EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 272-294 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Global existence of solutions for a system modelling electromigration of ions through biological cell membranes with L1 data Nour Eddine Alaa1,∗, Fatima Aqel2 1,2 Department of Mathematics, Laboratory LAMAI, Faculty of Sciences and Technology of Marrakech, University Cadi Ayyad, Abdelkarim Elkhattabi avenue, Marrakech, Morocco. Abstract. The aim of this work is to show the existence of weak solutions and supersolutions for a nonlinear system modelling Ions migration through biological cells membranes with L1- Data. In the first step, we describe the mathematical model after that we define an approximating scheme. Under simplifying assumptions on the model equation, we prove some L1 a priori estimates, then we prove that the solution of the truncated system converges to the solution of our main problem. 2010 Mathematics Subject Classifications: 74K15, 34A34, 35A01, 35A09, 35B45, 35D30, 35K57, 54D30 Key Words and Phrases: Weak solution, Truncated functions, Supersolution and subsolution, Global existence. 1. Introduction Mathematical models is an abstract model that uses mathematical language to describe the behaviour of a system. Mathematical models are used particularly in the natural sciences and engineering disciplines such as physics, biology, and electrical engineering in order to solve a complicated or the difficult nonlinear systems [1, 8, 13, 14, 7, 2], so one of the models that we are interested in is the ions electro-migration through biological cell membranes. Recently some several authors have introduced this model [15, 12, 10, 5, 19, 11]. Concerning those who have obtained the numerical results, here are some references [3, 4, 6]. These kinds of models have been studied by many researchers in the biophysical litter- ature, [12, 10]. For more understanding this model, we will begin by a simple description of this phenomena that arise across membranes. A membrane, in simple terms, may be defined as a phase that acts as a barrier to prevent mass movement but allows restricted and/or regulated passage of one or several ∗Corresponding author. Email addresses: n.alaa@uca.ac.ma (N. Alaa), aqel.fatima@gmail.com (F. Aqel) http://www.ejpam.com 272 c© 2017 EJPAM All rights reserved. N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 273 species through it. It can be a solid or liquid containing ionized or ionizable groups, or it can be completely un-ionized. Functionally, all membranes are active when used as barriers to separate two other phases unless they are too porous or too fragile. Passing through the barrier of a cell is to move materials into and out of a cell and this is important in cell communication and normal cell function. For example the cell of the nervous system function properly. The ions water, proteins, macromolecules and Nutrients need to be able to pass in and out of the cells. The first proposal that cellular membranes might contain a lipid bilayer was made in 1925 by two Dutch scientists, E. Gorter and F. Grendel [9], these two reaserchers extracted the membrane lipids from a known number of red blood cells, corresponding to a known surface area of plasma membrane. They then determined the surface area occupied by a monolayer of the extracted lipid spread out at an air-water interface. The surface area of the lipid monolayer turned out to be twice that occupied by the erythrocyte plasma membranes, leading to the conclusion that the membranes consisted of lipid bilayers rather than monolayers. The cells of the nervous system form networks. They are like all the cells that are able to function because they can control the substances inside of the cell and out of the cell, all this materials move in and out of the cell by passing to the plasma membrane. The plasma membrane surround the cell and separate the interior (intracellular)from the exterior (extracellular) of the cell environment. The impermeability of the cell membranes is composed of a lipid bilayer, which is a universal component of all cell membrane, its role is critical because its structural com- ponents provide the barrier that marks the boundaries of a cell. The structure is called a lipid bilayer, because it is composed of two layers of fat cells organized in two sheets. The lipid bilayer is typically nanometers thick and surrounds all the cells providing the cell membrane structure. The phospholipids organize themselves in a bilayer to hide their hydrophobic tail regions and expose the hydrophilic regions to water. This organization is spontaneous, meaning it is a natural process and does not require energy. This structure forms the layer that is the wall between the inside and outside of the cell. One of the mechanisms for getting in and out of the cell, we have the diffusion across the lipid bilayer. Since membranes are held together weak forces, certain molecules can slip between the lipids in the bilayer and across from one side to the other. This spontaneous process is termed diffusion. This process allows molecules, that are small and lipophilic (lipid soluble),including most drugs, to easily enter and exit cells. More on this later. The electrochemical equilibrium of the electro-diffusion system is the result of deli- cate balance between concentration gradients and electrostatic forces and requires a true compromise; microscopic electro-neutrality does not hold in a boundary layer around the location of membrane impermeability. This implies the presence of excess positive or neg- ative changes on either side of the membrane and causes a nonzero electrostatic potential difference across the membrane. In turn, a portion of the permeable salt is excluded from the compartment confining the large, charge-carrying protein, which causes a nonzero concentration gradient across the membrane which is the key component of the biological world. N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 274 In this work, we consider a class of models of ions migration through biological cell membranes. where the concentrations satisfy the Nernst Planck flux equation, including a kinetic reaction terms and the potential is given by the Poisson equation, for all 1 ≤ i ≤ NS ∂ωi ∂t − di∆ωi −midiv(ωi∇φ) = Si(ω, φ) on QT −ε∆φ = F (ω1, .., ωNS) on QT −di ∂ωi ∂υ −miωi ∂φ ∂υ = 0 in ΣT φ(t, x) = 0 in ΣT φ(0, x) = φ0(x) on Ω ωi(0, x) = ωi,0(x) on Ω (1) where Ω denotes an open and bounded subset of RN with smooth boundary ∂Ω. For each i, ωi is the concentration of the i species which has diffusion coefficients di which are non- negative inside the channel and a valency zi. φ is the electrical potential which describes the Coulomb interaction in a mean-field approximation, mi is the electric mobility that depends on the the universal gas constant, the charge carried by a mole of each species, the diffusion coefficient and also on the local temperature. The normal exterior derivative on ∂Ω is denoted by ∂υ and ∆ denotes the Laplacian op- erator on Ω. Also, we have QT =]0, T [×Ω and ΣT =]0, T [×∂Ω with T is a nonnegative constant. We set F (ω) = NS∑ i=1 ziωi 1 + ε NS∑ i=1 ωi − f where ω = (ω1, .., ωNS), f is the fixed charges concentration and the dimensionless param- eter ε is given by √ ε = λD l , the l denotes the reference length scale and λD is the Debye screening length of the reference solution defined by the following [18] λD = ( εskT 2e2ω ) 1 2 where εs is the dielectric permittivity of the solution (roughly equal to that of the solvent) and assume to be constant, k denotes the Boltzmann constant, T the absolute temperature, e the elementary charge and ω is a reference concentration of ions. In order to describe our result and to more illustre it, we have the following example. We are interested in the suicide substrate system, represented by Walsh and al. [19] E + S k1 k−1 X →k2 Y →k3 E + P, Y →k4 Ei where E, S and P stand for enzyme, substrate, and product, respectively; X and Y , enzyme- substrate intermediates; Ei, inactivated enzyme; and the ks are positive rate constants. N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 275 We denote the concentrations of the reactants by ω1 = [E], ω2 = [S], ω3 = [X], ω4 = [Y ], ω5 = [Ei], ω6 = [P ]. Then, the basic suicide substrate reaction model becomes ∂ω1 ∂t − d1∆ω1 −m1div(ω1∇φ) = −k1ω1ω2 + k−1ω3 + k3ω4 on QT ∂ω2 ∂t − d2∆ω2 −m2div(ω2∇φ) = −k1ω1ω2 + k−1ω3 on QT ∂ω3 ∂t − d3∆ω3 −m3div(ω3∇φ) = k1ω1ω2 − (k−1 + k2)ω3 on QT ∂ω4 ∂t − d4∆ω4 −m4div(ω4∇φ) = k2ω3 − (k3 + k4)ω4 on QT ∂ω5 ∂t − d5∆ω5 −m5div(ω5∇φ) = k4ω4 on QT ∂ω6 ∂t − d6∆ω6 −m6div(ω6∇φ) = k3ω4 on QT −di ∂ωi ∂υ −miωi ∂φ ∂υ = 0 in ΣT for all 1 ≤ i ≤ 6 −ε∆φ = NS∑ i=1 ziωi 1 + ε NS∑ i=1 ωi − f on QT φ(t, x) = 0 in ΣT φ(0, x) = φ0(x) on Ω ωi(0, x) = ωi,0(x) on Ω for all 1 ≤ i ≤ 6 2. The main result 2.1. Assumptions At first, we introduce the notion of weak solution of the problem (1), so let us begin by giving some hypothesis on the nonlinearities and also the initial data. These two main properties are ensured by the following assumptions. For all i ∈ {1, ..., NS} and ∀r ∈ [0,+∞)NS , the nonnegativity of solutions is preserved if and only if the quasi-positive condition is verified (H1) Si(r1, r2, ..., ri−1, 0, ri+1, ..., rNS) ≥ 0, for all r = (r1, r2, ..., rNS) ∈ [0,+∞)NS Furthermore, we restrict ourselves to the case of nonnegative solutions satisfying the tri- angular structure, which means that (H2) : { ∑ 1≤i≤NS Si(r) ≤ C(1 + ∑ 1≤i≤NS ri) ∀r ∈ [0,+∞)NS where C ≥ 0, and ∀i = 1, ..., NS N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 276 Since we allow the nonlinearities to depend on (t, x), let us assume that for all i = 1, ..., NS (H3) :  Si : QT × [0,+∞)NS → R is measurable; Si(., 0) ∈ L1(QT ) ∃K : QT × [0,+∞)→ [0,+∞) with ∀M > 0, K(.,M) ∈ L1(QT ) and a.e (t, x) ∈ QT , ∀r, r̂ ∈ [0,+∞)NS with |r|, |r̂| ≤M, |Si(t, x, r)− Si(t, x, r̂)| ≤ K(t, x,M)|r − r̂|. Then, we make the following assumptions ωi,0 ∈ L1(Ω), such that ωi,0 ≥ 0. (2) and φ0 ∈ L∞(Ω). (3) and there exists a function Θ ∈ L∞(QT ), such that{ |F (t, x, r)| ≤ Θ(t, x) a.e. (t, x) ∈ QT , ∀r ∈ [0,∞)NS , (4) Now, we clarify in which sense we want to solve our problem. In the following, we define the notion of weak solution. Definition 1. (ω, φ) = (ω1, ..., ωNS , φ) is said to be a weak solution of (1) if, for all 1 ≤ i ≤ NS ω ∈ C([0, T ];L1(Ω)NS) ∩ L1(0, T,W 1,1(Ω)NS), φ ∈ L∞(0, T,W 1,∞ 0 (Ω)), Si(ω, φ) ∈ L1(QT ) for all v ∈ C1(QT ) such that v(T, .) = 0 − ∫ QT ωi ∂v ∂t + di ∫ QT ∇ωi∇v +mi ∫ QT ωi∇φ∇v − ∫ Ω ωi(0, x)v(0, x)dx = ∫ QT Si(ω, φ)v for all θ ∈ D(Ω) and t ∈]0, T [∫ Ω ε∇φ∇θ = ∫ Ω F (ω)θ φ(0, x) = φ0(x) in Ω ωi(0, x) = ωi,0(x) in Ω (5) The principal result of this paper is the following theorem Theorem 1. We assume that (H1) − (H3), (2), (3) and (4) hold. Then there exists a weak solution (ω, φ) of (1) satisfying ωi ≥ 0 in QT for all 1 ≤ i ≤ NS. 3. Proof of the main result In this paper, we organized the steps of our work as follows. At first, we will prove the nonnegativity and the L1 bound of solutions uniformly in time, after that, we will show that the nonlinear terms are also bounded in L1(QT ), where here we add some assumptions on the nonlinearities, in order to obtain the desire estimation. The second main purpose is to give an approximate problem using the truncated functions not only on the nonlinearities but also on the initial data, where we will be inspired from the method of Michel Pierre [16]. N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 277 3.1. Existence of global weak supersolutions for bounded L1−nonlinearities Now, we need to approximate the system (1). For this, we truncate the nonlinear terms Si as follows Sni = TnoSi where the truncated function Tn : R→ R is given by Tn(σ) = σ if σ ∈ (−σn, n), Tn(σ) = −σn if σ < −σn and Tn(σ) = n if σ > n where σn = (NS)n. Also we need to truncate the initial data, so, we set ωni,0 = inf{ωi,0, n} for all i = 1, ..., NS. The next step is to consider an approximated system of (1), namely classical solutions (ωn, φn) = (ω1,n, ..., ωNS,n, φn) of for all 1 ≤ i ≤ NS ∂ωi,n ∂t − di∆ωi,n −midiv(ωi,n∇φn) = Sni (ωn, φn) on QT −di ∂ωi,n ∂υ −miωi,n ∂φn ∂υ = 0 in ΣT −ε∆φn = F (ωn) on QT φn(t, x) = 0 in ΣT φn(0, x) = φ0(x) on Ω ωi,n(0, x) = ωni,0(x) on Ω (6) Where Sni are essentially truncations of the nonlinearities Si and ωni,0 tends to ωi,0 in L1(Ω). Moreover, we assume that Sni have the same properties (H1)− (H3) as Si, and we choose the nonlinearities in such a way that they will be uniformly bounded for each n. First of all, we will need to prove the nonnegativity of ωn, for that we introduce the function Zn = (Z1,n, Z2,n, ..., ZNS,n) which is defined by Zi,n = ωi,ne mi di φn 1 ≤ i ≤ NS and we have, pi,n = e mi di φn and qi,n = 1 pi,n , where the terms (qi,n)1≤i≤NS , (pi,n)1≤i≤NS are uniformly bounded by a constant that not depends on n. Then, the concentrations (Zi,n)1≤i≤NS and the potential φn will satisfy the following system  ∂(qi,nZi,n) ∂t − didiv(qi,n∇Zi,n) = Sni (Zn, φn) in QT −ε∆φn = F (qnZn) in QT ∂Zi,n ∂υ = 0 on ΣT φn(t, x) = 0 on ΣT φn(0, x) = φ0(x) on Ω Zi,n(0, x) = Zni,0(x) on Ω (7) N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 278 where Sni = TnoŜi and the nonlinearities Ŝni are defined in RNS by Ŝi(r) = Ŝi(r1, r2, .., rNS) = { Si(r1, r2, ..., rNS) if (r1, r2, ..., rNS) ∈ [0,+∞)NS Si(r1, ..., rj−1, 0, rj+1, ..., rm) if rj ≤ 0. (8) Now, we introduce the function sign− defined on R by sign−r = { −1 if r < 0 0 if r ≥ 0 as sign− is an increasing function, we consider the convex function jε ∈ C2(R) such that j ′ ε(r)→ sign−r when ε→ 0 Also we put v = j ′ ε(Zi) as a test function in (7), then we have∫ T 0 ∫ Ω ∂(qi,nZi,n) ∂t j ′ ε(Zi,n) = −di ∫ T 0 ∫ Ω qi,n∇Zi,n∇(j ′ ε(Zi,n)) + ∫ T 0 ∫ Ω Sni (Zn, φn)j ′ ε(Zi,n) we denote by I1 and I2 the two members in the right side of previous equality, and by using the convexity of the function jε, we deduce that I1 = −di ∫ T 0 ∫ Ω qi,n∇Zi,n∇(j ′ ε(Zi,n)) = −di ∫ T 0 ∫ Ω qi,n|∇Zi,n|2j ” ε (Zi,n) ≤ 0. Concerning the second member I2, we define the second term I2, then, we deduce lim ε→0 I2 = lim ε→0 ∫ T 0 ∫ Ω Sni (Zn, φn)j ′ ε(Zi,n) = lim ε→0 ∫ [Zi,n≥0] Sni (Zn, φn)j ′ ε(Zi,n) + lim ε→0 ∫ [Zi,n<0] Sni (Zn, φn)j ′ ε(Zi,n) = lim ε→0 ∫ [Zi,n<0] Sni (Zn, φn)j ′ ε(Zi,n). By using (H1), we have lim ε→0 I2 = − ∫ [Zi,n<0] Sni (Zn, φn) = − ∫ [Zi,n<0] Tn(Si(Z1,n, ..., Zi−1,n, 0, Zi+1,n, ..., ZNS,n, φ)) ≤ 0 Then, we obtain lim ε→0 ∫ T 0 ∫ Ω ∂(qi,nZi,n) ∂t j ′ ε(Zi,n) ≤ 0 N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 279 which means that by passing to the limit, we obtain∫ T 0 ∫ Ω ∂(qi,nZi,n) ∂t sign−(Zi,n) ≤ 0 therefore ∫ T 0 ∫ Ω ∂(qi,nZi,n)− ∂t ≤ 0 this implies the following inequality∫ Ω (qi,nZi,n)−(t, x) ≤ ∫ Ω (qi,nZi,n)−(0, x) as (qi,nZi,n)(0, x) ≥ 0 for almost everywhere then we deduce∫ Ω (qi,nZi,n)−(t, x) ≤ 0. Finally (qi,nZi,n)−(t, x) = 0 and then Zi,n ≥ 0, for all i = 1, ..., NS. 3.1.1. A priori estimate First, we start by proving the following lemmas, where we are going to use the fact that ωi,n = qi,nZi,n. Lemma 1. Assume that (H2) and (7) are satisfied. Then we have the following result∫ Ω ∑ 1≤i≤NS (qi,nZi,n)(t) ≤ etC ∫ Ω ∑ 1≤i≤NS (qi,0Zi,0) + k(etC − 1) Proof. We sum the NS equations. Then, we have∑ 1≤i≤NS ∂(qi,nZi,n) ∂t − div( ∑ 1≤i≤NS diqi,n∇Zi,n) = ∑ 1≤i≤NS Sni (qnZn) and by using (H2), we have the existence of a positive constant denoted by C such that∑ 1≤i≤NS ∂(qi,nZi,n) ∂t − div( ∑ 1≤i≤NS diqi,n∇Zi,n) ≤ C(1 + ∑ 1≤i≤NS qi,nZi,n) Now, we set Vn(t) = ∑ 1≤i≤NS (qi,nZi,n)(t) , so by integrating on Ω, we obtain ∫ Ω ∂Vn(t) ∂t − ∫ ∂Ω ∑ 1≤i≤NS diqi,n ∂Zi,n ∂υ ≤ C ∫ Ω (1 + Vn(t)) N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 280 Since ∂Zi,n ∂υ = 0 for all i = 1, ..., NS, we have∫ Ω ∂Vn(t) ∂t ≤ ∫ Ω C[1 + Vn(t)] Here, we integrate over (0, t), for each t in the existence interval, we get∫ QT ∂ ∂s (Vn(s)e−sC) ≤ ∫ QT Ce−sC∫ Ω Vn(t)e−tC ≤ ∫ Ω Vn(0) + 1 C ∫ Ω C(1− e−tC) and we put k = meas(Ω), which give us the following estimate∫ Ω Vn(t) ≤ etC ∫ Ω Vn(0) + k(etC − 1). According to the definition of the initial data, It follows that the total mass ∫ Ω Vn(t) is bounded on any interval. Remark 1. Let φn be the unique solution of the elliptic problem −ε∆φn = F (qnZn) on QT φn(t, x) = 0 on ΣT φn(0, x) = φ0(x) on Ω, (9) where φn is the solution of the Poisson equation. Lemma 2. There exists a constant C depends only on T and on the L∞−norm of φ0, such that ||φn||L∞(0,T ;W 1,∞ 0 (Ω)) ≤ C. Proof. We have ∀t ∈]0, T [, φn is the unique solution of the elliptic problem (9) Where φn satisfies φn(t, x) = ∫ Ω H(s, x)θn(t, s)ds and θn is given by θn(t, s) = F (t, s, qnZn), s ∈ Ω, where H denotes the Green’s function associated to (9) . Then we have ||F (t, s, qnZn)||L∞(QT ) ≤ C hence ||φn||L∞(0,T ;W 1,∞ 0 (Ω)) ≤ C. N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 281 Concerning the nonlinearities (Sni )1≤i≤NS , we may indeed show the L1 bounded of those nonlinearities Sni for all T . The proof needs to give more restrictive assumptions on the nonlinear terms (Sni )1≤i≤NS . We assume that There exists a lower triangular invertible matrix A = (ai,j)1≤i,j≤NS with nonnegative coefficients, such that ∃b ∈ (0,+∞)NS , ∀(t, x, r) ∈ (0, T )× Ω× [0,+∞)NS AS(t, x, r) ≤ (1 + ∑ 1≤i≤NS ri)b (10) Where S(r) = (S1(r), S2(r), ..., SNS(r)). Proposition 1. Assume that (H1) and (10) hold. Then, if (Zn, φn) is solution of (7) on (0, T ), there exists a nonnegative constant denoted by C such that, for all 1 ≤ i ≤ NS and for all n ≥ 1 ∫ QT |Sni (Zn, φn)|dtdx ≤ C < +∞. (11) Proof. We denote by C0 any constant depending only on the initial data and T . Then for all t ∈ [0, T ], we have ∫ Ω(qi,nZi,n)(t) ≤ C0 for all 1≤ i ≤ NS. Now, we take the equation verified by qi,nZi,n and we sum the NS equations to obtain that, for 1 ≤ i ≤ NS and for 1 ≤ j ≤ i, we have i∑ j=1 ai,j ∂(qj,nZj,n) ∂t − i∑ j=1 ai,j [djdiv(qj,n∇Zj,n)] = i∑ j=1 ai,jS n j (Zn, φn) we multiply this equation by ϕ = 1 and integrating on QT . Indeed, we have∫ QT i∑ j=1 ai,j ∂(qj,nZj,n) ∂t − ∫ ΣT i∑ j=1 ai,jdjqj,n ∂Zj,n ∂υ dσ = ∫ QT i∑ j=1 ai,jS n j (Zn, φn) we use the boundary conditions, then we obtain∫ QT i∑ j=1 ai,j ∂(qj,nZj,n) ∂t = ∫ QT i∑ j=1 ai,jS n j (Zn, φn), therefore i∑ j=1 ai,j ∫ Ω (qj,nZj,n)(T ) = i∑ j=1 ai,j ∫ QT Snj (Zn, φn) + i∑ j=1 ai,j ∫ Ω (qj,nZj,n)(0, x) the nonnegativity of solutions gives us − i∑ j=1 ai,j ∫ QT Snj (Zn, φn) ≤ i∑ j=1 ai,j ∫ Ω (qj,nZj,n)(0, x). (12) N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 282 Now, we use (10). This leads us to the following estimate∫ QT hi(qnZn) ≤ i∑ j=1 ai,j ∫ Ω (qj,nZj,n)(0, x) + ∫ QT bi(1 + NS∑ i=1 qi,nZi,n) (13) where hi(qnZn) = − i∑ j=1 ai,jS n j (Zn, φn) + bi(1 + NS∑ i=1 qi,nZi,n) By using (12) and (13), we obtain || i∑ j=1 ai,jS n j (Zn, φn)||L1(QT ) ≤ C. Therefore, for 1 ≤ i ≤ NS ||Sni (Zn, φn)||L1(QT ) ≤ C. Before continuing the proof of the main result. First of all, we define the following the following set D = {ψ ∈ C∞(Q̄T ); ψ ≥ 0; ψ(0, T ) = 0} (14) or, we choose all the Dirichlet condition as follows D = {ψ ∈ C∞(Q̄T ) ; ψ ≥ 0; ψ(., T ) = 0, ψ = 0 on ΣT }. (15) Definition 2. The function (ω, φ) = (ω1, ..., ωNS , φ) is called a supersolution of problem (1), if ω ∈ C([0, T ];L1(Ω)NS) ∩ L1(0, T ;W 1,1(Ω)NS), φ ∈ L∞(0, T ;W 1,∞ 0 (Ω)), S(ω, φ) ∈ L1(QT )NS , and for all ψ ∈ D, − ∫ Ω ωi,0ψ(0) + ∫ QT [−ψtωi + di∇ωi∇ψ +miωi∇φ∇ψ] ≥ ∫ QT Si(ω, φ)ψ for all θ ∈ D(Ω) and t ∈]0, T [∫ Ω ε∇φ∇θ = ∫ Ω F (ω)θ φ(0, x) = φ0(x) in Ω (16) Theorem 2. Let (ωn, φn) = (ω1,n, ..., ωNS,n, φn) be a nonnegative solution to the ap- proximate system (6) satisfying (11) and the hypothesis of Theorem 1. Then up to a subsequence of (ωn) also denoted by ωn converges in L1(QT )NS and almost everywhere in QT to a supersolution of system (1) given by (16), which is equivalent to the following definition Z ∈ C([0, T ];L1(Ω)NS) ∩ L1(0, T ;W 1,1(Ω)NS), φ ∈ L∞(0, T ;W 1,∞ 0 (Ω)), S(Z, φ) ∈ L1(QT )NS , and for all ψ ∈ D, − ∫ Ω(qi,0Zi,0)ψ(0) + ∫ QT [−ψt(qiZi) + diqi∇Zi∇ψ] ≥ ∫ QT Si(Z, φ)ψ for all θ ∈ D(Ω) and t ∈]0, T [∫ Ω ε∇φ∇θ = ∫ Ω F (qZ)θ φ(0, x) = φ0(x) in Ω (17) N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 283 Proof. (Existence of Supersolution) Lemma 3. [17] Let Ω be an open bounded subset of RN with smooth boundary and ωi,n solution of (6). Then for every T > 0, the mapping (ωni,0, S n i ) ∈ L1(Ω)× L1(QT )→ ωi,n ∈ L1(QT ). (18) is compact from L1(Ω) × L1(QT ) into L1(QT ), and even into L1(0, T ;W 1,1(Ω)), and for the compactness of the trace, we use the continuity of the trace operator from W 1,1(Ω) into L1(∂Ω). Then the trace mapping (ωni,0, S n i )→ (ωi,n)|ΣT ∈ L1(ΣT ) is also compact. According to the a priori estimate (11) and to the compactness lemma, there exists ω ∈ L1(QT )NS with ∇ω ∈ [L1(QT )N ]NS such that, up to a subsequence, one may assume that { ωn → ω in L1(QT )NS and a.e in QT , ∇ωn → ∇ω in [L1(QT )N ]NS ω ∈ L1(0, T ;W 1,1(Ω))NS . (19) Now, we need to prove that the nonlinearities (Sni )1≤≤NS and (Si)1≤i≤NS belong to L1. So, first we define the function ηnM such that ηnM = sup 0≤|r|≤M, 1≤i≤NS |Sni (t, x, r)− Si(t, x, r)| where ηnM → 0 almost everywhere in QT and ηnM → 0 in L1(QT ). Indeed, since ηnM ≤ sup 0≤|r|≤M, 1≤i≤NS χ[−σn 0. Then, for all k > 0 and n ≥ 1 (di − ε) ∫ [|qi,nZi,n|≤k] |qi,n∇Zi,n|2 ≤ Ck2 + k[ ∫ Ω |qi,0Zi,0|+ ∫ QT |Sni (Zn, φn)|]. (22) where C is a constant depending only on ε, |Ω| and T . Proof. We may suppose Sni is a regular function. To show how the estimate (22) is easy to obtain, we introduce the function jk(r) = ∫ r 0 Tk(s)ds where Tk(s) is the projection of s onto [−k, k]. Multiplying the following equation by Tk(qi,nZi,n) ∂(qi,nZi,n) ∂t − didiv(qi,n∇Zi,n) = Sni (Zn, φn) and integrating by parts on QT , we get∫ QT ∂t(jk(qi,nZi,n))− di ∫ QT Tk(qi,nZi,n)div(qi,n∇Zi,n) = ∫ QT Tk(qi,nZi,n)Sni (Zn, φn) Using now the boundary conditions to obtain∫ QT ∂t(jk(qi,nZi,n)) + di ∫ QT ∇Tk(qi,nZi,n)(qi,n∇Zi,n) = ∫ QT Tk(qi,nZi,n)Sni (Zn, φn) which implies that∫ QT ∂t(jk(qi,nZi,n)) + di ∫ QT T ′ k(qi,nZi,n)|qi,n∇Zi,n|2 = ∫ QT Tk(qi,nZi,n)Sni (Zn, φn) +mi ∫ QT T ′ k(qi,nZi,n)(qi,n∇Zi,n)(qi,nZi,n)∇φn the integration over (0, T ) yields to the equality above∫ Ω jk(qi,nZi,n)(t) + di ∫ QT T ′ k(qi,nZi,n)|qi,n∇Zi,n|2 = ∫ Ω jk(q n i,0Z n i,0) + ∫ QT Tk(qi,nZi,n)Sni (Zn, φn) + mi ∫ QT T ′ k(qi,nZi,n)(qi,n∇Zi,n)(qi,nZi,n)∇φn. N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 285 Since T ′ k(qi,nZi,n) = 1 for all |qi,nZi,n| ≤ k and by using the following estimates jk(qi,nZi,n)(t) > 0, Tk(qi,nZi,n) ≤ k and jk(q n i,0Z n i,0) ≤ k|qni,0Zni,0|, we obtain di ∫ [|qi,nZi,n|≤k] |qi,n∇Zi,n|2 ≤ k[ ∫ QT |Sni (Zn, φn)|+ ∫ Ω |qni,0Zni,0|] + mi ∫ QT T ′ k(qi,nZi,n)(qi,n∇Zi,n)(qi,nZi,n)∇φn. As ||∇φn||L∞(QT ) ≤ C, and by using Young’s inequality di ∫ [|qi,nZi,n|≤k] |qi,n∇Zi,n|2 ≤ k[ ∫ QT |Sni (Zn, φn)|+ ∫ Ω |qni,0Zni,0|] + Cε ∫ [|qi,nZi,n|≤k] |qi,nZi,n|2 + ε ∫ [|qi,nZi,n|≤k] |qi,n∇Zi,n|2 consequently (di − ε) ∫ [|qi,nZi,n|≤k] |qi,n∇Zi,n|2 ≤ k[ ∫ QT |Sni (Zn, φn)|+ ∫ Ω |qi,0Zi,0|] + Cεk 2. Continuing the proof of Theorem 2 Now, we fix η ∈ (0, 1) and we introduce Vi,n = qi,nZi,n + ∑ 1≤j≤NS j 6=i η(qj,nZj,n), also we denote Ui,n = Tk(Vi,n). Since we need to differentiate twice Tk, we replace Tk by a C2− regularized function, such that Tk(r) = r if 0 ≤ r ≤ k − 1 T ′ k(r) = 0 if r ≥ k 0 ≤ T ′ k(r) ≤ 1 if r ≥ 0 −1 ≤ T ” k (r) ≤ 0 if r ≥ 0 when k → +∞, we have Tk(r) → r a.e T ′ k(r) → 1 a.e T ” k (r) → 0 a.e this enables us to state the main result of this section which means that in order to finish the proof of Theorem 2, we propose to pass to the limit when n tends to infinity, then N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 286 η → 0 and after that k → +∞, for this we need to use the hypothesis on the truncated function. Our goal now is to continue the proof of the second theorem. The first step is to fix an η ∈ (0, 1), Then, for all i = 1, ..., NS, we set Ci,n = ∑ j 6=i qj,nZj,n, Vi,n = qi,nZi,n + ηCi,n, Ui,n = Tk(Vi,n). First, we have −∆Ui,n = −div(T ′ k(Vi,n)∇Vi,n) = −T ” k (Vi,n)|∇Vi,n|2 − T ′ k(Vi,n)∆Vi,n and ∂Ui,n ∂t = T ′ k(Vi,n) ∂Vi,n ∂t . Then, we obtain ∂Ui,n ∂t − di∆Ui,n = T ′ k(Vi,n) ∂Vi,n ∂t − diT ′ k(Vi,n)∆Vi,n − diT ” k (Vi,n)|∇Vi,n|2 = T ′ k(Vi,n)[ ∂Vi,n ∂t − di∆Vi,n]− diT ” k (Vi,n)|∇Vi,n|2 = T ′ k(Vn)[ ∂(qi,nZi,n) ∂t + η ∂Ci,n ∂t − di∆(qi,nZi,n)− ηdi∆(Ci,n)] − diT ” k (Vi,n)|∇Vi,n|2 and we have ∂(qi,nZi,n) ∂t − di∆(qi,nZi,n) = ∂(qi,nZi,n) ∂t − didiv(qi,n∇Zi,n)− didiv(Zi,n∇qi,n) = Sni (Zn, φn)− didiv(Zi,n∇qi,n) also, for j = 1, ..., NS, and j 6= i we get ∂(qj,nZj,n) ∂t − di∆(qj,nZj,n) = Snj (Zn, φn) + (dj − di)div(qj,n∇Zj,n)− didiv(Zj,n∇qj,n). This yields to the following ∂Ui,n ∂t − di∆Ui,n = T ′ k(Vi,n)[Sni (Zn, φn) + η ∑ j 6=i Snj (Zn, φn)− didiv(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) +η ∑ j 6=i (dj − di)div(qj,n∇Zj,n)]− diT ” k (Vi,n)|∇Vi,n|2 Therefore ∂Ui,n ∂t −di∆Ui,n = [Yi,n+ηXi,n]−diT ′ k(Vi,n)div(Zi,n∇qi,n+η ∑ j 6=i Zj,n∇qj,n)−diT ” k (Vi,n)|∇Vi,n|2 N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 287 where Yi,n = T ′ k(Vi,n)[Sni (Zn, φn) + η ∑ j 6=i Snj (Zn, φn)] Xi,n = T ′ k(Vi,n) ∑ j 6=i (dj − di)div(qj,n∇Zj,n) we may write for ψ ∈ D :∫ QT ψ[ ∂Ui,n ∂t − di∆Ui,n] + di ∫ QT ψT ′ k(Vi,n)div(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) = ∫ QT ψ[Yi,n + ηXi,n]− di ∫ QT ψT ” k (Vi,n)|∇Vi,n|2. After an integration by parts, we obtain − ∫ Ω ψ(0)Ui,n(0) + ∫ QT [−ψtUi,n + di∇ψ∇Ui,n]− di ∫ ΣT ψ ∂Ui,n ∂υ −di ∫ QT ∇(ψT ′ k(Vi,n))(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) +di ∫ ΣT ψT ′ k(Vi,n)(Zi,n∂υqi,n + η ∑ j 6=i Zj,n∂υqj,n) = ∫ QT ψ[Yi,n + ηXi,n]− di ∫ QT ψT ” k (Vi,n)|∇Vi,n|2. Using the homogeneous Neumann boundary conditions, it is clear that − ∫ Ω ψ(0)Ui,n(0) + ∫ QT [−ψtUi,n + di∇ψ∇Ui,n]− di ∫ QT ∇(ψT ′ k(Vi,n))(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) = ∫ QT ψ[Yi,n + ηXi,n]− di ∫ QT ψT ” k (Vi,n)|∇Vi,n|2, (23) this lead us to the following − ∫ Ω ψ(0)Ui,n(0) + ∫ QT [−ψtUi,n + di∇ψ∇Ui,n]− di ∫ QT ∇ψT ′k(Vi,n)(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) −di ∫ QT ψT ” k (Vi,n)∇Vi,n(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) ≥ ∫ QT ψ[Yi,n + ηXi,n] (24) Where −di ∫ QT ψT ” k (Vi,n)|∇Vi,n|2 ≥ 0. Then (24) could be written as follows − ∫ Ω ψ(0)Ui,n(0) + ∫ QT [−ψtUi,n + di∇ψ∇Ui,n]− di ∫ QT ∇ψT ′k(Vi,n)(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) +mi ∫ QT ψT ” k (Vi,n)∇Vi,n(Zi,nqi,n∇φn) ≥ ∫ QT ψ[Yi,n + ηXi,n]− ηdi ∫ QT ψT ” k (Vi,n)∇Vi,n( ∑ j 6=i Zj,n∇qj,n) (25) We keep k and η fixed. We know that Ui,n converges in L1(QT ) and a.e to Ui,k where Ui,k = Tk(Vi), Vi = qiZi + η ∑ j 6=i qjZj N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 288 and then it is clear that the reaction terms converge only a.e. The point is that, Since T ′ k(r) = 0 for r > k then, Yi,n is equal to zero on the set [Vi,n > k]. But on the complement of this set, we have qi,nZi,n ≤ k, ∀j 6= i, qj,nZj,n ≤ k η By using the dominated convergence theorem, we can find that as n→ +∞ Yi,n → Yi,k = T ′ k(Vi)[Si(Z, φ) + η ∑ j 6=i Sj(Z, φ)] in L1(QT ). In addition we have ∇Ui,n converges in L1(QT ). So from here on, everything looks good but this is not sufficient, in other words we are not able yet to pass to the limit in (25) and still to control the terms −di ∫ QT ψT ” k (Vi,n)∇Vi,n( ∑ j 6=i Zj,n∇qj,n) and ∫ QT ψXi,n, so this is the main point of the following lemma Lemma 5. There exists C depending only on k, ψ and the initial data such that | ∫ QT ψXi,n| ≤ Cη −1 2 where η < 1. Proof. We have Xi,n = T ′ k(Vi,n) ∑ j 6=i (dj −di)div(qj,n∇Zj,n) and for ψ ∈ D, we integrate by parts on QT , then we use the boundary conditions to obtain∫ QT ψXi,n = ∫ QT ψT ′ k(Vi,n)[ ∑ j 6=i (dj − di)div(qj,n∇Zj,n)] = − ∫ QT ∇(ψT ′ k(Vi,n))[ ∑ j 6=i (dj − di)qj,n∇Zj,n] therefore − ∫ QT ψXi,n = ∫ QT [∇ψT ′k(Vi,n) + ψT ” k (Vi,n)∇Vi,n][ ∑ j 6=i (dj − di)qj,n∇Zj,n] In the following, we denote by C > 0 any constant depending only on the initial data and k, ψ but not n, η, then, we use Holder’s inequality, this yields | ∫ QT ∇ψT ′k(Vi,n)(qj,n∇Zj,n)| ≤ C{ ∫ [Vi,n≤k] |qj,n∇Zj,n|2} 1 2 ||∇ψ||L2(QT ) and we have | ∫ QT ψT ” k (Vi,n)∇Vi,n(qj,n∇Zj,n)| ≤ C||ψ||L∞(QT ){ ∫ [Vi,n≤k] |qj,n∇Zj,n|2} 1 2 { ∫ [Vi,n≤k] |∇Vi,n|2} 1 2 N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 289 and here we bound the last term of those inequalities as follows, first, we have ∇Vi,n = qi,n∇Zi,n + η[ ∑ j 6=i qj,n∇Zj,n]− [ mi di qi,nZi,n + η[ ∑ j 6=i mj dj qj,nZj,n]]∇φn Note that [Vi,n ≤ k] is included in [qi,nZi,n ≤ k], [qj,nZj,n ≤ k η ] for all j 6= i . From lemma 2 and by using the result of lemma 4, we have∫ [Vi,n≤k] |qi,n∇Zi,n|2 ≤ C, ∀j 6= i, ∫ [Vi,n≤k] |qj,n∇Zj,n|2 ≤ C η this implies | ∫ QT ψT ” k (Vi,n)∇Vi,n(qj,n∇Zj,n)| ≤ Cη− 1 2 . Hence | ∫ QT ∇ψT ′k(Vi,n)(qj,n∇Zj,n)| ≤ Cη− 1 2 finally we obtain the desired result, which means that | ∫ QT ψXi,n| ≤ Cη −1 2 still now to bound the first term, so first of all, we have −di ∫ QT ψT ” k (Vi,n)∇Vi,n( ∑ j 6=i Zj,n∇qj,n) = di ∫ QT ψT ” k (Vi,n)∇Vi,n( ∑ j 6=i mj dj Zj,nqj,n∇φn) By applying the same steps as before, we obtain | − di ∫ QT ψT ” k (Vi,n)∇Vi,n( ∑ j 6=i Zj,n∇qj,n)| ≤ Cη −1 2 . Now, thanks to the boundedness of φn in L∞(0, T,W 1,∞ 0 (Ω)), we conclude the existence of φ belongs to L∞(0, T,W 1,∞ 0 (Ω)), such that ∇φn → ∇φ for the topology σ(L∞(QT ), L1(QT )). (26) Since T ′ k has a compact support and ||T ′k||L∞(QT ) ≤ 1 and T ′ k(Vi,n) tends to T ′ k(Vi) a.e in QT , we get T ′ k(Vi,n)∇φn → T ′ k(Vi)∇φ for the topology σ(L∞(QT ), L1(QT )). Next, Let us show that T ′ k(Vi,n)(qi,nZi,n)∇φn → T ′ k(Vi)(qiZi)∇φ in D′(QT ) N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 290 For this reason, we will prove that T ′ k(Vi,n)(qi,nZi,n)∇φn → T ′ k(Vi)(qiZi)∇φ for the topology σ(L1(QT ), L∞(QT )). (27) So let v ∈ L∞(QT ), we have∫ T 0 ∫ Ω ((qi,nZi,n)T ′ k(Vi,n)∇φn − (qiZi)T ′ k(Vi)∇φ)vdxdt = ∫ T 0 ∫ Ω ((qi,nZi,n)− (qiZi))T ′ k(Vi,n)∇φnvdxdt + ∫ T 0 ∫ Ω (qiZi)(T ′ k(Vi,n)∇φn − T ′ k(Vi)∇φ)vdxdt Concerning the first term, we see that | ∫ T 0 ∫ Ω ((qi,nZi,n)−(qiZi))T ′ k(Vi,n)∇φnvdxdt| ≤ ||v||L∞(QT )||∇φn||L∞(QT )||(qi,nZi,n)−(qiZi)||L1(QT ) then by using the L1 convergence of qi,nZi,n, we obtain∫ T 0 ∫ Ω ((qi,nZi,n)− (qiZi))T ′ k(Vi,n)∇φnvdxdt→ 0 Since T ′ k(Vi,n)∇φn converges to T ′ k(Vi)∇φ for the topology σ(L∞(QT ), L1(QT )), we get the following result T ′ k(Vi,n)(Zi,n∇qi,n + η ∑ j 6=i Zj,n∇qj,n) converges to T ′ k(Vi)(Zi∇qi + η ∑ j 6=i Zj∇qj) for the topology σ(L1(QT ), L∞(QT )). Otherwise, we know that from (19) and (26), we obtain −ε∆φn → −ε∆φ in D ′ (QT ). Furthermore, F (t, x, qnZn)→ F (t, x, qZ) a.e in QT According to (4) and by applynig the Lebesgue convergence theorem, we obtain −ε∆φn(t, .)→ −ε∆φ(t, .) = F (t, ., qZ) strongly in L1(Ω). Now, let us look at the convergence of the term mi ∫ QT T ” k (Vi,n)∇Vi,n(Zi,nqi,n∇φn)ψ. First, we can notice that on the set [Vi,n ≤ k] ⊂ [qi,nZi,n ≤ k], the terms Zi,n∇qi,n are bounded in L∞(QT ). Indeed, on one hand |qi,nZi,n| ≤ k and on the other hand ||∇φn||L∞(QT ) ≤ C (see lemma 2). Then, we deduce that for all 1 ≤ i ≤ NS, ||qi,nZi,n∇φn||L∞(QT ) ≤ C(k). which imply that for a subsequence still denoted by Zi,n∇qi,n Zi,n∇qi,n ⇀ β converges weak-* in L∞(QT ). Such that β ∈ L∞(QT ). Then, since T ” k has a compact support and by using the pointwise convergence of Tk”(Vi,n) N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 291 to T ” k (Vi) as n tends to zero, the bounded character of T ” k and the weak-* convergence of Zi,n∇qi,n, we conclude that T ” k (Vi,n)Zi,n∇qi,n ⇀ T ” k (Vi)β converges weak-* in L∞(QT ). Moreover, we recall that ∇Vi,n converges to ∇Vi strongly in L1(QT ) and a.e in QT . Where ∇Vi = ∇(qiZi + η ∑ 1≤i≤NS,j 6=i qjZj). Finally, we obtain the desired result T ” k (Vi,n)∇Vi,nZi,n∇qi,n → T ” k (Vi)∇Viβ for the topology σ(L1(QT ), L∞(QT )). Now, we can let n tends to +∞ in (24). By using the strong convergence in L1(QT ) of Tk(Vi,n) to Tk(Vi) and the L1 convergence of the initial data, we obtain − ∫ Ω ψ(0)Ui,k(0) + ∫ QT [−ψtUi,k + di∇ψ∇Ui,k]− di ∫ QT ∇ψT ′k(Vi)(Zi∇qi + η ∑ j 6=i Zj∇qj) −di ∫ QT ψT ” k (Vi)∇Viβ ≥ ∫ QT ψT ′ k(Vi)[Si(Z, φ) + η ∑ j 6=i Sj(Z, φ)] + ε(i, η, k, ψ) (28) where ε(i, η, k, ψ) ≥ −C(k, ψ)η −1 2 so that lim η→ inf 0 ε(i, η, k, ψ) ≥ 0. Let η tends to 0 in the above inequality. Since Ui,k = Tk(Vi) = Tk(qiZi + η ∑ j 6=i (qjZj)) converges to Tk(qiZi) strongly in L1(QT ) and T ′ k(qiZi + η ∑ j 6=i (qjZj)) remains uniformly bounded by 1 and T ′ k(qiZi + η ∑ j 6=i (qjZj)) tends a.e to T ′ k(qiZi). Then, by passing to the limit in the sense of distributions, we found − ∫ Ω ψ(0)Tk(qi,0Zi,0) + ∫ QT [−ψtTk(qiZi) + di∇ψ∇Tk(qiZi)]− di ∫ QT ∇ψT ′k(qiZi)(Zi∇qi) −di ∫ QT ψT ” k (Vi)∇Viβ ≥ ∫ QT ψT ′ k(Vi)Si(Z, φ). (29) Finally, let k → +∞. Since T ” k (qiZi) → 0 a.e in QT , Tk(qiZi) tends to (qiZi) in L1(QT ), T ′ k(qiZi) tends a.e to 1 and Si ∈ L1(QT ), we can pass to the limit and obtain − ∫ Ω (qi,0Zi,0)ψ(0)+ ∫ QT [−ψt(qiZi)+di∇(qiZi)∇ψ] +mi ∫ QT ∇ψ(qiZi∇φ) ≥ ∫ QT ψSi(Z, φ) (30) finally, we obtain − ∫ Ω (qi,0Zi,0)ψ(0) + ∫ QT [−ψt(qiZi) + diqi∇Zi∇ψ] ≥ ∫ QT Si(Z, φ)ψ (31) N. Alaa, F. Aqel / Eur. J. Pure Appl. Math, 10 (2) (2017), 272-294 292 3.2. Global existence of weak solutions Theorem 3. Let us consider system (1) together with (14) or (15), with (H1) and (H3), with Zi,0 ∈ L1(Ω) such that Zi,0 ≥ 0, for all 1 ≤ i ≤ NS. We assume the structure (H1) + (H2) hold together with the a priori estimate (11). Then, system (1) has a weak solution on (0, T ) (i.e equality holds in (16) or (17)). Proof. By Theorem 2, up to a subsequence, the approximate solution (qnZn) converge to a weak supersolution. Now, we try to prove that this supersolution is also a subsolution and according to this two results we will deduce that the supersolution (resp. subsolution) is a weak solution of the main system that we have considered in the beginning. From the compactness lemma 3, we have (ωn,∇ωn)→ (ω,∇ω) in [L1(QT )]NS × [[L1(QT )]N ]NS . Where ωn = qnZn and ω = qZ. Then, for all ψ ∈ D nonnegative test function and all i = 1, ..., NS, we have − ∫ Ω (qi,0Zi,0)ψ(0)+ ∫ QT [−ψt(qiZi)+di∇(qiZi)∇ψ]+mi ∫ QT (qiZi)∇φ∇ψ ≥ ∫ QT Si(Z, φ)ψ where Si(Z, φ) ∈ L1(QT ). In the following step, we introduce the notations and Wn = NS∑ i=1 qi,nZi,n, Tn = NS∑ i=1 diqi,nZi,n, Yn = NS∑ i=1 miqi,nZi,n, W = NS∑ i=1 qiZi, T = NS∑ i=1 diqiZi, Y = NS∑ i=1 miqiZi, We sum the equations of the approximate problem then we get − ∫ Ω ψ(0)Wn(0) + ∫ QT [−ψtWn +∇Tn∇ψ] + ∫ QT Yn∇φn∇ψ = ∫ QT NS∑ i=1 Sni (Zn, φn)ψ and from structure (H2), we have NS∑ i=1 Sni (Zn, φn) ≤ C(1 +Wn) which means C(1 +Wn)− NS∑ i=1 Sni (Zn, φn) ≥ 0 and we already know that Wn → W in L1(QT ) Sni (Zn, φn) → Si(Z, φ) a.e in QT REFERENCES 293 using Fatou’s lemma, this leads to∫ QT −ψ NS∑ i=1 Si(Z, φ) ≤ lim n→ inf +∞ ∫ QT −ψ NS∑ i=1 Sni (Zn, φn) (32) thus − ∫ Ω ψ(0)W (0) + ∫ QT [−ψtW +∇T∇ψ] + ∫ QT Y∇φ∇ψ ≤ ∫ QT ψ NS∑ i=1 Si(Z, φ) finally, we obtain the suitable result. 4. References ACKNOWLEDGEMENTS The authors thank the readers of European Journal of Pure and Applied Mathematics, for making our journal successful. References [1] N. Alaa and F. Aqel. 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