Journal of Inequalities in Pure and
Applied Mathematics
WEAK PERIODIC SOLUTIONS OF SOME QUASILINEAR PARABOLIC
EQUATIONS WITH DATA MEASURES
volume 3, issue 3, article 46,
2002.
N. ALAA AND M. IGUERNANE
Faculté des Sciences et Téchniques Gueliz,
Département de Mathématiques et Informatique.
B.P.618 Marrakech-Maroc.
EMail: alaa@fstg-marrakech.ac.ma
Faculté des Sciences Semlalia,
Département de Mathématiques.
B.P.2390 Marrakech-Maroc.
EMail: iguernane@ucam.ac.ma
Received 03 August, 2001;
accepted 17 April, 2002.
Communicated by: D. Bainov
Abstract
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School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
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Abstract
The goal of this paper is to study the existence of weak periodic solutions for
some quasilinear parabolic equations with data measures and critical growth
nonlinearity with respect to the gradient. The classical techniques based on
Cα -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.The results are applied to reaction-diffusion systems
arising from chemical kinetics.
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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2000 Mathematics Subject Classification: 35K55, 35K57, 35B10, 35D05, 31C15.
Key words: Quasilinear equations, Periodic, Parabolic, Convex nonlinearities, Data
measures, Nonlinear capacities.
Contents
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Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Necessary Conditions for Existence . . . . . . . . . . . . . . . . . . . . .
2.1
No Existence in Superquadratic Case . . . . . . . . . . . . .
2.2
Regularity Condition on the Data f . . . . . . . . . . . . . . .
An Existence Result for Subquadratic Growth . . . . . . . . . . . . .
3.1
Statement of the Result . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1
Assumption . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2
The Main Result . . . . . . . . . . . . . . . . . . . . . .
3.2
Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . .
4
7
7
10
14
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3.2.1
3.2.2
Approximating Problem . . . . . . . . . . . . . . .
A Priori Estimates and Passing to the Limit
A Priori Estimate . . . . . . . . . . . . . . . . . . . . . . .
Passing to the Limit . . . . . . . . . . . . . . . . . . . . .
4
Application to a Class of Reaction-Diffusion Systems . . . . . .
References
15
17
17
22
26
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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1.
Introduction
Periodic behavior of solutions of parabolic boundary value problems arises from
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. [4], [5], [14], [15], [18], [19], [20], [23], [24], [25]) 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 [1], [2], [3], [6], [7], [9],
[8], [10], [11], [16], [21], [22], etc.).
In this work we are concerned with the periodic parabolic problem

ut − ∆u = J (t, x, u, ∇u) + λf in QT





P
u (t, x) = 0
on T
(1.1)





u (0, x) = u(T, x)
in Ω,
N
where Ω is an open bounded
subset of R , N ≥ 1, with smooth boundary ∂Ω,
P
QT = ]0, T [ × Ω, T = ]0, T [ × ∂Ω, T > 0, λ are given numbers, −∆
denotes the Laplacian operator on L1 with Dirichlet boundary conditions, the
perturbation J : QT × R × RN → [0, +∞[ is measurable and continuous with
respect to u and ∇u, and f is a given nonnegative measure on QT .
The work by Amann [4] is concerned with the problem (1.1) under the hypothesis that f is regular enough and the growth of the nonlinearities J with
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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respect to the gradient is sub-quadratic, namely
J (t, x, u, ∇u) ≤ c (|u|) |∇u|2 + 1 .
He obtained the existence of maximal and minimal solutions in C 1 (Ω) by using
the method of sub- and super-solutions and Schauder’s fixed point theorem in a
suitable Banach space (see also [5], [12]).
In this work we are interested in situations where f is irregular and where
the growth of J with respect to ∇u is arbitrary and, in particular, larger than
|∇u|2 for large |∇u| . The fact that f is not regular requires that one deals with
“weak” solutions for which ∇u, ut and even u 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 − ∆u + au = |∇u|p + λf in QT





P
u (t, x) = 0
on
(1.2)
T





u (0, x) = u(T, x)
in Ω,
where |·| denotes the RN -euclidean norm, a ≥ 0 and p ≥ 1.
If p ≤ 2, the method of sub- and super-solutions can be applied to prove
existence in (1.2) if f is regular enough. For instance if a > 0 and f ∈ C α (QT ),
then (1.2) has a solution since w ≡ 0 is a sub-solution and w(t, x) = v(x),
where v is a solution of the elliptic problem

p
 av − ∆v = |∇v| + λ kf k∞ in Ω

v=0
on ∂Ω
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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is a super-solution of (1.2) (see Amann [4]). The situation is quite different if
p > 2 : for instance a size condition is necessary on λf to have existence in
(1.2) even f is 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 + N2 of the problem,
indeed as proved in Section 2.2, existence in (1.2) with p > p∗ requires that f
be regular enough.
We prove in Section 3, that existence of a nonnegative weak periodic super
solution implies existence of nonnegative weak periodic solution in the case of
sub quadratic growth. Obviously, the classical approach fails to provide existence since f is not regular enough and new techniques must be applied. We
describe some of them here. Finally in Section 4, the results are applied to
reaction-diffusion systems arising from chemical kinetics.
To finish this paragraph, we recall the following notations and definitions:
Notations:
C0∞ (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 u ∈ C(]0, T [ ; L1 (Ω)), we say that u(0) = u(T ) in Mb (Ω)
if for all ϕ ∈ Cb (Ω) ,
Z
lim (u(t, x) − u(T − t, x))ϕdx = 0.
t→0
Ω
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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2.
Necessary Conditions for Existence
Throughout this section we are given
(2.1)
f a nonnnegative finite measure on ]0, T [ × Ω
(2.2)
and J : [0, T ] × Ω × RN → [0, +∞[ is such that
J is measurable, almost everywhere (t, x) , r 7−→ J (t, x, r)
is continuous, convex.
(2.3)
(2.4)
∀r ∈ RN , J (·, ·, r) is integrable on ]0, T [ × Ω.
J (t, x, 0) = min J (t, x, r) , r ∈ RN = 0.
For λ ∈ R, we consider the problem

u ∈ L1 (0, T ; W01,1 (Ω)) ∩ C(]0, T [ ; L1 (Ω)), u ≥ 0 in QT








 J (t, x, ∇u) ∈ L1loc (QT ) ,
(2.5)
0


ut − ∆u ≥ J (t, x, ∇u) + λF
in D (QT )







u(0) = u(T )
in Mb (Ω) .
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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2.1.
No Existence in Superquadratic Case
We prove in this section, if J (·, ·, r) is superquadratic at infinity, then there
exists λ∗ < +∞ such that (2.5) does not have any periodic solution for λ > λ∗ .
The techniques used here are similar to those in [1] for the parabolic problem
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with initial data measure. A rather sharp superquadratic condition on J is given
next where the (t, x)-dependence is taken into account. We assume
(2.6)
(2.7)
(2.8)
There exists ]ε, τ [ open in ]0, T [ , p > 2, and a constant c0 > 0
such that J (t, x, r) ≥ c0 |r|p almost every where (t, x) ∈ ]ε, τ [ × Ω
Z
f > 0.
]ε,τ [×Ω
Theorem 2.1. Assume that (2.1) – (2.4), (2.6) – (2.8) hold. Then there exists
λ∗ < +∞ such that (2.5) does not have any solution for λ > λ∗ .
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
Proof. Assume u is a solution of (2.5). By (2.6) and (2.7), we have
0
ut − ∆u ≥ c0 |∇u|p + λf in D (]ε, τ [ × Ω) .
Title Page
Let ϕ ∈ C0∞ (]0, T [ × Ω) , ϕ ≥ 0 and ϕ (ε) = ϕ (τ ) = 0. Multiply (2.9) by ϕ
and integrate to obtain
Z τZ
Z τZ
(2.10)
λ
fϕ ≤
∇u∇ϕ − c0 |∇u|p ϕ − uϕt .
Contents
(2.9)
ε
Ω
ε
Ω
Taking into account the equality
ϕt = −∆ (Gϕt ) = −∆ (Gϕ)t ,
where G is the Green Kernel on Ω. We obtain from (2.10)
Z τZ
Z τZ
λ
fϕ ≤
∇u∇ϕ − c0 |∇u|p ϕ − ∇u∇ (Gϕ)t
ε
Ω
ε
Ω
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this can be extended for all ϕ ∈ C 1 ([0, T ] ; L∞ (Ω)) ∩ L∞ 0, T ; W01,∞ (Ω) ,
ϕ ≥ 0 and ϕ (ε) = ϕ (τ ) = 0. We obtain
Z τZ
Z τZ
|∇ϕ − ∇ (Gϕ)t |
p
(2.11) λ
fϕ ≤
ϕ |∇u|
− c0 |∇u| dxdt
ϕ
ε
Ω
ε
Ω
if we recall Young’s inequality ∀s ∈ R sr ≤ c0 |r|p + c |s|q , p1 + 1q = 1. We see
that (2.11) implies
 Z τZ
Z τZ
|∇ϕ − ∇ (Gϕ)t |q


λ
fϕ ≤ c
dxdt



ϕq−1
ε
Ω
ε
Ω


(2.12)
1,∞
1
∞
∞
∀ϕ
∈
C
([0,
T
]
;
L
(Ω))
∩
L
0,
T
;
W
(Ω)

0






ϕ ≥ 0 and ϕ (ε) = ϕ (τ ) = 0.
Let us prove that this implies that λ is finite (hence the existence of λ∗ ). We
choose ϕ (t, x) = (t − ε)q (τ − t)q Φ (x) , Φ is a solution of

 −∆Φ (x) = λ1 Φ (x) , Φ > 0 in Ω
Weak Periodic Solutions of
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N. Alaa, M. Iguernane
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
Φ (x) = 0
in ∂Ω,
where λ1 is the first eigenvalue of −∆ in Ω. We then have from (2.12)
Z τZ
λ
(t − ε)q (τ − t)q Φ (x) f
ε
Ω
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τ
Z (t − ε)q (τ − t)q ∇Φ (x) − q (t − ε)q−1 (τ − t)q−1
≤c
λ1
ε
Ω
q h
i
× (τ + ε − 2t) ∇Φ (x) (t − ε)q(q−1) (τ − t)q(q−1) Φ (x)q−1 dxdt
Z τZ
Z τZ
(t − ε)q (τ − t)q |∇Φ (x)|q
|∇Φ (x)|q
≤ c1
dxdt
+
c
2
q−1 dxdt
Φ (x)q−1
ε
Ω
ε
Ω Φ (x)
Z
it provides
Z
τ
Z
q
q
Z
(t − ε) (τ − t) Φ (x) f ≤ c3
λ
ε
Ω
Ω
|∇Φ (x)|q
dx.
Φ (x)q−1
By the definition of Φ we have Φ ∈ W01,∞ (Ω) and Φα1(x) ∈ L1 (Ω) for all α < 1.
R
q
Since p > 2 then α = q − 1 < 1, therefore Ω |∇Φ(x)|
q−1 dx < ∞. This completes
Φ(x)
the proof.
2.2.
Regularity Condition on the Data f
We consider the following problem

1,1
1
u
∈
L
0,
T
;
W
(Ω)
∩ C(]0, T [ ; L1 (Ω)),

0







 J (., u, ∇u) ∈ L1loc (QT )
(2.13)
0


ut − ∆u ≥ J (t, x, u, ∇u) + λf in D (QT )







u(0) = u(T )
in Mb (Ω)
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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where f, J satisfy (2.1) – (2.4) and
(2.14)
there exists p > 1 , c1 , c2 > 0, J (t, x, s, r) ≥ c1 |r|p − c2.
Theorem 2.2. Assume that (2.1) – (2.4), (2.14) hold. Assume (2.13) has a
solution for some λ > 0. Then the measure f does not charge the set of Wq2,1 capacity zero p1 + 1q = 1 .
Remark 2.1. We recall that a compact set K in QT is of Wq2,1 -capacity zero
if there exists a sequence of C0∞ (Ω)-functions ϕn greater than 1 on K and
converging to zero in Wq2,1 . The above statement means that
Z
2,1
(2.15)
K compact, Wq -capacity (K) = 0 ⇒
f =0
N
2
+1 or p >
1+ N2 ,
Remark 2.2. The natural question is now the following. Let 1 ≤ p < 1 + N2
and f ∈ M+
b (QT ), does there exist u solution of (2.13) and if this solution
exists is it unique? It will make the object of a next work.
Proof of Theorem 2.2. From (2.13), (2.14), we get the following inequality
(2.16)
ut − ∆u ≥ c1 |∇u|p − c2 + λf
in D0 (QT ) .
Let K be a compact set of Wq2,1 -capacity zero and ϕn a sequence of C0∞ (QT )functions such that
(2.17)
ϕn ≥ 1 on K, ϕn → 0 in
Wq2,1
N. Alaa, M. Iguernane
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K
Obviously, this is not true for any measure f as soon as q <
(see, e.g. [7] and the references therein for more details.)
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
and a.e in QT , 0 ≤ ϕn ≤ 1.
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Multiplying (2.16) by χn = ϕqn leads to
Z TZ
Z TZ
(2.18) λ
χn f + c1
χn |∇u|p
0
Ω
0
Ω
Z TZ
Z TZ
∂χn
≤ c2
χn − u
+
∇χn ∇u.
∂t
0
Ω
0
Ω
We use ∇χn = qϕq−1
n ∇ϕn , and Young’s inequality to treat last integral above:
Z TZ
Z TZ
Z TZ
p
(2.19)
∇χn ∇u ≤ ε
χn |∇u| + cε
|∇ϕn |q
0
Ω
0
Ω
0
Ω
Due to (2.17), passing to the limit in (2.18), (2.19) with ε small enough easily
leads to
Z
(2.20)
λ
f = 0.
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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Contents
K
Remark 2.3. The result obtained here is valid if one replaces in (2.13) the
operator −∆ by ∆ that is to say for the equation

1,1
1
u
∈
L
0,
T
;
W
(Ω)
∩ C(]0, T [ ; L1 (Ω)),

0







 J (t, x, u, ∇u) ∈ L1loc (QT )


ut + ∆u ≥ J (t, x, u, ∇u) + λf







u(0) = u(T )
in Mb (Ω)
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0
in D (QT )
J. Ineq. Pure and Appl. Math. 3(3) Art. 46, 2002
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or also for the equation

p

 ut − ∆u + |∇u| = λf in QT



P
u (t, x) = 0
on T





u (0, x) = u(T, x)
in Ω.
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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3.
3.1.
An Existence Result for Subquadratic Growth
Statement of the Result
3.1.1. Assumption
lem (1.1).
First, we clarify in which sense we want to solve prob-
Definition 3.1. A function u is called a weak periodic solution of (1.1) if

u ∈ L2 (0, T, H01 (Ω)) ∩ C ([0, T ] , L2 (Ω)) ,








 J (t, x, u, ∇u) ∈ L1 (QT )
(3.1)
0


ut − ∆u = J (t, x, u, ∇u) + f in D (QT )







u (0) = u(T ) ∈ L2 (Ω) ,
where f is a nonnegative, integrable function and
(3.2) J : QT × R × RN → [0, +∞[ is a Caratheodory function, that means:
(t, x) 7−→ J (t, x, s, r) is measurable
(s, r) 7−→ J (t, x, s, r) is continuous for almost every (t, x)
(3.3) J is nondecreasing with respect to s and convex with respect to r.
(3.4) J (t, x, s, 0) = min J (t, x, s, r) , r ∈ RN = 0
(3.5) J (t, x, s, r) ≤ c (|s|) |r|2 + H (t, x) ,
1
where c : [0, +∞[ → [0, +∞[ is nondecreasing and H ∈ L (QT ) .
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
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Definition 3.2. We call weak periodic sub-solution (resp. super-solution) of
(1.1) a function u satisfying (4.1) with “ = ” replaced by “ ≤ ” (resp. ≥ ).
3.1.2.
The Main Result
We state now the main result of this section
Theorem 3.1. Suppose that hypotheses (3.2) – (3.5) hold and problem (1.1) has
a nonnegative weak super-solution w. Then (1.1) has a weak periodic solution
u such that: 0 ≤ u ≤ w.
3.2.
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
Proof of the Main Result
3.2.1. Approximating Problem Let n ≥ 1 and ̂n (t, x, s, ·) be the Yoshida’s
approximation of the function J (t, x, s, ·) which increases almost every where
to J (t, x, s, ·) as n tends to infinity and satisfies the following properties
̂n ≤ J,
and
k̂n,r (t, x, s, r)k ≤ n.
Let
Jn (t, x, s, r) = ̂n (t, x, s, r) 1[w≤n] (t, x, s, r) ,
where w is a super-solution of (1.1).
It is easily seen that Jn satisfies hypotheses (3.2) – (3.5).
Moreover
(3.6)
Jn ≤ J1[w≤n]
and
Jn ≤ Jn+1 .
On the other hand, since f ∈ L1 (QT ) , we can construct a sequence fn in
L∞ (QT ) such that
fn ≤ fn+1 ,
kfn kL1 (QT ) ≤ kf kL1 (QT )
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and fn converge to f in D0 (QT ) as n tends to infinity.
Let
Fn = fn 1[w≤n] ,
wn = min (w, n) ,
and consider the sequence (un ) defined by: u0 = w0 = 0,

un ∈ L2 (0, T ; H01 (Ω)) ∩ L∞ (0, T ; L2 (Ω))



0
(3.7)
unt − ∆un = Jn (t, x, un−1 , ∇un ) + Fn in D (QT )



un (0) = un (T ) ∈ L2 (Ω) .
We will show by induction that (3.7) has a solution such that
(3.8)
Weak Periodic Solutions of
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Equations with Data Measures
N. Alaa, M. Iguernane
0 ≤ un−1 ≤ un ≤ wn .
To do this, we first consider the linear periodic problem

u ∈ L2 (0, T, H01 (Ω)) ∩ L∞ (0, T, L2 (Ω)) , u ≥ 0 in QT



(3.9)
ut − ∆u = F1 in D0 (QT )



u (0) = u (T ) ∈ L2 (Ω) .
This problem has a solution u1 (see [17, Theorem 6.1, p. 483]). We remark that
w1 is a supersolution of (3.9) and thanks to the maximum principle, we have
w1 − u1 ≥ 0 on QT , hence there exists u1 such that
0 ≤ u0 ≤ u1 ≤ w1 .
Suppose that (3.8) is satisfied for n − 1.
Then from (3.6), un−1 is a weak sub-solution of (3.7). Let us prove that wn is
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a weak super-solution of (3.7). Indeed, by the definition of wn and the monotonicity of Jn we have

wn ∈ L2 (0, T, H01 (Ω)) ∩ L∞ (0, T, L2 (Ω))



wnt − ∆wn ≥ Jn (t, x, un−1 , ∇wn ) + Fn in D0 (QT )



wn (0) = wn (T ) ∈ L2 (Ω) .
Hence (3.7) has a solution un such that un−1 ≤ un ≤ wn (see [11]), which
proves (3.8) by induction.
3.2.2.
A Priori Estimates and Passing to the Limit
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 3.2. Let u, v ∈ L2 (0, T ; H01 (Ω)), such that

0 ≤ u ≤ v in QT




0


ut − ∆u ≥ 0 in D (QT )



0
(3.10)
vt − ∆v ≥ 0 in D (QT )





u (0) = u (T ) ∈ L2 (Ω)




v (0) = v (T ) ∈ L2 (Ω) .
Then, there exists a constant c2 > o, such that
Z
Z
2
|∇u| ≤ c2
|∇v|2 .
QT
QT
Weak Periodic Solutions of
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Lemma 3.3. Let un be a solution of (3.7), then there exists a constant c3 > o,
such that
Z
Jn (t, x, un−1 , ∇un ) dxdt ≤ c3 .
QT
Lemma 3.4. Let u ∈ L2 (0, T, H01 (Ω)) , such that

0
ut − ∆u = ρ in D (QT )



ρ ∈ MB+ (QT )



u (0) = u (T ) ∈ L2 (Ω) .
Weak Periodic Solutions of
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N. Alaa, M. Iguernane
Then
uρ ∈ L1 (QT )
Z
Z
uρ ≤
and
QT
|∇u|2 .
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QT
Lemma 3.5. Let u, un ∈ L2 (0, T, H01 (Ω)) , such that
(3.11)
0 ≤ un ≤ u in QT and u (0) = u (T ) ∈ L2 (Ω)
(3.12)
un * u weakly in L2 0, T, H01 (Ω)
(3.13)

0

unt − ∆un = ρn in D (QT )


un (0) = un (T ) ∈ L∞ (Ω)


 ρ ∈ L2 (Q ) , ρ ≥ 0 and kρ k
n
T
n
n L1 (QT ) ≤ c
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where c is a constant independent of n. Then un → u strongly in L
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2
(0, T, H01
(Ω)) .
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Proof of Lemma 3.2. Since u ∈ L2 (0, T, H01 (Ω)) and ∆u ∈ L2 (0, T, H −1 (Ω)) ,
then
Z
|∇u|2 = h−∆u, ui ,
QT
where h·, ·i denotes the duality product between L2 (0, T ; H01 (Ω)) and
L2 (0, T ; H −1 (Ω)) .Z
uut = 0 and 0 ≤ u ≤ v, then
Moreover, we have
QT
Z
|∇u|2 = hut − ∆u, ui ≤ hut − ∆u, vi
QT
≤ − h∆u, vi − h∆u, vi
Z
≤ 2
∇u∇v.
QT
N. Alaa, M. Iguernane
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Using Young’s inequality we obtain
Z
Z
Z
1
2
2
|∇u| ≤
|∇u| + c
|∇u|2 ,
2
QT
QT
QT
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where c is a positive constant .
Close
Proof of Lemma 3.3. Remark that
Z
Z
Jn (t, x, un−1 , ∇un ) dxdt =
QT
Weak Periodic Solutions of
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Equations with Data Measures
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Jn (t, x, un−1 , ∇un ) dxdt
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QT ∩[un ≤1]
Z
Jn (t, x, un−1 , ∇un ) dxdt.
+
QT ∩[un >1]
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We note
Z
Jn (t, x, un−1 , ∇un ) dxdt
I1 =
QT ∩[un ≤1]
and
Z
Jn (t, x, un−1 , ∇un ) dxdt.
I2 =
QT ∩[un >1]
Hypothesis (3.5) yields
Z
I1 ≤ c (1)
|∇un |2 + H (t, x) dxdt.
QT
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
N. Alaa, M. Iguernane
1
But H ∈ L (QT ) and 0 ≤ un ≤ w, then Lemma 3.2, implies that there exists a
constant c4 such that
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I1 ≤ c4 .
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(3.14)
On the other hand, we have
Z
I2 ≤
JJ
J
un Jn (t, x, un−1 , ∇un ) dxdt.
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QT
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Multiplying the equation in (3.9) by un and integrating by part yields:
Z
I2 ≤
|∇un |2 .
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QT
Using Lemma 3.2 and inequality (3.14), we complete the proof.
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Proof of Lemma 3.4. Consider the sequence um = min (u, m) . It is clear that
um ∈ L2 (0, T, H01 (Ω)) . Moreover um converge to u in L2 (0, T, H01 (Ω)) and
satisfies the equation

um ∈ L2 (0, T, H01 (Ω))





0
(3.15)
umt − ∆um ≥ ρ1[u<m] in D (QT )





um (0) = um (T ) ∈ L∞ (Ω) .
Multiply (3.15) by um and integrate by part on QT , we obtain
um , ρ1[u<m] = hum , umt − ∆um i
Z
Z
1
2
=
umt +
|∇um |2
2
QT
Z QT
=
|∇um |2 .
QT
N. Alaa, M. Iguernane
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JJ
J
QT
Thanks to Fatou’s lemma, we deduce
Z
Z
uρ =
Weak Periodic Solutions of
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Equations with Data Measures
|∇u|2 .
QT
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Proof of lemma 3.5. By relations (3.11) – (3.13), there exists ρ ∈ M+
b (QT ) ,
such that,

0
 ut − ∆u = ρ in D (QT )
(3.16)

u (0) = u (T ) ∈ L2 (Ω) .
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However,
Z
Z
2
|∇u − ∇un | = −
(u − un ) ∆ (u − un )
QT
QT
Z
Z
=−
u∆ (u − un ) +
un ∆ (u − un )
QT
QT
Z
Z
Z
=
∇u∇ (u − un ) −
∇un ∇u −
un ∇un
QT
QT
QT
Z
Z
Z
=
∇u∇ (u − un ) −
∇un ∇u −
u (unt − ∆un ) .
QT
QT
Moreover, by Lemma 3.4, we have
Z
Z
u (unt − ∆un ) ≤
QT
Hence
N. Alaa, M. Iguernane
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2
|∇u| .
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QT
Z
lim
n→+∞
QT
Weak Periodic Solutions of
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Equations with Data Measures
|∇u − ∇un |2 dxdt = 0.
QT
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Passing to the Limit According to Lemma 3.3 and estimate (3.8), (un )n is
bounded in L2 (0, T ; H01 (Ω)) . Therefore there exists u ∈ L2 (0, T ; H01 (Ω)) , up
to a subsequence still denoted (un ) for simplicity, such that
un → u strongly in L2 (QT ) and a.e. in QT
un * u weakly in L2 (0, T ; H01 (Ω)) .
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However, Lemma 3.5 implies that the last convergence is strong in
L2 (0, T ; H01 (Ω)) . Then to ensure that u is a solution of problem (1.1), it suffices to prove that
(3.17)
Jn (·, ·, un−1 , ∇un ) → J (·, ·, u, ∇u) in L1 (QT ) .
It is obvious by Lemma 3.2 and the strong convergence of un in L2 (0, T, H01 (Ω))
that
Jn (·, ·, un−1 , ∇un ) → J (·, ·, u, ∇u) a.e in QT .
To conclude that u is a solution of (1.1), we have to show, in view of Vitali’s
theorem that (Jn )n is equi-integrable in L1 (QT ) .
Let K be a measurable subset of QT , > 0 and k > 0, we have
Z
Z
Jn (t, x, un−1 , ∇un ) dxdt =
Jn (t, x, un−1 , ∇un ) dxdt
K
K∩[un ≤k]
Z
+
Jn (t, x, un−1 , ∇un ) dxdt.
K∩[un >k]
We note that
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Z
Jn (t, x, un−1 , ∇un ) dxdt
I1 =
K∩[un ≤k]
and
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Z
Jn (t, x, un−1 , ∇un ) dxdt.
I2 =
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K∩[un >k]
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To deal with the term I2 , we write
Z
1
I2 ≤
un Jn (t, x, un−1 , ∇un ) dxdt
k K
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which yields from the equation satisfied by un in (3.7)
Z
1
I2 ≤
(un ut − un ∆un ) dxdt
k
K
Z
1
|∇un |2 dxdt.
≤
k
K
0
By Lemma 3.2, there exists a constant c2 > 0 such that
Weak Periodic Solutions of
Some Quasilinear Parabolic
Equations with Data Measures
0
(3.18)
c
I2 ≤ 2 .
k
Then, there exists k0 > 0,such that, if k > k0 then
(3.19)
I2 ≤ .
3
By hypothesis (3.5), we have for all k > k0
Z
I1 ≤ c (k)
|∇un |2 + H (t, x) dxdt.
K∩[un ≤k]
The sequence |∇un |2 n is equi-integrable in L1 (QT ) . So there exists δ1 > 0
such that if |K| ≤ δ1 , then
Z
(3.20)
c (k)
|∇un |2 dxdt ≤ .
3
K∩[un ≤k]
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On the other hand H ∈ L1 (QT ) , therefore there exists δ2 > 0, such that
Z
(3.21)
c (k)
H (t, x) dxdt ≤ ,
3
K∩[un ≤k]
whenever |K| ≤ δ2 .
Choose δ0 = inf (δ1 , δ2 ) , if |K| ≤ δ0 , we have
Z
Jn (t, x, un−1 , ∇un ) dxdt ≤ .
K
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4.
Application to a Class of Reaction-Diffusion Systems
We will see in this section how to apply the result established below to a class
of raction-diffusion systems of the form

ut − ∆u = −J (t, x, v, ∇u) + F (t, x) in QT








in QT
 vt − ∆v = J (t, x, v, ∇u) + G (t, x)
(4.1)
P


u=v=0
on T







u (0) = u (T ) , v (0) = v (T )
in Ω,
where Ω is an openPbounded subset of RN , N ≥ 1, with smooth boundary ∂Ω,
QT = ]0, T [ × Ω, T = ]0, T [ × ∂Ω T > 0, F , G are integrable nonnegative
functions and J satisfies hypotheses (H1 ) − (H4 ).
Definition 4.1. A couple (u, v) is said to be a weak solution of the system (4.1)
if

u, v ∈ L2 (0, T ; H01 (Ω)) ∩ C ([0, T ] ; L2 (Ω))








 ut − ∆u = −J (t, x, v, ∇u) + F (t, x) in QT
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

vt − ∆v = J (t, x, v, ∇u) + G (t, x) in QT







u (0) = u (T ) , v (0) = v (T ) ∈ L2 (Ω) .
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Theorem 4.1. Under the hypotheses (3.2) – (3.5), and F, G ∈ L2 (QT ), system
(4.1) has a nonnegative weak periodic solution.
To prove this result, we introduce the function w solution of the following
linear problem

w ∈ L2 (0, T ; H01 (Ω)) ∩ C ([0, T ] ; L2 (Ω))





0
(4.2)
wt − ∆w = F + G in D (QT )





w (0) = w (T ) ∈ L2 (Ω) .
It is well known that (4.2) has a unique solution, see [17].
Consider now the equation

v ∈ L2 (0, T, H01 (Ω)) ∩ C ([0, T ] , L2 (Ω))





0
(4.3)
vt − ∆v = J (t, x, v, ∇w − ∇v) + G in D (QT )





v (0) = v (T ) ∈ L2 (Ω) .
It is clear that solving (4.1) is equivalent to solve (4.3) and set u = w − v.
Proof of Theorem 4.1. We remark that w is a supersolution of (4.3). Then by a
direct application of Theorem 3.1, problem (4.3) has a solution.
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Equations with Data Measures
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Weak Periodic Solutions of
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