
b′i (ui ) |ui − vi | dx .
FINITE DIMENSIONAL ATTRACTOR
299
Proof. Wet set wi = ui − vi , we have
(3.2)
d
(bi (ui ) − bi (vi )) − div ai (∇ui ) − ai (∇vi ) + fi (ui ) − fi (vi )
dt
∂H
∂H
(u) − δi
(v) .
= δi
∂ui
∂ui
Multiplying (3.2) by wi and integrating over Ω to obtain
Z
1 d
b′i (ui )wi2 dx + (ai (∇ui ) − ai (∇vi ), ∇wi ) + (fi (ui ) − fi (vi ), wi )
2 dt Ω
1Z
∂H
∂H
∂ui 2
(u) − δi
(v), wi +
w dx
b′′i (ui )
= δi
∂u
∂ui
2 Ω
∂t i
Z i
∂vi
(3.3)
wi dx.
b′i (ui ) − b′i (vi )
−
∂t
Ω
By the assumption above, we have
(3.4)
2
X
i=1
2
X
|∇wi |2 ,
ai (∇ui ) − ai (∇vi ), ∇wi ≥ c
i=1
2 Z
2 Z
1X
X
∂vi
∂ui 2
b′i (ui ) − b′i (vi )
wi dx −
wi dx
b′′i (ui )
2 i=1 Ω
∂t
∂t
Ω
i=1
2 Z X
∂ui ∂vi 2
≤c
+
|wi |
∂t
∂t
Ω
i=1
2 Z X
∂ui ∂vi 2
≤c
+
|wi |
∂t
∂t
i=1 Ω
≤c
(3.5)
≤c
2 X
∂ui ∂vi 2
+
kwi kL4 (Ω)
∂t
∂t
i=1
2
X
i=1
(3.6)
2
X
i=1
(3.7)
|wi | |∇wi | ≤
2
2
X
MX
2
2
|wi | ,
|∇wi | + c
2 i=1
i=1
|(fi (ui ) − fi (vi ), wi )| ≤ c
2
X
2
|wi | ,
i=1
2 2
X
X
∂H
∂H
2
|wi | .
(u) − δi
(v), wi ≤ c
δi
∂u
∂u
i
i=1
i=1
We finally deduce from (3.4)–(3.7) that
Z
2
2
2
X
X
X
d
|ui |2 .
|∇wi |2 ≤ c′
b′i (ui )wi2 dx + c
(3.8)
dt
Ω
i=1
i=1
i=1
300
H. EL OUARDI
1
Set ωi = [bi (ui )] 2 wi , we deduce from (3.1) and Gronwall’s lemma that,
2
X
|ωi (t2 )|2 ≤ exp (M2 (t2 − t1 )
2
X
|ωi (t1 )|2
i=1
i=1
(3.9)
2
X
|ωi (t1 )|2 ,
≤ exp(2)
for 0 ≤ t2 − t1 ≤
i=1
We fix s ∈ 0, l =
2
X
1
M2
2
|ωi (l)| + c
and integrate (3.1) over t ∈ (s, l) to obtain, owing to (4.5)
2 Z
X
i=1
i=1
2l
2
|∇ωi | dt ≤ c
′
s
2 Z
X
≤ c′
2
X
2l
2
|ωi | dt +
s
i=1
(3.10)
which yields
2 Z
X
i=1
2l
2
|∇ωi | dt ≤ c′
2
|ωi (s)| ≤ c′
l
2
X
l
2
|ωi (s)|
2
X
2
|ui (s)| ,
i=1
2
|wi (s)| .
i=1
Integrating (3.11) over s ∈ (0, l),
2 Z 2l
2 Z
X
X
2
(3.12)
|∇ui − ∇vi | dt ≤ c
i=1
2
X
i=1
i=1
(3.11)
2
.
M2
i=1
2l
2
|ui − vi | dt .
0
Proposition 4. Let (ϕ0 , ψ0 ) ∈ A; (u1 , u2 ) and (v1 , v2 ) be two solutions of (P).
Then,
2
2 X
X
d
≤c
kui − vi kL2 (0,l;H(Ω)) .
(3.13)
(ui − vi ) 2
dt
L l,2l;H −1 (Ω)
i=1
i=1
Proof. We have
′
Z 2l
′
∂wi ∂wi
(u
)
=
sup
, ϕi > dt ,
< b (ui )
b i
∂t L2 (l,2l;H −1 (Ω))
∂t
l
where ϕi ∈ L2 (l, 2l; V )), kϕi kL2 (l,2l;V ) = 1.
Noting that
′
b (ui )
Furthermore,
(3.14)
2 Z
X
i=1
∂wi
= div ai (∇ui ) − ai (∇vi ) − fi (ui ) − fi (vi )
∂t
∂vi
∂H
∂H
(u) − δi
(v) − b′ (ui ) − b′ (vi )
.
+ δi
∂ui
∂ui
∂t
l
2l
|ai (∇ui ) − ai (∇vi )| |∇ϕi | dt ≤ c
2
X
i=1
kwi kL2 (l,2l;V ) ,
FINITE DIMENSIONAL ATTRACTOR
2 Z
X
i=1
l
2l
301
2
X
fi (ui ) − fi (vi ) ϕi dt ≤ c wi 2
L (l,2l;H)
i=1
(3.15)
≤c
2
X
kwi kL2 (l,2l;V ) ,
i=1
2 Z
X
i=1
(3.16)
l
2l
2
∂H
X
∂H kwi kL2 (l,2l;H)
(u) − δi
(v) |ϕi | dt ≤ c
δi
∂ui
∂u
i=1
≤c
2
X
kwi kL2 (l,2l;V ) ,
i=1
2 Z
X
i=1
(3.17)
l
2l
Z
2 Z 2l
X
′
∂vi ∂vi ′
|ϕi | dx
b (ui ) − b (vi ) dt
|wi | dt
|ϕi | dx ≤ c
∂t
∂t
Ω
Ω
l
i=1
Z
≤c
2
2 X
X
∂vi kwi kL2 (l,2l;V ) .
kwi kL2 (l,2l;V ) ≤ c
+∞
∂t L (l,2l;H)
i=1
i=1
We thus deduce from (3.14)–(3.17) that
2 2 Z 2l
X
X
∂wi ′
≤
|ai (∇ui ) − ai (∇vi )| |∇ϕi | dt
b (ui )
∂t L2 (l,2l;H −1 (Ω))
i=1
i=1 l
2 Z 2l
2 Z 2l X
X
∂H ∂H
(u) − δi
(v) |∇ϕi | dt
+
fi (ui ) − fi (vi ) |ϕi | dt +
δi
∂ui
∂ui
i=1 l
i=1 l
Z
2
2 Z 2l
X
X
′
b (ui ) − b′ (vi ) ∂vi |ϕi | dx ≤ c
kwi kL2 (l,2l;V ) ,
+
dt
∂t
Ω
i=1
i=1 l
and by (3.12), we obtain
2
2 X
X
∂wi ′
kwi kL2 (0,l;H) .
≤c
(3.18)
b (ui )
∂t L2 (l,2l;H −1 (Ω))
i=1
i=1
Theorem 3. The global attractor A associated with (P) has finite fractal dimension.
Proof. It follows from Proposition 1 and Proposition 2 that the semigroup associated with (P) possesses a bounded and positively invariant absorbing set B2
2
in H01 (Ω) ∩ H 2 (Ω) . More precisely, we will take B2 = ∪t≥t0 S(t)B2 , where
2
B2 is a bounded absorbing set in H 2 (Ω) and τ is such that t ≥ τ implies
2
S(t)B2 ⊂ B2 and where the closure is taken in the topology of H 2 (Ω) . We
use the l-trajectories method introduced by Málek and Pražák in [20].
Let l be
fixed as defined above, we introduce the space
of
trajectories
X
=
w
=
(u, v) :
l
(0, l) → H 2 , w is the solution of (P) on (0, l) . We endow Xl with the topology of
302
H. EL OUARDI
2
L2 (0, l; H) . We then set Bl = {w ∈ Xl , w(0) = (ϕ0 , ψ0 ) ∈ B2 }. By construction
2
B2 is weakly closed in H 2 (Ω) . This yields that Bl is a complete metric space,
and we define the operators Lt : Xl → Xl , t ≥ 0, by (Lt (w)) (s) = w(t+s), s ∈ [0, l],
where w is the unique solution of (P) such that w|[0.l] = w. We finally set L =Ll .
Finally if we express (3.12) in terms of l-trajectories, we get for all w1 , w2 ∈ Bl
(3.19)
kLw1 − Lw2 kL2 (0,l;V ) ≤ c kw1 − w2 kXl ,
and it follows from (3.13) that
d
≤ c kw1 − w2 kXl .
(3.20)
(Lw1 − Lw2 ) 2
dt
L (0,l;H −1 (Ω))
Furthermore, L is Lipschitz from Bl onto Xl , ∀t ≥ 0, and the mapping t → Lt w
is Lipschitz, ∀w ∈ Xl . Thanks to (3.19) and (3.20), it can be proved (see [20] for
the details of the proof) that the semigroup Lt possesses an exponential attractor
Ml on Bl , that Ml is compact for the topology of Xl , is positively invariant and
has finite fractal dimension.
References
[1] Alt, H. W., Luckhaus, S., Quasilinear elliptic and parabolic differential equations, Math.
Z. 183 (1983), 311–341.
[2] Bamberger, A., Etude d’une équation doublement non linéaire, J. Funct. Anal. 24 (1977),
148–155.
[3] Blanchard, D., Francfort, G., Study of doubly nonlinear heat equation with no growth
assumptions on the parabolic term, Siam. J. Anal. Math. 9 (5), (1988), 1032–1056.
[4] Diaz, J., de Thelin, F., On a nonlinear parabolic problem arising in some models related
to turbulent flows, Siam. J. Anal. Math. 25 (4), (1994), 1085–1111.
[5] Diaz, J., Padial, J. F., Uniqueness and existence of solutions in the BVt (Q) space to a
doubly nonlinear parabolic problem, Publ. Mat. 40 (1996), 527–560.
[6] Eden, A., Michaux, B., Rakotoson, J. M., Doubly nonlinear parabolic type equations as
dynamical systems, J. Dynam. Differential Equations 3 (1), (1991), 87–131.
[7] Eden, A., Michaux, B., Rakotoson, J. M., Semi-discretized nonlinear evolution equations
as discrete dynamical systems and error analysis, Indiana Univ. Math. J. 39 (3), (1990),
737–783.
[8] El Hachimi, A., de Thelin, F., Supersolutions and stabilisation of the solutions of the
equation ut − △p u = f (x, u), Part I, Nonlinear Anal., Theory Methods Appl. 12 (12),
(1988), 1385–1398.
[9] El Hachimi, A., de Thelin, F., Supersolutions and stabilisation of the solutions of the
equation ut − △p u = f (x, u), Part II, Publ. Mat. 35 (1991), 347–362.
[10] El Ouardi, H., de Thelin, F., Supersolutions and stabilization of the solutions of a nonlinear
parabolic system, Publ. Mat. 33 (1989), 369–381.
[11] El Ouardi, H., El Hachimi, A., Existence and attractors of solutions for nonlinear parabolic
systems, Electron. J. Qual. Theory Differ. Equ. 5 (2001), 1–16.
[12] El Ouardi, H., El Hachimi, A., Existence and regularity of a global attractor for doubly
nonlinear parabolic equations, Electron. J. Differ. Equ. (45) (2002), 1–15.
[13] El Ouardi, H., El Hachimi, A., Attractors for a class of doubly nonlinear parabolic systems,
Electron. J. Qual. Theory Differ. Equ. (1) (2006), 1–15.
FINITE DIMENSIONAL ATTRACTOR
303
[14] Dung, L., Ultimate boundedness of solutions and gradients of a class of degenerate parabolic
systems, Nonlinear Anal., Theory Methods Appl. 39 (2000), 157–171.
[15] Dung, L., Global attractors and steady states for a class of reaction diffusion systems, J.
Differential Equations 147 (1998), 1–29.
[16] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires,
Dunod, Paris 1969.
[17] Langlais, M., Phillips, D., Stabilization of solution of nonlinear and degenerate evolution
equatins, Nonlinear Anal., Theory Methods Appl. 9 (1985), 321–333.
[18] Ladyzhznskaya, O., Solonnikov, V. A., Outraltseva, N. N., Linear and quasilinear equations
of parabolic type, Trans. Amer. Math. Soc. XI, 1968.
[19] Marion, M., Attractors for reaction-diffusion equation: existence and estimate of their
dimension, Appl. Anal. 25 (1987), 101–147.
[20] Málek, J., Pražák, D., Long time behaviour via the method of l-trajectories, J. Differential
Equations 18 (2), (2002), 243–279.
[21] Miranville, A., Finite dimensional global attractor for a class of doubly nonlinear parabolic
equations, Central European J. Math. 4 (1), (2006), 163–182.
[22] Foias, C., Constantin, P., Temam, R., Attractors representing turbulent flows, Mem. Am.
Math. Soc. 314 (1985), 67p.
[23] Foias, C., Temam, R., Structure of the set of stationary solutions of the Navier-Stokes
equations, Commun. Pure Appl. Math. 30 (1977), 149–164.
[24] Simon, J., Régularité de la solution d’un problème aux limites non linéaire, Ann. Fac. Sci.
Toulouse Math. (6),(1981), 247–274.
[25] Schatzman, M., Stationary solutions and asymptotic behavior of a quasilinear parabolic
equation, Indiana Univ. Math. J. 33 (1984), 1–29.
[26] Raviart, P. A., Sur la résolution de certaines équations paraboliques non linéaires, J. Funct.
Anal. 5 (1970), 299–328.
[27] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, Appl. Math.
Sci. 68, Springer-Verlag 1988.
[28] Tsutsumi, M., On solutions of some doubly nonlinear degenerate parabolic systems with
absorption, J. Math. Anal. Appl. 132 (1988), 187–212.
Equipe Architures des Systèmes, Université Hassan II Ain Chock
Ensem, BP. 8118, Oasis Casablanca, Morrocco
E-mail : elouardi@ensem-uh2c.ac.ma