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volume 3, issue 3, article 46, 2002.

Received 03 August, 2001;

accepted 17 April, 2002.

Communicated by:D. Bainov

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Journal of Inequalities in Pure and Applied Mathematics

WEAK PERIODIC SOLUTIONS OF SOME QUASILINEAR PARABOLIC EQUATIONS WITH DATA MEASURES

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

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

060-01

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Weak Periodic Solutions of Some Quasilinear Parabolic Equations with Data Measures

N. Alaa,M. Iguernane

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J. Ineq. Pure and Appl. Math. 3(3) Art. 46, 2002

http://jipam.vu.edu.au

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 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.The results are applied to reaction-diffusion systems arising from chemical kinetics.

2000 Mathematics Subject Classification:35K55, 35K57, 35B10, 35D05, 31C15.

Key words: Quasilinear equations, Periodic, Parabolic, Convex nonlinearities, Data measures, Nonlinear capacities.

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 ex- amine 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

(1.1)











u_t−∆u=J(t, x, u,∇u) +λf inQ_T

u(t, x) = 0 on P

T

u(0, x) = u(T, x) inΩ,

whereΩis an open bounded subset ofR^N, N ≥1,with smooth boundary∂Ω, Q_T = ]0, T[ × Ω, P

T = ]0, T[ ×∂Ω, T > 0, λ are given numbers, −∆

denotes the Laplacian operator on L¹ with Dirichlet boundary conditions, the perturbationJ :Q_T ×R×R^N →[0,+∞[is measurable and continuous with respect touand∇u,andf is a given nonnegative measure onQ_T.

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

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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 inC¹(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 thatfis not regular requires that one deals with

“weak” solutions for which ∇u, u_t and even u itself are not bounded. As a consequence, the classical theory usingC^α-a priori estimates to prove existence fails. Let us make this more precise on a model problem like

(1.2)











u_t−∆u+au=|∇u|^p +λf in Q_T

u(t, x) = 0 on P

T

u(0, x) =u(T, x) in Ω, where|·|denotes theR^N-euclidean norm,a≥0andp≥1.

If p ≤ 2, the method of sub- and super-solutions can be applied to prove existence in (1.2) iffis regular enough. For instance ifa >0andf ∈C^α(Q_T), then (1.2) has a solution since w ≡ 0 is a sub-solution and w(t, x) = v(x), wherevis a solution of the elliptic problem







av−∆v =|∇v|^p+λkfk_∞ inΩ

v = 0 on∂Ω

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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 forλ > 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 thatf be regular enough.

We prove in Section3, 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^∞(Q_T) ={ϕ :Q_T →R, indefinitely derivable with compact support inQ_T} Cb(Ω) ={ϕ: Ω→R, continuous and bounded inΩ}

M_b(Q_T) ={µbounded Radon measure inQ_T}

M⁺_b (Q_T) = {µbounded nonnegative Radon measure inQ_T}.

Definition 1.1. Letu∈ C(]0, T[ ;L¹(Ω)), we say thatu(0) = u(T)inM_b(Ω) if for allϕ∈ C_b(Ω),

limt→0

Z

Ω

(u(t, x)−u(T −t, x))ϕdx= 0.

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2. Necessary Conditions for Existence

Throughout this section we are given

f a nonnnegative finite measure on ]0, T[×Ω (2.1)

andJ : [0, T]×Ω×R^N →[0,+∞[ is such that

J is measurable, almost everywhere (t, x), r7−→J(t, x, r) (2.2)

is continuous, convex.

∀r∈R^N, J(·,·, r) is integrable on ]0, T[×Ω.

(2.3)

J(t, x,0) = min

J(t, x, r), r ∈R^N = 0.

(2.4)

Forλ∈R,we consider the problem

(2.5)











u∈L¹(0, T;W₀^1,1(Ω))∩C(]0, T[ ;L¹(Ω)), u ≥0 inQ_T J(t, x,∇u)∈L¹_loc(Q_T),

u_t−∆u≥J(t, x,∇u) +λF inD⁰(Q_T)

u(0) = u(T) inM_b(Ω).

2.1. No Existence in Superquadratic Case

We prove in this section, if J(·,·, r) is superquadratic at infinity, then there existsλ^∗ <+∞such that (2.5) does not have any periodic solution forλ > 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 onJis given next where the(t, x)-dependence is taken into account. We assume

There exists ]ε, τ[ open in ]0, T[, p >2, and a constantc₀ >0 (2.6)

such thatJ(t, x, r)≥c₀|r|^p almost every where (t, x)∈]ε, τ[×Ω (2.7)

Z

]ε,τ[×Ω

f >0.

(2.8)

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 forλ > λ^∗.

Proof. Assumeuis a solution of (2.5). By (2.6) and (2.7), we have (2.9) u_t−∆u≥c₀|∇u|^p +λf inD⁰(]ε, τ[×Ω).

Let ϕ ∈ C₀^∞(]0, T[×Ω), ϕ ≥ 0and ϕ(ε) = ϕ(τ) = 0.Multiply (2.9) byϕ and integrate to obtain

(2.10) λ

Z τ ε

Z

Ω

f ϕ≤ Z τ

ε

Z

Ω

∇u∇ϕ−c₀|∇u|^pϕ−uϕ_t. Taking into account the equality

ϕt=−∆ (Gϕt) =−∆ (Gϕ)_t, whereGis the Green Kernel onΩ.We obtain from (2.10)

λ Z τ

ε

Z

Ω

f ϕ≤ Z τ

ε

Z

Ω

∇u∇ϕ−c₀|∇u|^pϕ− ∇u∇(Gϕ)_t

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this can be extended for allϕ ∈ C¹([0, T] ; L^∞(Ω))∩L^∞ 0, T;W₀^1,∞(Ω) , ϕ ≥0andϕ(ε) =ϕ(τ) = 0.We obtain

(2.11) λ Z τ

ε

Z

Ω

f ϕ≤ Z τ

ε

Z

Ω

ϕ

|∇u||∇ϕ− ∇(Gϕ)_t|

ϕ −c₀|∇u|^p

dxdt if we recall Young’s inequality∀s ∈Rsr ≤c₀|r|^p+c|s|^q, ¹_p +¹_q = 1.We see that (2.11) implies

(2.12)









 λ

Z τ ε

Z

Ω

f ϕ≤c Z τ

ε

Z

Ω

|∇ϕ− ∇(Gϕ)_t|^q ϕ^q−1 dxdt

∀ϕ ∈C¹([0, T] ;L^∞(Ω))∩L^∞ 0, T;W₀^1,∞(Ω) ϕ ≥0andϕ(ε) =ϕ(τ) = 0.

Let us prove that this implies that λ is finite (hence the existence ofλ^∗). We chooseϕ(t, x) = (t−ε)^q(τ −t)^qΦ (x),Φis a solution of







−∆Φ (x) =λ1Φ (x),Φ>0 inΩ

Φ (x) = 0 in∂Ω,

whereλ₁ is the first eigenvalue of−∆inΩ.We then have from (2.12) λ

Z τ ε

Z

Ω

(t−ε)^q(τ −t)^qΦ (x)f

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≤c Z τ

ε

Z

Ω

(t−ε)^q(τ−t)^q∇Φ (x)− q

λ₁ (t−ε)^q−1(τ−t)^q−1

×(τ +ε−2t)∇Φ (x)

q

h

(t−ε)^q(q−1)(τ −t)^q(q−1)Φ (x)^q−1i dxdt

≤c₁ Z τ

ε

Z

Ω

(t−ε)^q(τ −t)^q|∇Φ (x)|^q

Φ (x)^q−1 dxdt+c₂ Z τ

ε

Z

Ω

|∇Φ (x)|^q Φ (x)^q−1 dxdt it provides

λ Z τ

ε

Z

Ω

(t−ε)^q(τ −t)^qΦ (x)f ≤c₃ Z

Ω

|∇Φ (x)|^q Φ (x)^q−1 dx.

By the definition ofΦwe haveΦ∈W₀^1,∞(Ω)and_Φα¹(x) ∈L¹(Ω)for allα <1.

Sincep >2thenα=q−1<1,thereforeR

Ω

|∇Φ(x)|^q

Φ(x)^q−1dx <∞.This completes the proof.

2.2. Regularity Condition on the Data f

We consider the following problem

(2.13)











u∈L¹ 0, T;W₀^1,1(Ω)

∩C(]0, T[ ;L¹(Ω)), J(., u,∇u)∈L¹_loc(Q_T)

u_t−∆u≥J(t, x, u,∇u) +λf inD⁰(Q_T) u(0) =u(T) in Mb(Ω)

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wheref, Jsatisfy (2.1) – (2.4) and

(2.14) there existsp > 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 someλ > 0.Then the measuref does not charge the set ofW_q^2,1- capacity zero

1

p +¹_q = 1 .

Remark 2.1. We recall that a compact set K inQ_T is of W_q^2,1-capacity zero if there exists a sequence of C₀^∞(Ω)-functions ϕ_n greater than 1 on K and converging to zero inW_q^2,1. The above statement means that

(2.15) Kcompact,W_q^2,1-capacity(K) = 0

⇒ Z

K

f = 0

Obviously, this is not true for any measuref as soon asq < ^N₂ +1orp > 1+_N², (see, e.g. [7] and the references therein for more details.)

Remark 2.2. The natural question is now the following. Let 1 ≤ p < 1 + _N² and f ∈ M⁺_b (Q_T), does there existu solution of (2.13) and if this solution exists is it unique? It will make the object of a next work.

Proof of Theorem2.2. From (2.13), (2.14), we get the following inequality (2.16) ut−∆u≥c1|∇u|^p−c2+λf inD⁰(QT).

LetK be a compact set ofW_q^2,1-capacity zero andϕ_na sequence ofC₀^∞(Q_T)- functions such that

(2.17) ϕn ≥1onK, ϕn →0inW_q^2,1 and a.e inQT,0≤ϕn ≤1.

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Multiplying (2.16) byχ_n=ϕ^q_nleads to (2.18) λ

Z T 0

Z

Ω

χnf +c1

Z T 0

Z

Ω

χn|∇u|^p

≤c2

Z T 0

Z

Ω

χn−u∂χ_n

∂t + Z T

0

Z

Ω

∇χn∇u.

We use∇χ_n=qϕ^q−1_n ∇ϕ_n,and Young’s inequality to treat last integral above:

(2.19)

Z T 0

Z

Ω

∇χ_n∇u≤ε Z T

0

Z

Ω

χ_n|∇u|^p +c_ε Z T

0

Z

Ω

|∇ϕ_n|^q

Due to (2.17), passing to the limit in (2.18), (2.19) withε small enough easily leads to

(2.20) λ

Z

K

f = 0.

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











u∈L¹ 0, T;W₀^1,1(Ω)

∩C(]0, T[ ;L¹(Ω)), J(t, x, u,∇u)∈L¹_loc(Q_T)

u_t+ ∆u≥J(t, x, u,∇u) +λf inD⁰(Q_T) u(0) =u(T) in Mb(Ω)

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or also for the equation











u_t−∆u+|∇u|^p =λf inQ_T

u(t, x) = 0 on P

T

u(0, x) =u(T, x) inΩ.

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3. An Existence Result for Subquadratic Growth

3.1. Statement of the Result

3.1.1. Assumption First, we clarify in which sense we want to solve problem (1.1).

Definition 3.1. A functionuis called a weak periodic solution of (1.1) if

(3.1)











u∈L²(0, T, H₀¹(Ω))∩C([0, T], L²(Ω)), J(t, x, u,∇u)∈L¹(Q_T)

u_t−∆u=J(t, x, u,∇u) +f inD⁰(Q_T) u(0) =u(T)∈L²(Ω),

wheref is a nonnegative, integrable function and

(3.2) J :Q_T ×R×R^N →[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) J is nondecreasing with respect tosand convex with respect tor.

(3.3)

J(t, x, s,0) = min

J(t, x, s, r), r∈R^N = 0 (3.4)

J(t, x, s, r)≤c(|s|) |r|²+H(t, x) , (3.5)

wherec: [0,+∞[→[0,+∞[is nondecreasing andH ∈L¹(QT).

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Definition 3.2. We call weak periodic sub-solution (resp. super-solution) of (1.1) a functionusatisfying (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-solutionw.Then (1.1) has a weak periodic solution usuch that:0≤u≤w.

3.2. Proof of the Main Result

3.2.1. Approximating Problem Letn≥1andˆ_n(t, x, s,·)be the Yoshida’s approximation of the functionJ(t, x, s,·)which increases almost every where toJ(t, x, s,·)asntends to infinity and satisfies the following properties

ˆ

_n ≤J, and kˆ_n,r(t, x, s, r)k ≤n.

Let

J_n(t, x, s, r) = ˆ_n(t, x, s, r) 1[w≤n](t, x, s, r), wherewis a super-solution of (1.1).

It is easily seen thatJ_nsatisfies hypotheses (3.2) – (3.5).

Moreover

(3.6) J_n ≤J1[w≤n] and J_n≤J_n+1.

On the other hand, since f ∈ L¹(Q_T), we can construct a sequence f_n in L^∞(QT)such that

f_n≤f_n+1, kf_nk_L1(QT)≤ kfk_L1(QT)

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andf_nconverge tof inD⁰(Q_T)asntends to infinity.

Let

F_n=f_n1[w≤n], w_n= min (w, n), and consider the sequence(un)defined by: u0 =w0 = 0,

(3.7)











u_n ∈L²(0, T;H₀¹(Ω))∩L^∞(0, T;L²(Ω))

u_n_t −∆u_n =J_n(t, x, un−1,∇u_n) +F_ninD⁰(Q_T) un(0) =un(T) ∈L²(Ω).

We will show by induction that (3.7) has a solution such that

(3.8) 0≤un−1 ≤un ≤wn.

To do this, we first consider the linear periodic problem

(3.9)











u∈L²(0, T, H₀¹(Ω))∩L^∞(0, T, L²(Ω)), u≥0inQT

u_t−∆u=F₁ in D⁰(Q_T) u(0) =u(T)∈L²(Ω).

This problem has a solutionu₁(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 onQ_T,hence there existsu₁ such that

0≤u₀ ≤u₁ ≤w₁. Suppose that (3.8) is satisfied forn−1.

Then from (3.6), un−1 is a weak sub-solution of (3.7). Let us prove thatwnis

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a weak super-solution of (3.7). Indeed, by the definition of w_n and the mono- tonicity ofJnwe have











w_n ∈L²(0, T, H₀¹(Ω))∩L^∞(0, T, L²(Ω))

wnt −∆wn≥Jn(t, x, un−1,∇wn) +FninD⁰(QT) w_n(0) =w_n(T)∈L²(Ω).

Hence (3.7) has a solution u_n such that un−1 ≤ u_n ≤ w_n (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. Letu, v ∈L²(0, T;H₀¹(Ω)),such that

(3.10)











0≤u≤v inQ_T ut−∆u≥0 inD⁰(QT) v_t−∆v ≥0 inD⁰(Q_T) u(0) =u(T)∈L²(Ω) v(0) =v(T)∈L²(Ω). Then, there exists a constantc₂ > o,such that

Z

QT

|∇u|² ≤c₂ Z

QT

|∇v|².

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Lemma 3.3. Let u_nbe a solution of (3.7), then there exists a constantc₃ > o, such that

Z

QT

J_n(t, x, un−1,∇u_n)dxdt≤c₃. Lemma 3.4. Letu∈L²(0, T, H₀¹(Ω)),such that











u_t−∆u=ρ inD⁰(Q_T) ρ∈M_B⁺(Q_T)

u(0) =u(T)∈L²(Ω). Then

uρ ∈L¹(QT) and Z

QT

uρ≤ Z

QT

|∇u|². Lemma 3.5. Letu, u_n ∈L²(0, T, H₀¹(Ω)),such that

(3.11) 0≤u_n ≤uinQ_T andu(0) =u(T)∈L²(Ω)

(3.12) u_n* uweakly inL² 0, T, H₀¹(Ω)

(3.13)











u_n_t −∆u_n=ρ_ninD⁰(Q_T) u_n(0) =u_n(T)∈L^∞(Ω)

ρ_n∈L²(Q_T), ρ_n ≥0and kρ_nk_L1(QT) ≤c

wherecis a constant independent ofn. Thenun→ustrongly inL²(0, T, H₀¹(Ω)).

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Proof of Lemma3.2. Sinceu∈L²(0, T, H₀¹(Ω))and∆u∈L²(0, T, H⁻¹(Ω)),

then Z

QT

|∇u|² =h−∆u, ui,

where h·,·i denotes the duality product between L²(0, T;H₀¹(Ω)) and L²(0, T;H⁻¹(Ω)).

Moreover, we have Z

QT

uu_t = 0and0≤u≤v,then Z

QT

|∇u|² = hu_t−∆u, ui ≤ hu_t−∆u, vi

≤ − h∆u, vi − h∆u, vi

≤ 2 Z

QT

∇u∇v.

Using Young’s inequality we obtain Z

QT

|∇u|² ≤ 1 2

Z

QT

|∇u|²+c Z

QT

|∇u|², wherecis a positive constant .

Proof of Lemma3.3. Remark that Z

QT

J_n(t, x, un−1,∇u_n)dxdt= Z

QT∩[u_n≤1]

J_n(t, x, un−1,∇u_n)dxdt +

Z

QT∩[un>1]

J_n(t, x, u_n−1,∇u_n)dxdt.

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We note

I₁ = Z

QT∩[un≤1]

J_n(t, x, un−1,∇u_n)dxdt and

I2 = Z

QT∩[u_n>1]

Jn(t, x, un−1,∇un)dxdt.

Hypothesis (3.5) yields I₁ ≤c(1)

Z

QT

|∇u_n|²+H(t, x) dxdt.

ButH ∈L¹(Q_T)and0≤u_n ≤w,then Lemma3.2, implies that there exists a constantc₄ such that

(3.14) I₁ ≤c₄.

On the other hand, we have I₂ ≤

Z

QT

u_nJ_n(t, x, un−1,∇u_n)dxdt.

Multiplying the equation in (3.9) byu_nand integrating by part yields:

I₂ ≤ Z

Q_T

|∇u_n|².

Using Lemma3.2and inequality (3.14), we complete the proof.

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Proof of Lemma3.4. Consider the sequence u_m = min (u, m).It is clear that um ∈ L²(0, T, H₀¹(Ω)).Moreover um converge touin L²(0, T, H₀¹(Ω)) and satisfies the equation

(3.15)











u_m ∈L²(0, T, H₀¹(Ω))

u_m_t −∆u_m ≥ρ1_[u<m]inD⁰(Q_T) u_m(0) =u_m(T)∈L^∞(Ω).

Multiply (3.15) byu_mand integrate by part onQ_T,we obtain u_m, ρ1_[u<m]

=hu_m, u_m_t −∆u_mi

= 1 2

Z

QT

u²_m_t + Z

QT

|∇u_m|²

= Z

QT

|∇u_m|². Thanks to Fatou’s lemma, we deduce

Z

QT

uρ= Z

QT

|∇u|².

Proof of lemma3.5. By relations (3.11) – (3.13), there exists ρ ∈ M⁺_b (Q_T), such that,

(3.16)







u_t−∆u=ρinD⁰(Q_T) u(0) =u(T)∈L²(Ω).

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However, Z

QT

|∇u− ∇u_n|² =− Z

QT

(u−u_n) ∆ (u−u_n)

=− Z

QT

u∆ (u−u_n) + Z

QT

u_n∆ (u−u_n)

= Z

Q_T

∇u∇(u−u_n)− Z

Q_T

∇u_n∇u− Z

Q_T

u_n∇u_n

= Z

QT

∇u∇(u−u_n)− Z

QT

∇u_n∇u− Z

QT

u(u_n_t−∆u_n). Moreover, by Lemma3.4, we have

Z

QT

u(u_n_t −∆u_n)≤ Z

QT

|∇u|². Hence

n→+∞lim Z

QT

|∇u− ∇u_n|²dxdt= 0.

Passing to the Limit According to Lemma 3.3 and estimate (3.8), (u_n)_n is bounded inL²(0, T;H₀¹(Ω)).Therefore there existsu∈L²(0, T;H₀¹(Ω)),up to a subsequence still denoted(u_n)for simplicity, such that

u_n →u strongly inL²(Q_T) and a.e. inQ_T un * u weakly inL²(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 thatuis a solution of problem (1.1), it suf- fices to prove that

(3.17) Jn(·,·, un−1,∇un)→J(·,·, u,∇u) inL¹(QT).

It is obvious by Lemma3.2and the strong convergence ofu_ninL²(0, T, H₀¹(Ω)) that

J_n(·,·, un−1,∇u_n)→J(·,·, u,∇u) a.e inQ_T.

To conclude that uis a solution of (1.1), we have to show, in view of Vitali’s theorem that(Jn)_nis equi-integrable inL¹(QT).

Let K be a measurable subset ofQ_T, >0andk > 0,we have Z

K

J_n(t, x, un−1,∇u_n)dxdt= Z

K∩[u_n≤k]

J_n(t, x, un−1,∇u_n)dxdt +

Z

K∩[un>k]

J_n(t, x, u_n−1,∇u_n)dxdt.

We note that

I₁ = Z

K∩[un≤k]

J_n(t, x, un−1,∇u_n)dxdt and

I₂ = Z

K∩[un>k]

J_n(t, x, un−1,∇u_n)dxdt.

To deal with the termI₂,we write I₂ ≤ 1

k Z

K

u_nJ_n(t, x, un−1,∇u_n)dxdt

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which yields from the equation satisfied byu_nin (3.7) I2 ≤ 1

k Z

K

(unut−un∆un)dxdt

≤ 1 k

Z

K

|∇u_n|²dxdt.

By Lemma3.2, there exists a constantc⁰₂ >0such that

(3.18) I2 ≤ c⁰₂

k. Then, there existsk₀ >0,such that, ifk > k₀ then

(3.19) I2 ≤

3. By hypothesis (3.5), we have for allk > k₀

I₁ ≤c(k) Z

K∩[un≤k]

|∇u_n|²+H(t, x) dxdt.

The sequence |∇u_n|²

n is equi-integrable inL¹(Q_T).So there existsδ₁ > 0 such that if|K| ≤δ₁,then

(3.20) c(k)

Z

K∩[un≤k]

|∇u_n|²

dxdt≤ 3.

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On the other handH ∈L¹(Q_T),therefore there existsδ₂ >0,such that

(3.21) c(k)

Z

K∩[un≤k]

H(t, x)dxdt≤ 3, whenever|K| ≤δ₂.

Chooseδ₀ = inf (δ₁, δ₂),if|K| ≤δ₀, we have Z

K

J_n(t, x, un−1,∇u_n)dxdt≤.

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4. Application to a Class of Reaction-Diffusion Sys- tems

We will see in this section how to apply the result established below to a class of raction-diffusion systems of the form

(4.1)











u_t−∆u=−J(t, x, v,∇u) +F(t, x) inQ_T v_t−∆v =J(t, x, v,∇u) +G(t, x) inQ_T

u=v = 0 on P

T

u(0) =u(T), v(0) =v(T) inΩ,

whereΩis an open bounded subset ofR^N,N ≥ 1,with smooth boundary∂Ω, Q_T = ]0, T[×Ω,P

T = ]0, T[×∂ΩT >0, F,Gare integrable nonnegative functions andJ 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)) u_t−∆u=−J(t, x, v,∇u) +F (t, x) inQ_T v_t−∆v =J(t, x, v,∇u) +G(t, x) inQ_T u(0) =u(T), v(0) =v(T)∈L²(Ω).

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Theorem 4.1. Under the hypotheses (3.2) – (3.5), andF, G∈L²(Q_T), system (4.1) has a nonnegative weak periodic solution.

To prove this result, we introduce the function wsolution of the following linear problem

(4.2)











w∈L²(0, T;H₀¹(Ω))∩C([0, T] ;L²(Ω)) w_t−∆w=F +G inD⁰(Q_T)

w(0) = w(T)∈L²(Ω).

It is well known that (4.2) has a unique solution, see [17].

Consider now the equation

(4.3)











v ∈L²(0, T, H₀¹(Ω))∩C([0, T], L²(Ω))

v_t−∆v =J(t, x, v,∇w− ∇v) +G inD⁰(Q_T) v(0) =v(T) ∈L²(Ω).

It is clear that solving (4.1) is equivalent to solve (4.3) and setu=w−v.

Proof of Theorem4.1. We remark thatwis a supersolution of (4.3). Then by a direct application of Theorem3.1, problem (4.3) has a solution.

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