Weak Periodic Solutions of Some Quasilinear Parabolic Systems ...

WEAK PERIODIC SOLUTIONS OF
SOME QUASILINEAR PARABOLIC SYSTEMS WITH
DATA MEASURES**
* Address for correspondence:
Cadi Ayyad University Faculté des Sciences et Techniques Gueliz
Département de Mathématiques et Informatique
B.P.618 Marrakech, Morocco.
e-mail: alaa@fstg-marrakech.ac.ma
† Cadi Ayyad University Faculté des Sciences Semlalia
Département de Mathématiques, B.P.2390 Marrakech, Morocco.
e-mail: iguernane@ucam.ac.ma
N. Alaa* and M. Iguernane†
Cadi Ayyad University
Morocco
N. Alaa and M. Iguernane
ﺔــﺻﻼﺨﻟا
ﻊﻣ ﺔﯿﻄﺨﻟا ﮫﺒﺷ ﺔﯿﻟﻮﻟﺪﮭﻟا ﺔﻤﻈﻧﻷا ﺾﻌﺒﻟ ﺔﻔﯿﻌﺿ ﺔﯾرود لﻮﻠﺣ دﻮﺟو ﺔﺳارد ﻰﻟإ ﺔﻗرﻮﻟا هﺬھ فﺪﮭﺗ
.ﺔـﻘـﺘﺸﻤﻠﻟ ﺔﺒﺴﻨﻟﺎﺑ ﻲﻄﺧ ﺮﯿﻏ جﺮﺣ ءﺎﻤﻧو ، ﺔﺳﺎﻘﻣ تﺎﯿﻄﻌﻣ
ﻖﺳﺎﻨﺗ ماﺪﻌﻧﻻ ً اﺮﻈﻧ ﺎﮭﺗﺎﻘﺘﺸﻣ و لﻮﻠﺤﻟا تﺎﻤﯿﯿﻘﺗ ﻰﻠﻋ ةﺪﻤﺘﻌﻤﻟا ﺔﯿﻜﯿﺳﻼﻜﻟا ﺔﻘﯾﺮﻄﻟا ﻖﯿﺒﻄﺗ ﺎﻨھ ﺎﻨﻨﻜﻤﯾ ﻻو
ABSTRACT
: ﻦﯿﺘﯿﻟﺎﺘﻟا ﻦﯿﺘﺠﯿﺘﻨﻟا ﻰﻟإ تّدأ ىﺮﺧأ ﺐﯿﻟﺎﺳأ ﻰﻠﻋ ﺎﻧﺪﻤﺘﻋا ﻞﺑﺎﻘﻤﻟﺎﺑو ،ﺔﺳﺎﻘﻤﻟا تﺎﯿﻄﻌﻤﻟا
.ﺔﻔﯿﻌﺿ ﺔﯾرود لﻮﻠﺣ ﻰﻠﻋ لﻮﺼﺤﻟا ﻞﺟأ ﻦﻣ ﺔﺳﺎﻘﻤﻟا تﺎﯿﻄﻌﻤﻟا ﻰﻠﻋ ﺔﻣزﻻ طوﺮﺷ دﻮﺟو •
.يرود ﻒﯿﻌﺿ ﻞﺣ دﻮﺟﻮﻟ ﻲﻔﻜﯾ يرود قﻮﻓ ﻒﯿﻌﺿ ﻞﺣ دﻮﺟو •
The goal of this paper is to study existence of weak periodic solutions for some
quasilinear parabolic systems with data measures and critical growth nonlinearity with
respect to the gradient. The classical techniques based on C α -estimates for the solution
or its gradient cannot be applied because of the lack of regularity and a new approach
must be considered. Various necessary conditions are obtained on the data for existence.
The existence of at least one weak periodic solution is proved under the assumption that
a weak periodic super solution is known.
Key Words: quasilinear systems, periodic, parabolic, convex nonlinearities, data
measures, nonlinear capacities.
MSC 2000 classification: 35K55, 35K57, 35B10, 35D05, 31C15.
** This work was supported by the
“Action Integrée MA/33/02” program between the
University Cadi Ayyad FST Gueliz of Marrakech-Morocco and
l’ENS of Cachan, Bretagne, France
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 257

258
N. Alaa and M. Iguernane
WEAK PERIODIC SOLUTIONS OF SOME QUASILINEAR PARABOLIC SYSTEMS
WITH DATA MEASURES
1. INTRODUCTION
Periodic behavior of solutions of parabolic boundary value problems arises in many biological, chemical,
and physical systems, and various methods have been proposed for the study of the existence and qualitative
property of periodic solutions. Most of the work in the earlier literature is devoted to scalar semilinear parabolic
equations under either Dirichlet or Neumann boundary conditions (cf. [1–13]). All these works examine the
classical solutions. In recent years attention has been given to weak solutions of parabolic equations under linear
boundary conditions, and different methods for the existence problem have been used (cf [14–26]).
In this work we are concerned with the periodic parabolic problem
⎧
⎪⎨
⎪⎩
ut − d1∆u = f (t, x, u, v, ∇u, ∇v) + λµ in QT
vt − d2∆v = g (t, x, u, v, ∇u, ∇v) + λ in QT
u (t, x) = v(t, x) = 0 on
T
u (0, x) = u(T, x) and v (0, x) = v(T, x) in Ω
where Ω is an open bounded subset of R N , N ≥ 1, with smooth boundary ∂Ω, QT = [0, T ] × Ω,
T = [0, T ] × ∂Ω, T > 0, d1, d2 > 0, λ are given numbers, −∆ denotes the Laplacian operator on L 1
with Dirichlet boundary conditions, the perturbation f, g : QT × R × R × R N × R N → [0, +∞) is measurable
and continuous with respect to u, v, ∇u and ∇v, and µ, are given nonnegative measures on QT .
The work by Amann [1] is concerned with the problem (1.1) under the hypothesis that µ, are regular enough
and the growth of the nonlinearities f and g with respect to the gradient is sub-quadratic, namely
|f (t, x, u, v, ∇u, ∇v)| + |g (t, x, u, v, ∇u, ∇v)| ≤ c (|u| + |v|) |∇u| 2 + |∇v| 2
+ 1 .
We obtain the existence of maximal and minimal solutions in C 1 (Ω) by using the method of sub- and supersolutions
and Schauder’s fixed point theorem in a suitable Banach space (see also [2], [27]).
In this work we are interested in situations where µ, are irregular and where the growth of f and g with
respect to (∇u, ∇v) is arbitrary and, in particular, larger than |∇u| 2 + |∇v| 2 for large |∇u| + |∇v|. The fact that
µ, are not regular requires that one deals with “weak” solutions for which ∇u, ∇v, ut, vt and even u, v itself are
not bounded. As a consequence the classical theory using C α -a priori estimates to prove existence fails. Let us
make this more precise on a model problem like
⎧
⎪⎨
⎪⎩
ut − d1∆u + au = |∇v| p + λµ in QT
vt − d2∆v + bv = |∇u| q + λ in QT
u (t, x) = v(t, x) = 0 on
T
u (0, x) = u(T, x) and v (0, x) = v(T, x) in Ω ,
where |.| denotes the R N -euclidean norm a, b ≥ 0 and p, q ≥ 1.
2
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005
(1.1)
(1.2)

N. Alaa and M. Iguernane
If p, q ≤ 2, the method of sub- and super-solutions can be applied to prove existence in (1.2) if µ and
are regular enough. For instance, if a, b > 0 and µ, ∈ C α (QT ), then (1.2) has a solution since w ≡ 0 is a
sub-solution and w(t, x) = ( w, w) is a super-solution of (1.2) (see Amann [1]), where ( w, w) is a solution of the
elliptic system
⎧
⎪⎨
⎪⎩
a w − d1∆ w = |∇ w| p + λ µ ∞
b w − d2∆ w = |∇ w| q + λ ∞
in Ω
in Ω
w = w on ∂Ω .
The situation is quite different if p, q > 2 : for instance a size condition is necessary on λµ and λ to have
existence in (1.2) even when µ and are very regular, indeed we prove in Section 2.1 that there exists λ∗ < +∞
such that (1.2) does not have any periodic solution for λ > λ∗ . On the other hand we obtain another critical
value p∗ = 1 + 2
N of the problem, indeed as proved in Section 2.2, existence in (1.2) with p, q > p∗ requires that
µ and be regular enough.
We prove in Section 3 that existence of a nonnegative weak periodic super solution implies existence of a
nonnegative weak periodic solution in the case of sub quadratic growth. Obviously, the classical approach fails
to provide existence since µ and are not regular enough and new techniques must be applied. We describe
some of them here.
To finish this paragraph, we recall the following notations and definitions:
Notations:
C ∞ 0 (QT ) = {ϕ : QT → R, indefinitely derivable with compact support in QT }
Cb (Ω) = {ϕ : Ω → R, continuous and bounded in Ω}
Mb (QT ) = {µ bounded Radon measure in QT }
M +
b (QT ) = {µ bounded nonnegative Radon measure in QT } ;
Definition 1.1. Let σ ∈ C([0, T ] ; L1 (Ω)), we say that σ(0) = σ(T ) in Mb (Ω) if
for all ϕ ∈ Cb (Ω) , lim (σ(t, x) − σ(T − t, x))ϕdx = 0.
t→0
Ω
2. NECESSARY CONDITIONS FOR EXISTENCE
Through this section we are given
(H1) µ, are nonnegative finite measure on (0, T ) × Ω
and f, g : [0, T ] × Ω × R N × R N → [0, +∞[ are such that
(H2) f, g are measurable, almost everywhere (t, x) , (r, s) −→ f (t, x, r, s) , g(t, x, r, s) are continuous, convex.
(H3) ∀r, s ∈ R N , f (., ., r, s) , g (., ., r, s) are integrable on [0, T ] × Ω.
(H4) f (t, x, 0, 0) = g(t, x, 0, 0) = 0.
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 259
3

260
N. Alaa and M. Iguernane
4
For λ ∈ R, we consider the problem
⎧
⎪⎨
⎪⎩
u, v ∈ L 1 (0, T ; W 1,1
0 (Ω)) ∩ C([0, T ] ; L 1 (Ω)), u, v ≥ 0 in QT
f (t, x, ∇u, ∇v) , g (t, x, ∇u, ∇v) ∈ L 1 loc (QT )
ut − d1∆u ≥ f (t, x, ∇u, ∇v) + λµ in D
(QT )
vt − d2∆v ≥ g (t, x, ∇u, ∇v) + λ in D
(QT )
u(0) = u(T ) and v(0) = v(T ) in Mb (Ω) .
2.1. No Existence in Superquadratic Case
We prove in this section, if (f (., ., r, s) , g (., ., r, s)) are superquadratic at infinity, then there exists λ ∗ < +∞
such that (2.1) does not have any periodic solution for λ > λ ∗ . The techniques used here are similar to those
in [14] for the parabolic problem with initial data measure. A rather sharp superquadratic condition on (f, g)
is given next where the (t, x)-dependence is taken into account. We assume that there exists [ε, τ] open in
[0, T ] , p, q > 2, and a constant c0 > 0 such that
(H5) f (t, x, r, s) + g(t, x, r, s) ≥ c0(|r| p + |s| q ) almost every where (t, x) ∈ [ε, τ] ×Ω,
(H6)
[ε,τ]×Ω
µ + > 0.
Theorem 2.1. Assume that (H1) − (H6) hold. Then there exists λ ∗ < +∞ such that (2.1) does not have any
solution for λ > λ ∗ .
Proof.
Assume (u, v) is a solution of (2.1). By (H5), we have for ϕ ∈ C ∞ 0 ([0, T ] × Ω) , ϕ ≥ 0, and ϕ (ε) = ϕ (τ) = 0:
and integrate to obtain
< ut − d1∆u, ϕ > ≥ < f + λµ, ϕ > in D
([ε, τ] × Ω)
< vt − d2∆v, ϕ > ≥ < g + λ, ϕ > in D
([ε, τ] × Ω)
τ
τ
λ (µ + )ϕdxdt ≤
Taking into account the equality
ε
Ω
ϕt = −∆ (Gϕt) = −∆ (Gϕ) t ,
ε
Ω
{d1∇u∇ϕ + d2∇v∇ϕ − c0(|∇u| p +
|∇v| q )ϕ − (u + v)ϕt}dxdt .
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005
(2.1)
(2.2)
(2.3)

where G is the Green Kernel on Ω, we obtain from (2.3):
≤
τ
λ (µ + )ϕdxdt
τ
ε
ε
Ω
Ω
(d1∇u∇ϕ + d2∇v∇ϕ − c0(|∇u| p + |∇v| q )ϕ − ∇(u + v)∇ (Gϕ) t ) dxdt ;
this can be extended for all ϕ ∈ C1 ([0, T ] ; L∞ (Ω)) ∩ L∞
We obtain:
τ
λ (µ + )ϕdxdt ≤
ε
Ω
τ
ε
+ϕ
Ω
0, T ; W 1,∞
0
ϕ ∇u d1∇ϕ − ∇ (Gϕ) t − c0 |∇u|
ϕ
p
∇v d2∇ϕ − ∇ (Gϕ) t − c0 |∇v|
ϕ
q
dxdt ,
N. Alaa and M. Iguernane
(Ω) , ϕ ≥ 0, and ϕ (ε) = ϕ (τ) = 0.
if we recall Young inequality ∀s, r ∈ R, ∀α > 0, there exists c > 0 such that sr ≤ c0 |r| α + c |s| α
We see that (2.4) implies
⎧
τ
τ
λ (µ + )ϕdxdt ≤ c
⎪⎨ ε Ω
ε Ω
⎪⎩
∀ϕ ∈ C 1 ([0, T ] ; L ∞ (Ω)) ∩ L ∞
ϕ ≥ 0 and ϕ (ε) = ϕ (τ) = 0.
|d1∇ϕ − ∇ (Gϕ) t | p
ϕ p −1
0, T ; W 1,∞
0
+ |d2∇ϕ − ∇ (Gϕ) t | q
ϕq
dxdt
−1
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 261
(Ω)
Let us prove that this implies that λ is finite (hence the existence of λ∗ ).
We chose ϕ (t, x) = (t − ε) p +q
(τ − t) p +q
Φ (x) , Φ is a solution of
−∆Φ (x) = λ1Φ (x) , Φ > 0 in Ω
Φ (x) = 0 in ∂Ω
where λ1 is the first eigenvalue of −∆ in Ω. We then have from (2.5)
where
τ
λ
ε
Ω
τ
≤ c
ε
Ω
τ
c
ε
Ω
(t − ε) p +q
(τ − t) p +q
Φ (x) (µ + )dxdt
|d1N1 (t, x) − N2 (t, x)| p
(t − ε) (p +q )(p −1) (τ − t) (p +q )(p −1) Φ (x) p −1
dxdt +
|d2N1 (t, x) − N2 (t, x)| q
(t − ε) (p +q )(q −1) (τ − t) (p +q )(q −1) Φ (x) q −1 dxdt,
N1 (t, x) = (t − ε) p +q
(τ − t) p +q
∇Φ (x)
5
(2.4)
, 1 1
α + α = 1.
(2.5)

262
N. Alaa and M. Iguernane
6
and
Which gives that
it provides
N2 (t, x) = p + q
≤ c1
τ
λ
c2
c3
c4
≤ c5
ε
τ
ε
τ
ε
τ
ε
τ
ε
ε
Ω
Ω
Ω
Ω
Ω
τ
λ
Ω
Ω
λ1
(t − ε) p +q −1 (τ − t) p +q −1 (τ + ε − 2t) ∇Φ (x) .
(t − ε) p +q
(τ − t) p +q
Φ (x) (µ + )dxdt
(t − ε) (p +q ) (τ − t) (p +q ) |∇Φ (x)| p
Φ (x) p −1
(t − ε) q
(τ − t) q
|∇Φ (x)| p
Φ (x) p dxdt+
−1
(t − ε) (p +q ) (τ − t) (p +q ) |∇Φ (x)| q
Φ (x) q −1
(t − ε) p
(τ − t) p
|∇Φ (x)| q
Φ (x) q dxdt
−1
(t − ε) p +q
(τ − t) p +q
|∇Φ (x)| p
Φ (x) p −1
dx + c6
Ω
Φ (x) (µ + ) dxdt
|∇Φ (x)| q
Φ (x) q −1 dx,
dxdt+
dxdt+
where ci, i = 1, ..., 5 are some positives constant. By the definition of Φ we have Φ ∈ W 1,∞
0 (Ω) and
1
Φα (x) ∈ L1 (Ω) for all α < 1. Since p, q > 2 then α1 = q − 1 < 1 and α2 = q − 1 < 1 therefore
|∇Φ (x)| p
Φ (x) p
|∇Φ (x)|
dx < ∞ and
−1 q
Φ (x) q dx < ∞. This finishes the proof.
−1
Ω
2.2. Regularity Condition on the Data µ and ν
We consider the following problem
⎧
⎪⎨
⎪⎩
Ω
u, v ∈ L 1 0, T ; W 1,1
0 (Ω) ∩ C( 0, T ); L 1 (Ω) ,
f (., ∇u, ∇v) , g (., ∇u, ∇v) ∈ L 1 loc (QT )
ut − ∆u ≥ f (t, x, ∇u, ∇v) + λµ in D
(QT )
vt − ∆v ≥ g (t, x, ∇u, ∇v) + λ in D
(QT )
u(0) = u(T ) and v(0) = v(T ) in Mb (Ω) .
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005
(2.6)

where f, g, µ, satisfy (H1) – (H4) and there exists p, q > 1, c1, c2 > 0 such that
f (t, x, r, s) + g (t, x, r, s) ≥ c1 (|r| p + |s| q ) − c2.
For simplicity, we suppose in this subsection that p = max(p, q).
N. Alaa and M. Iguernane
Theorem 2.2. Assume that (H1)−(H4), (2.7) hold. Assume (2.6) hasa solution for some λ > 0. Then the
.
measure µ, does not charge the set of W 2,1
p -capacity zero.
1 1
p + p = 1
Remark 2.3. We recall that a compact set K in QT is of W 2,1
p -capacity zero if there exists a sequence of
C∞ 0 (Ω)-functions ϕn greater that 1 on K and converging to zero in W 2,1
p . The above statement means that
K compact, W 2,1
p -capacity (K) = 0
K
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 263
7
(2.7)
⇒ µ + = 0. (2.8)
Obviously, this is not true for any measure µ, g as soon as p, q > 1 + 2
N or p , q < N
2 + 1, (see, e.g. [19] and the
references there for more details).
Remark 2.4. The natural question is now the following. For 1 ≤ p, q < 1 + 2
N
there exist (u, v) solution of (2.6) and if this solution exists is it unique?
Proof of Theorem 2.2.
From (2.6) , (2.7), we get the following inequality
, and µ, ∈ M+
b (QT ), does
(u + v) t − ∆ (u + v) ≥ c1 (|∇u| p + |∇v| q ) − c2 + λ (µ + ) in D (QT ) . (2.9)
Let K be a compact set of W 2,1
p -capacity zero and ϕn a sequence of C ∞ 0 (QT )-functions such that
ϕn ≥ 1 on K, ϕn → 0 in W 2,1
p and a.e in QT , 0 ≤ ϕn ≤ 1. (2.10)
Multiplying (2.9) by χn = ϕ p
n leads to
T
λ χn (µ + ) dxdt + c1
0
Ω
T
≤ c2
0 Ω
T
0
Ω
χn (|∇u| p + |∇v| q ) dxdt (2.11)
χn − (u + v) ∂χn
T
dxdt + ∇χn∇ (u + v) dxdt. (2.12)
∂t
By the definition of χn and by the fact that 0 ≤ ϕn ≤ 1 on QT , it is easy to see that there exists a sub-sequence
also denoted by ϕn such that
0
∂ϕn
∂t −→ 0 a.e in QT , ∂χn
−→ 0 a.e in Lp (QT ) and
∂t ∂χn
−→ 0 a.e in Lq (QT ) .
∂t
Ω

264
N. Alaa and M. Iguernane
8
Then
lim
T
n→∞
0
Ω
(u + v) ∂χn
dxdt = 0.
∂t
We use ∇χn = pϕp−1 n ∇ϕn, and Young’s inequality to treat last integral above:
and
T
0
T
0
T
∇χn∇udxdt ≤ ε χn |∇u| p dxdt + cε
Ω
Ω
0
0
Ω
T
∇χn∇vdxdt ≤ ε
Ω
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005
T
(χn) q
p |∇v| q dxdt + c ε
But q ≥ p , then there exists a constant c
ε > 0 such that
T
0
T
∇χn∇vdxdt ≤ ε
Ω
0
Ω
(χn) q
p |∇v| q dxdt + c
ε
0
Ω
T
0
0
T
|∇ϕn| p
Ω
Ω
|∇ϕn| q
|∇ϕn| p
dxdt (2.13)
dxdt. (2.14)
dxdt. (2.15)
Due to (2.10), passing to the limit in (2.11) , (2.13) , and (2.15) with ε small enough easily leads to
λ
K
(µ + ) = 0. (2.16)
Remark 2.5. This result remains valid when the operator ∆ is replaced by −∆, which is to say
⎧
⎪⎨
⎪⎩
or also for the equation
⎧
⎪⎨
u, v ∈ L1
0, T ; W 1,1
0 (Ω) ∩ C((0, T ) ; L1 (Ω)),
f (t, x, ∇u, ∇v) , g (t, x, ∇u, ∇v) ∈ L 1 loc (QT )
ut + ∆u ≥ f (t, x, ∇u, ∇v) + λµ in D
(QT )
vt + ∆v ≥ g (t, x, ∇u, ∇v) + λ in D
(QT )
u(0) = u(T ) in Mb (Ω)
ut − ∆u + |∇u| p = λf in QT
u (t, x) = 0 on
T
⎪⎩ u (0, x) = u(T, x) in Ω.

3. AN EXISTENCE RESULT FOR SUBQUADRATIC GROWTH
3.1. Assumptions and Statement of the Result
First, we clarify in which sense we want to solve problem (1.1).
Definition 3.1. A function (u, v) is called a weak periodic solution of (1.1) if
⎧
⎪⎨
⎪⎩
u, v ∈ L 2 0, T, H 1 0 (Ω) ∩ C [0, T ] , L 2 (Ω) ,
f (t, x, u, v, ∇u) , g (t, x, u, v, ∇v) ∈ L 1 (QT )
ut − ∆u = f (t, x, u, v, ∇u) + µ in D
(QT )
vt − ∆v = g (t, x, u, v, ∇v) + in D
(QT )
u (0) = u(T ) ∈ L 2 (Ω) and v (0) = v(T ) ∈ L 2 (Ω) .
In (3.1) µ, are nonnegative, integrable functions and
(H7) f, g: QT × R × R × R N → [0, +∞) are Caratheodory functions.
(H8) f (t, x, s1, s2, 0) = g (t, x, s1, s2, 0) = 0
(H9) f(t, x, s1, s2, r) + g(t, x, s1, s2, r) ≤ c(|s1| + |s2|) |r| 2 + H(t, x) ,
where c : [0, +∞[ → [0, +∞[ is nondecreasing and H ∈ L 1 (QT ).
(H10) f, g are nondecreasing with respect to (s1, s2) and convex with respect to r.
Remark 3.2. (H7) means that:
(t, x) −→ f (t, x, s1, s2, r) and (t, x) −→ g (t, x, s1, s2, r) are measurable
(s1, s2, r) −→ f (t, x, s1, s2, r) and (s1, s2, r) −→ g (t, x, s1, s2, r)
are continuous for almost every (t, x) .
N. Alaa and M. Iguernane
Definition 3.3. We call a weak periodic sub-solution (resp, super-solution) of (3.1) a function (u, v) satisfying
(3.1) with “ = ” replaced by “ ≤ ” (resp. ≥ ).
We state now the main result of this section.
Theorem 3.4. Suppose that hypotheses (H7) – (H10) hold and problem (3.1) has a nonnegative weak supersolution
( w, w). Then (3.1) has a weak periodic solution (u, v) such that: 0 ≤ u ≤ w and 0 ≤ v ≤ w.
In the sequel we will give proof of this theorem, we begin with an approximating problem, then we give some
a priori estimate, and we end by passing to the limit.
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 265
9
(3.1)

266
N. Alaa and M. Iguernane
10
3.2. Approximating Problem
For n ≥ 1, we consider fn (t, x, s1, s2, .) (resp gn (t, x, s1, s2, .)), the Yosida-approximation of f (t, x, s1, s2, .)
(resp g (t, x, s1, s2, .)), which increases to f (t, x, s1, s2, .) (resp g (t, x, s1, s2, .)) as n tends to ∞ and satisfies
fn (t, x, s1, s2, r) ≤ f (t, x, s1, s2, r) and fnr (t, x, s1, s2, r) ≤ n
gn (t, x, s1, s2, r) ≤ f (t, x, s1, s2, r) and gnr (t, x, s1, s2, r) ≤ n.
Where fnr (resp gnr) denotes a section of the subdifferential of fn (resp gn) with respect to r. We set
fn (t, x, s1, s2, r) = fn (t, x, s1, s2, r) 1 [ w≤n]
gn (t, x, s1, s2, r) = gn (t, x, s1, s2, r) 1 [ w≤n] ,
where ( w, w) is a supersolution of (3.1). Then fn, gn satisfy (H7) – (H10) and
fn ≤ f1 [ w≤n], fn ≤ fn+1
gn ≤ g1 [ w≤n], gn ≤ gn+1.
Consider the sequence (un, vn) , n ∈ N ∗ , defined by: (u0, v0) = (0, 0),
with
⎧
⎪⎨
⎪⎩
un, vn ∈ L 2 0, T ; H 1 0 (Ω) ∩ C [0, T ] , L 2 (Ω) ∩ L ∞ (QT )
unt − ∆un = fn (t, x, un−1, vn−1, ∇un) + µn
vnt − ∆vn = gn (t, x, un−1, vn−1, ∇vn) + n
in D
(QT )
in D
(QT )
un (0) = un (T ) ∈ L 2 (Ω) and vn (0) = vn (T ) ∈ L 2 (Ω) ,
µn = µ × 1 [ w

To do this, we first consider the periodic problem
⎧
⎪⎨
⎪⎩
u1 ∈ L 2 0, T ; H 1 0 (Ω) ∩ C [0, T ] , L 2 (Ω) ∩ L ∞ (QT )
u1t − ∆u1 = f1 (t, x, 0, 0, ∇u1) + µ1 in D
(QT )
u1 (0) = u1 (T ) ∈ L 2 (Ω) .
N. Alaa and M. Iguernane
We remark that w1 is a supersolution of (3.4) and thanks to the result in [3], there exists u1 such that
0 ≤ u0 ≤ u1 ≤ w1.
In the same manner, there exists a solution v1 of
such that
⎧
⎪⎨
⎪⎩
v1 ∈ L 2 0, T ; H 1 0 (Ω) ∩ C [0, T ] , L 2 (Ω) ∩ L ∞ (QT )
v1t − ∆v1 = g1 (t, x, 0, 0, ∇v1) + 1 in D
(QT )
v1 (0) = v1 (T ) ∈ L 2 (Ω)
0 ≤ v0 ≤ v1 ≤ w1.
Suppose that (3.3) is satisfied for n − 1. We have
and
( wn) t − ∆ wn ≥ fn (t, x, wn, wn, ∇ wn) + µn
≥ fn (t, x, un−1, vn−1, ∇ wn) + µn,
(un−1) t − ∆un−1 = fn−1 (t, x, un−2, vn−2, ∇un−1) + µn−1
≤ fn (t, x, un−1, vn−1, ∇un−1) + µn.
Then, wn (respectively un−1) is a weak super-solution (respectively sub-solution) of
⎧
⎪⎨
un ∈ L 2 0, T ; H 1 0 (Ω) ∩ C [0, T ] , L 2 (Ω) ∩ L ∞ (QT )
unt − ∆un = fn (t, x, un−1, vn−1, ∇un) + µn in D
(QT )
⎪⎩ un (0) = un (T ) ∈ L2 (Ω) .
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 267
11
(3.4)

268
N. Alaa and M. Iguernane
12
Again, thanks to the result in [3], there exists un such that
un−1 ≤ un ≤ wn ≤ w.
In the same manner, there exists a solution vn of
such that
⎧
⎪⎨
⎪⎩
vn ∈ L 2 0, T ; H 1 0 (Ω) ∩ C [0, T ] , L 2 (Ω) ∩ L ∞ (QT )
vnt − ∆vn = gn (t, x, un−1, vn−1, ∇vn) + n in D
(QT )
vn (0) = vn (T ) ∈ L 2 (Ω) ,
vn−1 ≤ vn ≤ wn ≤ w.
Which proves (3.3) by induction.
3.3. A priori Estimate
In this section, we are going to give several technical results as lemmas that will be very useful for the proof
of the main result.
Lemma 3.5. Let u, w ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ C [0, T ] , L 2 (Ω) , such that
Then
⎧
⎪⎨
⎪⎩
QT
0 ≤ u ≤ w in QT
ut − ∆u ≥ 0 in D
(QT )
wt − ∆w ≥ 0 in D
(QT )
u (0) = u (T ) ∈ L 2 (Ω)
w (0) = w (T ) ∈ L 2 (Ω) .
|∇u| 2
≤ 4
QT
|∇w| 2 .
Lemma 3.6. Let un, vn be a solution of (3.2), then there exists a constant c8 > 0, such that
⎧
⎪⎨
⎪⎩
QT
QT
|f (t, x, un−1, vn−1, ∇un)| dxdt ≤ c8
|g (t, x, un−1, vn−1, ∇vn)| dxdt ≤ c8.
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005
(3.5)

Lemma 3.7. Let u, un ∈ L 2 0, T, H 1 0 (Ω) ∩ C [0, T ] , L 2 (Ω) , such that
N. Alaa and M. Iguernane
0 ≤ un ≤ u in QT and u (0) = u (T ) ∈ L 2 (Ω) (3.6)
un u weakly in L 2 0, T, H 1 0 (Ω)
⎧
⎪⎨
⎪⎩
unt − ∆un = ρn in D
(QT )
un (0) = un (T ) ∈ L 2 (Ω)
ρn ∈ L 1 (QT ) , ρn ≥ 0 and ρn L 1 (QT )
where c is a positive constant independent of n. Then un → u strongly in L 2 0, T, H 1 0 (Ω) .
Proof of Lemma 3.5.
Let us recall the definition of the Lebesgue–Steklov average uh of the solution u, for h > 0
uh (t, x) := 1
h
t+h
t
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 269
≤ c
13
(3.7)
(3.8)
u (s, x) ds. (3.9)
This function is well-defined by definition of weak solution and takes values in the Sobolev space H 1 0 (Ω).
Moreover, it is differentiable in time for all h > 0, and its derivative equals
that there exists a function ε (h), such that lim
h→0 ε (h) = 0 and
We have
QT −h
T −h
0
|∇uh| 2 ≤ ε (h) + 4
Ω
QT −h
|∇uh| 2 dxdt =
=
=
u (t + h) − u (t)
h
. We will prove
|∇wh| 2 . (3.10)
T −h
0
T −h
0
T −h
0
uh, −∆uh dt
uh, uht − ∆uh dt −
T −h
uh, uht − ∆uh dt + ε1 (h) ,
where , denotes the product duality between H 1 0 (Ω) and H −1 (Ω) and
ε1 (h) = −
T −h
0
uh, uht dt.
0
uh, uht dt

270
N. Alaa and M. Iguernane
14
It follows that
with
Then
It follows that
T −h
0
Ω
ε2 (h) =
T −h
0
T −h
0
Ω
Ω
|∇uh| 2 dxdt ≤
[whuh]
T −h
0
≤
dx.
≤ −
≤ −
T −h
0
T −h
0
T −h
0
T −h
0
wh, uht − ∆uh dt + ε1 (h)
whuht +
T −h
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005
0
whtuh +
|∇uh| 2 dxdt ≤ ε1 (h) + ε2 (h) + 1
2
Ω
∇wh∇uh + ε1 (h)
Ω
T −h
T −h
0 dx +
0
T −h
[whuh]
uh, wht − ∆wh dt + 2
T −h
|∇uh|
Ω
2 T −h
dxdt ≤ 2 (ε1 (h) + ε2 (h)) + 4
0
0
Ω
0
|∇uh| 2 dxdt + 2
Ω
|∇wh| 2 dxdt.
∇wh∇uh + ε1 (h)
Ω
∇wh∇uh + ε1 (h) + ε2 (h) ,
Ω
T −h
If we take ε (h) = 2 (ε1 (h) + ε2 (h)), we obtain (3.10). To finish the proof, it suffices to prove that
However
lim ε (h) = 0.
h→0
ε1 (h) = −
= −
T −h
0
T −h
0
uh, uht dt
uhuht
Ω
= − 1
2 T −h
uh 2
0
Ω
since uh ∈ C [0, T ] , L 2 (Ω) , we obtain
lim
h→0 ε1 (h) = 0.
dx
= − 1
2
uh (0) − u
2 Ω
2 h (T − h) dx,
0
Ω
|∇wh| 2 dxdt.

In the other part
lim
h→0 ε2 (h) = lim
h→0
(uh (T − h) wh (T − h) − uh (0) wh (0)) dx
Ω
=
= 0.
(uh (T ) wh (T ) − uh (0) wh (0)) dx
Ω
N. Alaa and M. Iguernane
Then the lemma yields.
Proof of Lemma 3.6.
Remark that
We note
QT
I1 =
I2 =
Hypothesis (H9) yields
|f (t, x, un−1, vn−1, ∇un)| dxdt =
I1 ≤ c (1)
QT ∩[un+vn≤1]
QT ∩[un+vn>1]
QT
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 271
QT ∩[un+vn≤1]
+
f (t, x, un−1, vn−1, ∇un) dxdt
f (t, x, un−1, vn−1, ∇un) dxdt
|∇un| 2
+ H (t, x) dxdt
QT ∩[un+vn>1]
f (t, x, un−1, vn−1, ∇un) dxdt
f (t, x, un−1, vn−1, ∇un) dxdt
But H ∈ L 1 (QT ) and 0 ≤ un ≤ w, then Lemma 3.5, implies that there exists a positive constant c4 such that
On the other hand, we have
I1 ≤ c4 . (3.11)
I2 ≤
QT
(un + vn) (f + g) (t, x, un−1, vn−1, ∇un) dxdt.
Addition of the two equations from the system (3.2), multiplying the sum by un + vn and integrating by parts
yields:
I2 ≤
QT
|∇un| 2 + |∇un| 2
dxdt.
Using Lemma 3.5 and inequality (3.11), we finish the proof.
15

272
N. Alaa and M. Iguernane
16
Proof of Lemma 3.7.
To prove that un → u strongly in L 2 0, T, H 1 0 (Ω) , it suffice to prove that
lim
n→∞
|∇un|
QT
2
dxdt ≤ |∇u|
QT
2 dxdt. (3.12)
For h > 0, let uh and (un) h the sequences given by (3.9). We have
QT
|∇un| 2 dxdt = lim
h→0
|∇ (un) h |
QT −h
2 dxdt
= lim
h→0
= lim
h→0
Since un ≤ u, then (un) h ≤ uh and we obtain
It follows that
QT
lim
n→∞
|∇un| 2 dxdt ≤ lim
h→0
QT
≤ lim
h→0
T −h
0
T −h
0
T −h
0
QT −h
|∇un| 2 dxdt ≤ lim
n→∞ lim
h→0
(un) h , −∆ (un) h
(un) h , ((un) h ) t − ∆ (un) h .
uh, ((un) h ) t − ∆ (un) h
∇uh∇ (un) h dxdt +
QT −h
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005
QT −h
∇uh∇ (un) h dxdt +
uh ((un) h ) t dxdt
QT −h
uh ((un) h ) t dxdt
It is easy to see that un u weakly in L 2 0, T, H 1 0 (Ω) , implies (un) h uh weakly in L 2 0, T, H 1 0 (Ω) . Then
we obtain
lim
n→∞
QT
|∇un| 2 dxdt ≤ lim
h→0 lim
n→∞
≤ lim
h→0
≤
≤
QT
QT
QT −h
QT −h
∇uh∇ (un) h dxdt +
∇uh∇uhdxdt +
QT −h
QT −h
uh (uh) t dxdt
|∇u| 2 dxdt + 1
2 lim
2
uh (T − h) − u
h→0
2 h (0)
|∇u| 2 dxdt,
.
uh ((un) h ) t dxdt
which give (3.12).
.

3.4. Passing to the Limit
N. Alaa and M. Iguernane
According to Lemma 3.5 and estimate (3.3), (un) n , (vn) n are bounded in L 2 0, T ; H 1 0 (Ω) . Therefore there
exists u, v ∈ L 2 0, T ; H 1 0 (Ω) , up to a subsequence still denoted (un, vn) for simplicity, such that
(un, vn) → (u, v) strongly in L 2 (QT ) × L 2 (QT ) and a.e. in QT × QT
(un, vn) (u, v) weakly in L 2 0, T ; H 1 0 (Ω) × L 2 0, T ; H 1 0 (Ω) .
But Lemma 3.7 implies that the last convergence is strong in L 2 0, T ; H 1 0 (Ω) × L 2 0, T ; H 1 0 (Ω) . Then to
ensure that (u, v) is a solution of problem , it suffice to prove that
f (., ., un−1, vn−1, ∇un) → f (., ., u, v, ∇u) in L 1 (QT )
g (., ., un−1, vn−1, ∇vn) → g (., ., u, v, ∇v) in L 1 (QT ) .
It is obvious by Lemma 3.6 and the strong convergence of un, vn in L 2 0, T, H 1 0 (Ω) that
f (., ., un−1, vn−1, ∇un) → f (., ., u, v, ∇u) a.e in QT
g (., ., un−1, vn−1, ∇vn) → g (., ., u, v, ∇v) a.e in QT .
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 273
17
(3.13)
To conclude that (u, v) is a solution of (3.1), we have to show, in view of Vitali’s theorem (cf [28]) that
f (., ., un−1, vn−1, ∇un) (respectively g (., ., un−1, vn−1, ∇vn)) is equi-integrable in L 1 (QT ).
Let K be a measurable subset of QT , ε > 0, and k > 0, we have
we note
K
I1 =
I2 =
f (x, t, un−1, vn−1, ∇un) dxdt =
K∩[un+vn≤k]
K∩[un+vn>k]
To deal with the term I2, we write
K∩[un+vn≤k]
+
K∩[un+vn>k]
f (x, t, un−1, vn−1, ∇un) dxdt
f (x, t, un−1, vn−1, ∇un) dxdt.
I2 ≤ 1
(un + vn) (f + g) (x, t, un−1, vn−1, ∇un) dxdt
k
K
f (x, t, un−1, vn−1, ∇un) dxdt
f (x, t, un−1, vn−1, ∇un) dxdt

274
N. Alaa and M. Iguernane
18
which yields from the system (3.7)
I2 ≤ 1
((un + vn) (un + vn) t − (un + vn) ∆ (un + vn)) dxdt
k
K
≤ 1
k
K
|∇un| 2 + |∇vn| 2
dxdt.
Thanks to Lemma 3.5, there exists a constant c7 > 0 such that
I2 ≤ c7
k
. (3.14)
Then, there exists k0 > 0, such that, if k > k0 then
I2 ≤ ε
. (3.15)
3
By hypothesis (H9), we have for all k > k0
The sequence
I1 ≤ c (k)
|∇un| 2
c (k)
n
K∩[un+vn≤k]
|∇un| 2
+ H (t, x) dxdt.
is equi-integrable in L 1 (QT ). So there exists δ1 > 0 such that if |K| ≤ δ1, then
K∩[un+vn≤k]
|∇un| 2
dxdt ≤ ε
. (3.16)
3
On the other hand H ∈ L 1 (QT ), therefore there exists δ2 > 0, such that
c (k)
whenever |K| ≤ δ2.
K∩[un+vn≤k]
Chose δ0 = inf (δ1, δ2), if |K| ≤ δ0, we have
In the same manner, we obtain
K
H (t, x) dxdt ≤ ε
. (3.17)
3
f (x, t, un−1, vn−1, ∇un) dxdt ≤ ε.
g (x, t, un−1, vn−1, ∇un) dxdt ≤ ε.
K
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005

ACKNOWLEDGMENT
The authors thank a referee for interesting comments and suggestions.
REFERENCES
N. Alaa and M. Iguernane
[1] H. Amann, “Periodic Solutions of Semilinear Parabolic Equations”, in Nonlinear Analysis. New York: Academic Press, 1978,
pp. 1–29.
[2] D.W. Bange, “Periodic Solution of a Quasilinear Parabolic Differential Equation”, J. Differential Equation, 17 (1975), pp. 61–72.
[3] P. Hess, “Periodic-Parabolic Boundary Value Problem and Positivity”, Pitman Res. Notes Math Ser. 247. New York: Longman
Scientific and Technical, 1991.
[4] T. Kusamo, “Periodic Solution of the First Boundary Value Problem for Quasilinear Parabolic Equations of Second Order”,
Funckcial. Ekvac, 9 (1966), pp. 129–137.
[5] B.P. Liu and C.V. Pao, “Periodic Solutions of Coupled Semilinear Parabolic Boundary Value Problems”, Nonlinear Anal.,
6 (1982), pp. 237–252.
[6] X. Lu and W. Feng, “Periodic Solution and Oscillation in a Competition Model with Diffusion and Distributed Delay Effect”,
Nonlinear Anal., 27 (1996), pp. 699–709.
[7] M. Nakao, “On Boundedness Periodicity and Almost Periodicity of Solutions of Some Nonlinear Parabolic Equations”,
J. Differential Equations, 19 (1975), pp. 371–385.
[8] A. Tineo, “Existence of Global Coexistence for Periodic Competition Diffusion Systems”, Nonlinear Anal., 19 (1992),
pp. 335–344.
[9] A. Tineo, “Asymptotic Behavior of Solutions of a Periodic Reaction–Diffusion System of a Competitor–Mutualist Model”,
J. Differential Equation, 108 (1994), pp. 326–341.
[10] L.Y. Tsai, “Periodic Solutions of Nonlinear Parabolic Differential Equations”, Bull. Inst. Math. Acad. Sinica, 5 (1977),
pp. 219–247.
[11] H.A. Levine, “The Role of Critical Exponents in Blowup Theorems”, SIAM Review, 32 (1990), pp. 262–288.
[12] H.A. Levine and B. Sleeman, “A System of Reaction–Diffusion Equations Arising in the Theory of Reinforced Random Walks”,
SIAM Journal Appl. Math, 57(3) (1997), pp 683–730.
[13] C.Y. Chan and K. Deng, “Impulsive Effects on Global Existence of Solutions of Semilinear Heat Equations”, Nonlinear Anal.
TMA, 26 (1996), pp. 1481–1489.
[14] N. Alaa, “Solutions Faibles d’Equations Paraboliques Quasilinéaires avec Données Initiales Mesures”, Ann. Math. Blaise Pascal,
3(2) (1996), pp. 1–15.
[15] N. Alaa and I. Mounir, “Global Existence for Reaction–Diffusion Systems with Mass Control and Critical Growth with Respect
to the Gradient”, J. Math. Anal. App., 253 (2001), pp. 532–557.
[16] N. Alaa and M. Pierre, “Weak Solutions of Some Quasilinear Elliptic Equations with Data Measures”, SIAM J. Math. Anal.,
24(1) (1993), pp. 23–35.
[17] N. Alaa and M. Iguernane, “Weak Periodic Solutions of Some Quasilinear Parabolic Equations with Data Measures”, JIPAM,
3(3) (2002), Article 46.
[18] P. Baras, “Semilinear Problem with Convex Nonlinearity”, in Recent Advances in Nonlinear Elliptic and Parabolic Problems,
Proc. Nancy 88. ed. Ph. Bénilan, M. Chipot, L.C. Evans, and M.Pierre. Pitman Res. Notes in Math., 1989.
[19] P. Baras and M. Pierre, “Problèmes Paraboliques Semi-Linéaires avec Données Mesures”, Applicable Analysis, 18 (1984),
pp. 11–149.
[20] L. Boccardo, F. Murat, and J.P. Puel, “Existence Results for some Quasilinear Parabolic Equations”, Nonlinear Anal., 13 (1989),
pp. 373–392.
[21] L. Boccardo and T. Gallouet, “Nonlinear Elliptic and Parabolic Equations Involving Measure Data”, Journal of Functional
Analysis, 87 (1989), pp. 149–768.
[22] A. Dalla’Aglio-L. Orsina, “Nonlinear Parabolic Equations with Natural Growth Conditions and L1 Data”, Nonlinear Analysis,
27 (1996), pp. 59–73.
[23] J. Deuel and P. Hess, “Nonlinear Parabolic Boundary Value Problems with Upper and Lower Solutions” Israel Journal of
Mathematics, 29(1), 1978.
[24] A.W. Leung and L.A. Ortega, “Existence and Monotone Scheme for Time-Periodic Nonquasimonotone Reaction–Diffusion
Systems. Application to Autocatalytic Chemistry”, J. Math. Anal. App., 221 (1998), pp. 712–733.
[25] C.V. Pao, “Periodic Solutions of Parabolic Systems with Nonlinear Boundary Conditions”, J. Math Anal. App., 234 (1999),
pp. 695–716.
July 2005 The Arabian Journal for Science and Engineering, Volume 30, Number 2A. 275
19

276
N. Alaa and M. Iguernane
20
[26] A. Porretta, “Existence Results for Nonlinear Parabolic Equations via Strong Convergence of Truncations”, Annali di Matematica
pura ed applicata (IV), 177 (1989), pp. 143–172.
[27] W. Feng and X. Lu, “Asymptotic Periodicity in Logistic Equations with Discrete Delays”, Nonlinear Anal., 26(1996),
pp. 171–178.
[28] W. Rudin, Analyse Réelle et Complexe. Masson, 1973.
Paper Received 21 December 2002; Revised 12 May 2004; Accepted 1 June 2005.
The Arabian Journal for Science and Engineering, Volume 30, Number 2A. July 2005