\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 04, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/04\hfil Existence of global solutions] {Existence of global solutions for reaction diffusion systems modeling the electrodeposition of alloys with initial data measures} \author[N. Alaa, F. Aqel \hfil EJDE-2017/04\hfilneg] {Noureddine Alaa, Fatima Aqel} \address{Nour Eddine Alaa \newline Laboratory LAMAI, Faculty of Science and Technology of Marrakech, University Cadi Ayyad \\ B.P. 549, Abdelkarim Elkhattabi avenue, Marrakech - 40000, Morocco} \email{n.alaa@uca.ac.ma} \address{Fatima Aqel \newline Laboratory LAMAI, Faculty of Science and Technology of Marrakech, University Cadi Ayyad \\ B.P. 549, Abdelkarim Elkhattabi avenue, Marrakech - 40000, Morocco} \email{aqel.fatima@gmail.com} \thanks{Submitted September 28, 2016. Published January 5, 2017.} \subjclass[2010]{74K15, 35D30, 34A34, 35A01, 35A09, 54D30} \keywords{Electrodeposition of alloys; weak solution; truncated functions; \hfill\break\indent global solution} \begin{abstract} In this work, we are interested in the mathematical model of reaction diffusion systems. The originality of our study is to work with concentrations appearing in reactors together with measure initial data. To validate this model, we prove the existence of global weak solutions. The ``j'' technique introduced by Pierre and Martin \cite{m1} is suitable for this type of solutions. However, its adaptation has some new technical difficulties that we have to overcome. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} A reaction diffusion system is a set of partial differential equations that can be understood to represent molecules reacting and diffusing over some space. They arise quite naturally in systems consisting of many chemical reactions or interacting components and they are widely used to describe some chemical, physical and biological systems where the principal ingredients of all these models are the second Fick's law and rate equation. In this paper we discuss the existence of global solutions for a reaction diffusion system modeling electrodeposition process with initial data measures. We start by a simple history about this model problem. Electrodeposition is an attractive method for the fabrication of thin metal films and layered structures \cite{g1}. Structures with a wide range of compositions, morphologies, and functionalities can be deposited by varying the large number of experimental parameters available in electrochemical methods. In addition, electrochemistry offers a low-cost alternative to more involved deposition techniques. Also, electrodeposition is an alternative method for fabricating nanoengineered materials. In general, electropolating is both an art and a science. Although based on several technologies and sciences, including chemistry, physics, chemical and electrical engineering. So the purposes of electropolating for which articles are electropolated are the appearence, protection, special surface properties and engineering or mechanical properties. To understand this process, we will begin by a brief definition of this model, where we include some previous researches obtained by \cite{a3,b5,d1,d2,l2,m2, m3} where the discussion in the following will be limited to one group of alloy codeposition. The classification of alloy codeposition systems developed by Brenner \cite{b5}, including normal, anomalous, and induced codeposition. In our work, we will be interested in the second group. The next paragraph is devoted to explain why most of researchers have classified the Nickel-Iron electrodeposition as anomalous codeposition. Fe-Ni deposition is classified as anomalous codeposition \cite{b5} because the discharge rate of the more noble compenent Ni inhibited, causing of the less noble metal $Fe$ at a higher deposition rate than nickel. According to Dahms \cite{d1} and Dahms and Croll \cite{d2}, Fe-Ni anomalous codeposition is due to the local $pH$ rise at the interface due to the parallel parasitic hydrogen evolution reaction. There are also some recent mathematical models that have been proposed for expalining this phenomena, include those by Hessami and Tobias\cite{h1}, Grande and Talbot \cite{g1} and Matlosz \cite{m2}, the mechanism of single-metal deposition of iron suggested by Bockris and Al \cite{b3} and of nickel suggested by Matulis and Slizys \cite{m3}. Hessami and Tobias \cite{h1} assumed that the electrodeposition of Fe-Ni occurs as a result of the reduction of both the bivalent metal ions, Ni$^{2+}$ and Fe$^{2+}$, the monohydroxide ions $Fe(OH)^{+}$ and Ni(OH)$^{+}$. According to this model, the dominant mechanism in the electrodeposition process was the reduction of divalent or bivalent ions rather than monohydroxide species. In this work, the ionic species of interest are: H$^+$, OH$^-$, Fe$^{2+}$, Ni$^{2+}$, FeOH$^{+}$ and NiOH$^{+}$ where the following homogeneous reactions are considered \begin{gather*} \text{H$_2$O $\rightleftharpoons$ H$^++$OH$^{-}$}\\ \text{FeOH$^+ \rightleftharpoons$ Fe$^{2+}+$OH$^{-}$}\\ \text{NiOH$^{+} \rightleftharpoons$ Ni$^{2+}+$OH$^{-}$} \end{gather*} Then, we have Grande and Talbot \cite{g1} who proposed a one dimensionnel diffusion model, where they determined the effect of buffuring and the hydrolysis reactions on predicted surface $PH$ and deposit composition. Their model includes the assumption that anomalous deposition of nickel and iron occurs due to the electrodeposition of their respective monohydroxide species. On the other hand, we have the study of Alaa et al. \cite{a2} who studied the existence of global solutions for a Model of Nickel Iron alloy electrodeposition on rotating disk with quadratic nonlinearities in one dimension space. There Model addresses dissociation, diffusion, electomigration, convection and deposition of multiple ion species, where they have presented a generalization of \cite{a3} to ensure the global existence of classical solutions and their positivity, where in \cite{a3}, the same researchers proved the existence and the positivity of weak solutions for their model problem without no restriction of growth on the nonlinear terms. In this work, we study the existence of global solutions in more general case. So, instead of studying the problem of electrodeposition of Nickel-Iron alloy, we will consider that our model is composed of $NS$ different species. We are interested in particular to the study of the following reaction diffusion systems of the type \begin{equation} \label{e1.1} \begin{gathered} \frac{\partial \omega _i}{\partial t}-d_i\Delta \omega _i-m_i \operatorname{div}(\omega _i\nabla \phi )=S_i(\omega ) \quad\text{in } Q_T, \\ -\Delta \phi =F(\omega) \quad\text{in } Q_T, \\ -d_i\frac{\partial \omega _i}{\partial \upsilon }-m_i\omega _i \frac{\partial \phi }{\partial \upsilon }=0\quad\text{on }\Sigma _T,\\ \phi (t,x)=0\quad\text{on }\Sigma _T, \\ \phi (0,x)=\phi _0(x) \quad \text{in }\Omega , \\ \omega _i(0,.)=\mu _i\quad\text{in }\Omega , \end{gathered} \end{equation} Where $\Omega $ denotes an open and bounded subset of $\mathbb{R}^{N}$, with smooth boundary $\partial \Omega$. The normal exterior derivative on $\partial\Omega$ is denoted by $\partial _{\upsilon }$ and we have $Q_T=]0,T[\times \Omega $, $\Sigma _T=]0,T[\times \partial \Omega $ with $T$ is a nonnegative constant. The components $\omega _i$'s represent the concentrations of $NS$ species considered during the electrodeposition, $z_i$ are the charges and $m_i$ is the electrical mobility, $d_i$ is the diffusion coefficients associated to each one of our species and $\phi $ designates the electric potential, $S_i$ are the reaction terms or production rate and $F$ is a bounded function in $L^{\infty}(Q_T)$ which depends on the concentrations $\omega=(\omega_1,\omega_2,\dots ,\omega_{NS})$ and also on the fixed charge concentration. We suppose that $S_i$ depends continuously on the $\omega _i$'s. We also assume that $d_i$ are nonnegative constants for each $i=1,\dots ,NS$. The layout of this work is as follows. We begin in the second section by defining notation and essential concepts, after that we consider our problem and we expose the principal result. The third paragraph is devoted to the proof of the principal result by passing through an approximate problem then obtaining the appropriate estimations of $\omega _i$, $\phi $ and $S_i$ to pass to the limit and prove that the solution of truncated system converges to the solution of our model problem \eqref{e1.1}. \subsection{Notion of weak solution} Throughout this paper we make the following assumptions: for all $i=1,\dots ,NS$, \begin{equation} \label{e1.2} \mu _i\in M_{b}^{+}(\Omega ), \end{equation} and \begin{equation} \label{e1.3} \phi_0\in L^\infty(\Omega). \end{equation} There exists $\Theta\in L^\infty(Q_T)$, such that \begin{equation} \label{e1.4} \begin{gathered} |F(t,x,r)|\leq \Theta(t,x) \quad \text{a.e. }(t,x)\in Q_T,\\ \forall r\in[0,+\infty)^{NS}. \end{gathered} \end{equation} The total mass control is preserved on time if for all $r\in[0,+\infty)^{NS}$ \begin{equation} \label{e1.5} \sum_{i=1}^{NS}S_i(r)\leq K\Big(\sum_{i=1}^{NS}r_i\Big)+N \quad \text{ where }K, N\geq 0\,. \end{equation} The nonnegativity of solutions is preserved if and only if the quasi-positivity condition is satisfied \begin{equation} \label{e1.6} S_i(r_1,\dots ,r_{i-1},0,r_{i+1},\dots ,r_{NS})\geq 0. \end{equation} After all, to expect the existence of global solutions in time, more structure must be required on $S_i$. Additional assumptions usually come from the underlying model. We assume the existence of a lower triangular invertible matrix $Q=(q_{ij})_{1\leq i, j\leq NS}$ with nonnegative coefficients, such that \begin{equation} \label{e1.7} \begin{gathered} \exists L,M\in (0,+\infty)^{NS}, \; \forall (t,x,r)\in (0,T)\times \Omega\times [0,+\infty)^{NS}, \\ QS(t,x,r)\leq L\Big(\sum_{1\leq i\leq NS}r_i\Big)+M\,, \end{gathered} \end{equation} where $S(r)=(S_1(r),S_2(r),\dots ,S_{NS}(r))$. To finish this paragraph, we recall the following notation and definitions: \begin{gather*} C_0^{\infty }(Q_T)=\{\varphi :Q_T\to \mathbb{R}\text{ indefinitely derivable with compact support in }Q_T\},\\ C_{b}(\Omega)=\{\varphi :\Omega \to \mathbb{R}\text{ a continuous and bounded function in }\Omega \},\\ M_{b}(\Omega )=\{\mu_i \text{ bounded Radon measure in }\Omega \},\\ M_{b}^{+}(\Omega )=\{\mu_i \text{ bounded nonnegative Radon measure in }\Omega \}. \end{gather*} \begin{definition} \label{def1.1} \rm Let $\omega_i\in C(]0,T[;L^1(\Omega))$ and $\mu_i\in M_b(\Omega)$. We say that $\omega_i(0,.)=\mu_i$ in $M_b(\Omega)$ if for every $\varphi\in C_b(\Omega)$, \begin{equation*} \lim_{t\to 0} \int_{\Omega}\omega_i(t,x)\varphi dx=\langle \mu_i,\varphi\rangle, \end{equation*} \end{definition} Now, we introduce the notion of weak solution that we will use in this work. \begin{definition} \label{def1.2} \rm A weak solution of problem \eqref{e1.1}, is a couple of functions $(\omega ,\phi )=(\omega _1,\omega _2,\dots ,\omega _{NS},\phi )$ such that $\omega\in C(]0,T[;L^{1}(\Omega )^{NS})\cap L^{1}(0,T;W^{1,1}(\Omega )^{NS})$, $\phi \in L^{\infty }(0,T;W_0^{1,\infty}(\Omega ))$ and $S_i(\omega )\in L^{1}(Q_T)$. For all $1\leq i\leq NS$, the couple $(\omega,\phi)$ satisfies \begin{equation} \label{e1.8} \begin{gathered} \frac{\partial \omega _i}{\partial t}-d_i\Delta \omega _i -m_i\operatorname{div}(\omega _i\nabla \phi )=S_i(\omega ) \quad\text{ in }D'(Q_T), -\Delta \phi =F(\omega) \quad \text{in } D'(Q_T) \\ \phi (0,x)=\phi _0(x) \quad \text{in } \Omega \\ \omega _i(0,.)=\mu _i \quad \text{in } M_{b}(\Omega ) \end{gathered} \end{equation} \end{definition} We mention here that if $\mu=(\mu_1,\mu_2,\dots ,\mu_{NS})$ belongs to $L^2(\Omega)^{NS}$. Then, we could talk about a strong solution which is defined in the following sense. \begin{definition} \label{def1.3} \rm A strong solution of problem \eqref{e1.1} is a couple of functions $(\omega ,\phi )=(\omega _1,\omega _2,\dots ,\omega _{NS},\phi )$ such that $\mu \in L^2(\Omega )^{NS}$, $\mu_i \geq 0$, $\omega \in C([0,T];L^2(\Omega )^{NS})\cap L^2(0,T;H^1(\Omega )^{NS})$, $\phi \in L^{\infty }(0,T;W_0^{1,\infty}(\Omega ))$ and $S_i(\omega )\in L^{1}(Q_T)$ for all $1\leq i\leq NS$ and satisfying: \begin{equation} \label{e1.9} \begin{gathered} \text{for all $v\in C^{1}(Q_T)$ such that $v(T,.)=0$,} \\ \begin{aligned} &-\int_{Q_T}\omega _i\frac{\partial v}{\partial t} +d_i\int_{Q_T}\nabla \omega _i\nabla v+m_i\int_{Q_T}\omega _i\nabla \phi \nabla v-\langle \mu_i,v(0,x)\rangle\\ &=\int_{Q_T}S_i(\omega )v \quad\text{for all $\theta \in D(\Omega )$ and $t\in ]0,T[$} \end{aligned} \\ \int_{\Omega } \nabla \phi \nabla \theta =\int_{\Omega }F(\omega) \theta \\ \phi (0,x)=\phi _0(x) \quad \text{in }\Omega , \\ \omega _i(0,x)=\mu _i(x)\quad \text{in } \Omega \,. \end{gathered} \end{equation} \end{definition} \section{Main results} \begin{theorem} \label{thm2.1} We assume that \eqref{e1.2}, \eqref{e1.3}, \eqref{e1.4}, \eqref{e1.5}, \eqref{e1.6} and \eqref{e1.7} hold. Then the problem \eqref{e1.1} has a weak solution $(\omega ,\phi )$ satisfying $\omega_i \geq 0$ in $Q_T$, for all $i=1,\dots ,NS$. \end{theorem} \subsection{Proof of main result} \subsubsection*{Approximating scheme} To approximate our problem, we truncate the initial data $(\mu_i)_{1\leq i\leq NS}$ as follows \begin{equation} \label{e2.1} \mu _i^{n}\in C_0^{\infty }(\Omega )\quad \text{such that }\mu _i^{n}\geq 0, \quad \|\mu _i^{n}\|_{L^{1}(\Omega )}\leq \|\mu _i\|_{M_{b}(\Omega )},\; \mu _i^{n}\to \mu _i \end{equation} in $M_{b}(\Omega )$. To each nonlinearity $S_i$ we associate the function $\hat{S}_i$ such that \begin{equation} \label{e2.2} \begin{aligned} \hat{S}_i(r) &=\hat{S}_i(r_1,r_2,\dots ,r_{NS}) \\ &=\begin{cases} S_i(r_1,r_2,\dots ,r_{NS})& \text{if }(r_1,r_2,\dots ,r_{NS})\in [ 0,+\infty )^{NS} \\ S_i(r_1,\dots ,r_{j-1},0,r_{j+1},\dots ,r_{m}) &\text{if }r_{j}\leq 0\,. \end{cases} \end{aligned} \end{equation} Also we consider the truncated function $\eta _n\in C_0^{\infty }(\mathbb{R}^{NS})$, satisfying \begin{gather*} 0 \leq \eta_n\leq 1, \\ \eta _n(r)=\begin{cases} 1 &\text{if } |r|\leq n \\ 0 &\text{if } |r|\geq n+1. \end{cases} \end{gather*} Then for $r \in \mathbb{R}^{NS}$ we define \begin{equation} \label{e2.3} S_i^{n}(r)=\eta _n(r)\hat{S}_i(r)\quad \text{for all }n\geq 1 \end{equation} \begin{proposition}[\cite{a4}] \label{prop2.2} We assume that \eqref{e1.1}, \eqref{e1.2}, \eqref{e1.3}, \eqref{e1.4}, \eqref{e1.5} and \eqref{e1.6} are satisfied. Then for each $n$ there exists a strong solution $(\omega _n,\phi _n)$ of \begin{equation} \label{e2.4} \begin{gathered} \frac{\partial \omega _{i,n}}{\partial t}-d_i\Delta \omega _{i,n}-m_i \operatorname{div}(\omega _{i,n}\nabla \phi _n)=S_i^{n}(\omega _n) \quad\text{in } Q_T, \\ -\Delta \phi_n =F(\omega_n)\quad\text{in }Q_T,\\ -d_i\frac{\partial \omega _{i,n}}{\partial \upsilon }-m_i\omega _{i,n} \frac{\partial \phi _n}{\partial \upsilon }=0\quad\text{in }\Sigma _T, \\ \phi _n(t,x)=0\quad\text{in }\Sigma _T, \\ \phi _n(0,x)=\phi _0(x)\quad\text{in }\Omega , \\ \omega _{i,n}(0,.)=\mu _i^{n}\quad\text{in }\Omega. \end{gathered} \end{equation} Here the solutions $\omega_{i,n}$ are nonnegative. \end{proposition} \section{A priori estimate} In the following section, we give some a priori estimates for proving that under suitable additional assumptions, the solution of \eqref{e2.4} converges to a solution of \eqref{e1.1} as $n$ tends to $\infty $. To ensure the existence of solution, we use the main structural assumptions on the nonlinearities. First we need to prove the following lemma. \begin{lemma} \label{lem3.1} Under the assumptions of Proposition \ref{prop2.2} we have \begin{equation} \label{e3.1} \int_{\Omega }\sum_{i=1}^{NS}|\omega _{i,n}(t)|\leq e^{tK}[\sum_{i=1}^{NS}\|\mu _i\|_{M_b(\Omega)}+NK^{-1}(1-e^{-tK})] \end{equation} \end{lemma} \begin{proof} We have \[ \frac{\partial }{\partial t}\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big) -\sum_{i=1}^{NS}d_i\Delta \omega _{i,n} -\operatorname{div}\Big(\sum_{i=1}^{NS}m_i\omega _{i,n}\nabla \phi _n\Big) =\sum_{i=1}^{NS}S_i^{n}(\omega_n) \] from \eqref{e1.5}, we there exist $K, N\geq 0$ such that \begin{equation} \label{e3.2} \frac{\partial }{\partial t}\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big) -\sum_{i=1}^{NS}d_i\Delta \omega _{i,n} -\operatorname{div}\Big(\sum_{i=1}^{NS}m_i\omega _{i,n}\nabla \phi _n\Big) \leq K\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big)+N. \end{equation} Integrating over $\Omega $ and using the boundary conditions, we obtain \[ \partial_t \int_{\Omega }W_n(t)\leq \int_{\Omega }[K(W_n)+N], \] where $W_n=\sum_{i=1}^{NS} \omega_{i,n}$. Hence \[ \int_{\Omega}\frac{\partial (W_ne^{-sK})}{\partial t} \leq \int_{\Omega}Ne^{-sK}. \] Integrating the previous inequality on $[0,t]$, for all $00$, there exists a constant $C$ depending on $T$, $L_i$, $M_i$, $q_{ij}$ and $\|\mu_i\|_{M_b(\Omega)}$ for all $1\leq j\leq i\leq NS$ such that \begin{equation*} \int_{Q_T}\sum_{1\leq i\leq NS}|S_i^{n}(\omega_n)|\,dt\,dx\leq C. \end{equation*} \end{lemma} \begin{proof} We denote by $C_0$ any constant depending only on the initial data and $T$. Then for all $t\in [ 0,T]$, we have $\int_{\Omega}(\omega_{i,n})(t)\leq C_0$ for all 1$\leq i\leq NS$. Now, we take the equation satisfied by $\omega_{i,n}$ and we sum the $NS$ equations to obtain that, for $1\leq i \leq NS$ and for $1\leq j\leq i$, we have \begin{equation*} \sum_{j=1}^{i}q_{ij}\frac{\partial (\omega_{j,n})}{\partial t} -\sum_{j=1}^{i}q_{ij}d_{j}\Delta\omega_{j,n} -\operatorname{div}\Big(\sum_{j=1}^{i}q_{ij}m_{j}\omega_{j,n}\nabla\phi_n\Big) =\sum_{j=1}^{i}q_{ij}S_{j}^{n}(\omega_n) \end{equation*} Integrating on $Q_T$, we have \begin{equation*} \sum_{j=1}^{i}q _{i,j}\int_{Q_T}\frac{\partial (\omega_{j,n})}{\partial t}-\sum_{j=1}^{i}q_{ij}\int_{\Sigma _T}(d_{j}\frac{\partial \omega_{j,n}}{\partial \upsilon }+m_j\omega_{j,n}\frac{\partial \phi_n}{\partial \upsilon}) =\sum_{j=1}^{i}q_{ij}\int_{Q_T}S_{j}^{n}(\omega_n)\,. \end{equation*} Then using the boundary conditions, we obtain \begin{equation*} \int_{Q_T}\sum_{j=1}^{i}q_{ij}\frac{\partial (\omega_{j,n})}{\partial t}=\int_{Q_T}\sum_{j=1}^{i}q_{ij}S_{j}^{n}(\omega_n); \end{equation*} therefore, \begin{equation*} \sum_{j=1}^{i}q_{ij}\int_{\Omega }\omega_{j,n}(T)=\sum_{j=1}^{i}q_{ij}\int_{Q_T}S_{j}^{n}(\omega_n) +\sum_{j=1}^{i}q_{ij}\int_{\Omega}\omega_{j,n}(0,x)\,. \end{equation*} The nonnegativity of solutions yields \begin{equation} \label{e3.3} -\sum_{j=1}^{i}q_{ij}\int_{Q_T}S_{j}^{n}(\omega_n) \leq \sum_{j=1}^{i}q_{ij}\int_{\Omega }\mu_{j}^n \leq \sum_{j=1}^{i}q_{ij}\|\mu_{j}\|_{M_b(\Omega)} \end{equation} This leads us to the estimate \begin{equation} \label{e3.4} \int_{Q_T}h_i(\omega_n) \leq \sum_{j=1}^{i}q_{ij}\|\mu_{j}\|_{M_b(\Omega)} +\int_{Q_T}M_i+L_i\Big(\sum_{l=1}^{NS}\omega_{l,n}\Big), \end{equation} where \begin{equation*} h_i(\omega_n)=-\sum_{j=1}^{i}q_{ij}S_{j}^{n}(\omega_n)+M_i +L_i\Big(\sum_{l=1}^{NS}\omega_{l,n}\Big) \end{equation*} From the above, we have \[ \sum_{j=1}^{i}q_{ij}S_{j}^{n}(\omega_n)=-h_i(\omega_n)+M_i+L_i(\sum_{l=1}^{NS}\omega_{l,n}). \] By using \eqref{e3.3}, \eqref{e3.4} and the previous equality, we obtain \[ \|\sum_{j=1}^{i}q_{ij}S_{j}^{n}(\omega_n)\|_{L^{1}(Q_T)} \leq C; \] therefore, for $1\leq i\leq NS$, we have $\|S_i^{n}(\omega_n)\|_{L^{1}(Q_T)}\leq C$. \end{proof} Let $\phi_n $ be the unique solution of the elliptic problem \begin{equation} \label{e3.5} \begin{gathered} - \Delta \phi_n =F(\omega_n) \quad \text{on } Q_T=(0,T)\times \Omega ,\\ \phi_n (t,x)=0\quad \text{on }\Sigma _T=(0,T)\times \partial \Omega, \\ \phi_n(0,x)=\phi_0(x)\quad \text{on }\Omega, \end{gathered} \end{equation} where $F(\omega)$ is a bounded function in $L^{\infty}(Q_T)$, $\phi_0$ is also bounded in $L^\infty(\Omega)$ and $\phi_n $ is the solution of equation \eqref{e3.5}. \begin{lemma} \label{lem3.4} There exists a constant $C$ depending on $T$ and on the $L^{\infty}$ norm of $\phi_0$, such that \begin{equation*} \|\phi_n \|_{L^{\infty }(0,T;W_0^{1,\infty }(\Omega ))}\leq C. \end{equation*} \end{lemma} \begin{proof} We have that for all $t\in ]0,T[$, the function $\phi_n$ is the unique solution of the elliptic problem \eqref{e3.5}, where \begin{gather*} \phi_n(t,x)=\int_{\Omega} G(s,x)\theta^n(t,s)ds, \\ \theta^n(t,s)=F(t,s,\omega_n), \textit{ \ }s\in\Omega \end{gather*} where $G$ denotes the Green's function associated to the Poisson equation. Then we have \[ \|F(t,s,\omega_n)\|_{L^{\infty}(Q_T)}\leq C, \] hence \[ \|\phi_n \|_{L^{\infty }(0,T;W_0^{1,\infty }(\Omega ))}\leq C. \] \end{proof} \begin{lemma} \label{lem3.5} Let $\omega_{i,n}$ be a solution of \eqref{e2.4}. Then, for every $T>0$, the mapping \begin{equation} \label{e3.6} (\mu_i^{n},S_i^{n})\in \ L^{1}(\Omega )\times L^{1}(Q_T)\mapsto \omega_{i,n}\in L^{1}(Q_T) \end{equation} is compact. Moreover, it is continuous from $L^{1}(\Omega )\times L^{1}(Q_T)$ to $C(]0,T[;L^{1}(\Omega ))$. Moreover, $(\omega_{i,n})_{1\leq i\leq NS}$ is compact in $L^1(0,T;W^{1,1}(\Omega))$ and for the trace compactness, we use the continuity of the trace operator from $W^{1,1}(\Omega)$ to $L^1(\partial\Omega)$, then the trace mapping is also compact in $L^1(\Sigma_T)$. \end{lemma} For more details, we refer the readers to \cite[Lemma 5.6]{p2}. \begin{proposition} \label{prop3.6} Under the hypothesis of Lemme 3.2, there exists $(\omega _i)_{1\leq i\leq NS}$ in $L^{1}(Q_T)$ with $\nabla\omega_i \in [L^1(Q_T)^{N}]^{NS}$ such that, up to a subsequence, we have the following convergence \begin{gather*} \omega _{i,n}\to \omega _i\quad \text{in $L^{1}(Q_T)$ and a.e. in }Q_T, \\ \nabla\omega _{i,n}\to \nabla\omega _i\quad \text{in $[L^{1}(Q_T)^N]^{NS}$ and a.e. in }Q_T. \end{gather*} \end{proposition} \section{Convergence} Now, we show that $(\omega ,\phi )$ is a solution of \eqref{e1.1}. From Proposition \ref{prop3.6}, for $i=1,\dots ,N$, we have \begin{equation} \label{e4.1} \begin{gathered} \omega _{i,n}\to \omega _i\quad \text{in }L^{1}(Q_T), \\ \nabla\omega _{i,n}\to \nabla\omega _i\quad \text{in }[L^{1}(Q_T)^N]^{NS} \end{gathered} \end{equation} and if we extract a new subsequence, then we can assume that it converges \begin{equation} \label{e4.2} \omega _{i,n}\to \omega _i\quad \text{almost everywhere in }Q_T, \end{equation} and \begin{equation} \label{e4.3} \mu _i^{n}\to \mu _i\quad \text{in }M_{b}(\Omega ) \end{equation} Since $S_i$ is continuous with respect to $\omega $ and by construction of $S_i^{n}$, we have \begin{equation} \label{e4.4} S_i^{n}(\omega _n)\to S_i(\omega )\quad \text{almost everywhere in }Q_T \end{equation} For the proof of Theorem \ref{thm2.1}, we pass to the limit in the approximate problem when $n$ tends to infinity. In fact, we need to prove that the convergence in \eqref{e4.4} holds in $L^{1}(Q_T)$. To fill up the gap between those two kinds of convergence, we state the following lemma. \begin{lemma} \label{lem4.1} Let $\sigma _n$ be a sequence in $L^{1}(Q_T)$ and $\sigma \in L^{1}(Q_T)$ such that \begin{gather} \label{e4.5} \sigma _n\to \sigma \text{ almost everywhere in }Q_T, \\ \label{e4.6} \sigma _n \text{ is uniformly integrable in } Q_T\,. \end{gather} Then $\sigma _n\to \sigma$ in $L^{1}(Q_T)$. \end{lemma} \begin{proof} Condition \eqref{e4.6} is implied by: for each $\varepsilon >0$, there exists $\theta >0$ such that \[ (K\subset Q_T\text{ measurable, } \operatorname{meas}(K)<\theta )\Longrightarrow \int_{K}|\sigma _n|<\varepsilon \quad \forall n \] In this proof, we have to show that $S_i^{n}(\omega _n)$ are not only bounded in $L^{1}(Q_T)$ but uniformly integrable. We can not realize this without an extra assumption on $(S_i)$ which is \eqref{e1.7}, which we assume in the first section. \end{proof} \begin{theorem} \label{thm4.2} Assume the conditions in Proposition \ref{prop2.2} and \eqref{e1.7} are satisfied. Then, for all $(\mu _i)_{1\leq i\leq NS}$ in $M_{b}(\Omega )$ and $\phi_0\in L^\infty(\Omega)$, there exists a weak solution $(\omega,\phi)$ to problem \eqref{e1.1}. \end{theorem} \begin{lemma}[\cite{l1}] \label{lem4.3} Let $\sigma _n$ be a sequence in $L^{1}(Q_T)$. Then the following instructions are equivalent \begin{itemize} \item[(i)] $\sigma _n$ is uniformly bounded in $L^{1}(Q_T)$; \item[(ii)] there exists a function $J:(0,\infty )\to (0,\infty )$ with $J(0^{+})=0$ and \begin{itemize} \item[(a)] $J$ is convex, $J'$ is concave, $J'\geq 0$; \item[(b)] $\lim_{r\to +\infty}\frac{J(r)}{r}=+\infty$; \item[(c)] $\sup_n\int_{Q_T}J(|\sigma _n|)<\infty$. \end{itemize} \end{itemize} \end{lemma} \textbf{Proof of Theorem \ref{thm4.2}} By Proposition \ref{prop2.2}, we have the existence of the solution $(\omega_n,\phi _n)$ to the approximate problem \eqref{e2.4}. Since $\phi _n$ is uniformly bounded in $L^{\infty }(0,T;W^{1,\infty}(\Omega ))$, we deduce the existence of $\phi \in L^{\infty}(0,T;W^{1,\infty }(\Omega ))$, such that \begin{equation} \label{e4.7} \nabla \phi _n\to \nabla \phi \quad \text{in the topology of }\sigma (L^{\infty }(Q_T),L^{1}(Q_T)) \end{equation} Next we show that \[ \omega _{i,n}\nabla \phi _n\to \omega _i\nabla \phi \quad\text{in }D'(Q_T) \] To this end, we will prove that \begin{equation} \label{e4.8} \omega _{i,n}\nabla \phi _n\to \omega _i\nabla \phi \quad\text{in the topology of }\sigma (L^{1}(Q_T),L^{\infty }(Q_T)) \end{equation} For $v\in L^{\infty }(Q_T)$, we have \begin{align*} &\int_{Q_T}v(\omega _{i,n}\nabla \phi _n-\omega _i\nabla \phi )\,dx\,dt \\ &=\int_{Q_T}v\nabla \phi _n(\omega _{i,n}-\omega _i)\,dx\,dt +\int_{Q_T}v\omega _i(\nabla \phi _n-\nabla \phi )\,dx\,dt \end{align*} Concerning the first term in this equality, we have \[ \big|\int_{Q_T}v\nabla \phi _n(\omega _{i,n}-\omega_i)\,dx\,dt\big| \leq \|v\|_{L^{\infty }(Q_T)}\|\nabla \phi _n\|_{L^{\infty }(Q_T)}\|\omega _{i,n}-\omega _i\|_{L^{1}(Q_T)} \] and by \eqref{e4.1}, we obtain \[ \int_{Q_T}v\nabla \phi _n(\omega _{i,n}-\omega_i)\,dx\,dt\to 0. \] The second term approcahes zero because $\nabla \phi _n$ converges to $\nabla \phi $ in the topology of $\sigma (L^{\infty }(Q_T),L^{1}(Q_T))$ and $v\omega _i\in L^{1}(Q_T)$. So from the convergence result, we obtain \[ \frac{\partial \omega _{i,n}}{\partial t}-d_i\Delta \omega _{i,n}-m_i \operatorname{div}(\omega _{i,n}\nabla \phi _n)\to \frac{\partial \omega _i}{\partial t}-d_i\Delta \omega _i-m_i\operatorname{div}(\omega _i\nabla \phi )\quad \text{in }D'(Q_T) \] From \eqref{e4.2} and \eqref{e4.7}, we have \[ - \Delta \phi _n\to - \Delta \phi \quad \text{in }D'(Q_T) \] Furthermore, \[ F(t,x,\omega_n)\to F(t,x,\omega) \quad\text{ a.e. in }Q_T. \] According to \eqref{e1.4} and by applying the Lebesgue theorem, we obtain \[ -\Delta \phi _n(t,.)\to - \Delta \phi (t,.) = F(t,.,\omega)\quad \text{strongly in }L^{1}(\Omega )\,. \] Now, we define $J$ as in the lemma \ref{lem4.3} where (ii)(c) is replaced by \begin{equation} \label{e4.9} \sup_n\int_{Q_T}J\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big)<\infty, \quad \sup_n\int_{\Omega}J\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\mu _{j}^{n}\big) <\infty\,. \end{equation} Which is possible by lemma \ref{lem4.3} since \[ \sum_{i=1}^{NS}\omega _{i,n}\to \sum_{i=1}^{NS}\omega _i\quad \text{in }L^{1}(Q_T) \] and \[ \sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\mu _{j}^{n} \to \sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\mu _{j}\quad \text{in }M_{b}(\Omega )\,. \] Putting \begin{equation} \label{e4.10} j(r)=\int_0^{r}\min (J'(s),(J^{\ast })^{-1}(s))ds, \end{equation} where $J^{\ast }$ is the conjugate function of $J$, $j$ satisfies (ii)(a) and (ii)(b) and we have \begin{equation} \label{e4.11} \forall r>0 \quad j(r)\leq J(r)\quad J^{\ast}(j'(r))\leq r\,. \end{equation} Our goal is to show that \begin{equation} \label{e4.12} \sup_n\int_{Q_T}j'\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij} \omega _{j,n}\Big)\Big(\sum_{i=1}^{NS}|S_i^{n}(\omega _n)|\big)<\infty\,. \end{equation} First of all, we indicate how the proof of the theorem can be completed. We see that estimation \eqref{e4.12} implies the uniform integrability of $(S_i^{n}(\omega _n))_{1\leq i\leq NS}$ in $Q_T$. Indeed, let $\varepsilon >0$ and $K$ be a measurable set of $Q_T$. Then \begin{align*} \int_{K}|S_i^{n}(\omega _n)| &\leq \int_{K\cap [ \sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\omega _{j,n}k ]}|S_i^{n}(\omega _n)| \\ &\leq I_1+I_2\,. \end{align*} For $I_1$, we find that \[ \sup_{0\leq \omega_{1,n}, \dots ,\omega_{NS,n}\leq k}|S_i^{n}(t,x,\omega_n)| \leq C(k), \] where the constant $C$ depends on $L_i$, $M_i$ for all $i=1,\dots ,NS$ and $k(\varepsilon)$, consequently \[ I_1\leq C(k(\varepsilon ))\operatorname{meas}(K)<\frac{\varepsilon }{2} \quad \text{if }\operatorname{meas}(K)<\frac{\varepsilon }{2[C(k (\varepsilon ))]}\,. \] We estimated the second integral as follows \[ I_2\leq \frac{1}{j'(k)}\int_{Q_T}j'\Big(\sum_{i=1}^{NS} \sum_{j=1}^{i} q_{ij}\omega _{j,n}\Big)|S_i^n(\omega_n)| \] Where we have used the convexity of the function $J$ and \eqref{e4.10} to obtain the desired result. Thanks to \eqref{e4.12}, this may be less than $\frac{\varepsilon }{2}$, by choosing $k=k (\varepsilon )$ sufficiently large and depend only on $\varepsilon $. This proves the uniform integrability of $S_i^{n}(\omega _n)$. We use now the almost everywhere convergence of $S_i^{n}(\omega _n)$ in $Q_T$, then, we obtain that $S_i^{n}(\omega _n)$ converges in $L^{1}(Q_T)$, which completes the proof of the theorem. Returning now to the proof of the estimate \eqref{e4.12}. We set \begin{equation} \label{e4.13} \begin{gathered} T_{1,n}=M_1+L_1\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big) -q_{1,1}S_1^n(\omega_n)\geq 0, \\ T_{2,n}=M_2+L_2\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big)-q_{2,1}S_1^n (\omega_n)-q_{2,2}S_2^n(\omega_n)\geq 0, \\ \dots \\ T_{NS,n}=M_{NS}+L_{NS}\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big)-q_{NS,1}S_1^n(\omega_n)- \dots -q_{NS,NS}S_{NS}^n(\omega_n)\geq 0 \end{gathered} \end{equation} which implies $$ \sum_{i=1}^{NS}T_{i,n}=\sum_{i=1}^{NS}M_i +\Big(\sum_{l=1}^{NS}\omega _{l,n}\Big)\sum_{i=1}^{NS}L_i -\sum_{i=1}^{NS}\sum_{j=1}^{i}q_{ij}S_{j}^n(\omega_n)\geq 0\,. $$ For every $j=1,\dots ,i$ and $i=1,\dots ,NS$ we have \[ \frac{\partial \omega _{j,n}}{\partial t}=d_{j}\Delta \omega _{j,n}+m_{j} \operatorname{div}(\omega _{j,n}\nabla \phi _n)+S_{j}^{n}(\omega _n)\,. \] We multiply the first equation by $(q_{ij})_{1\leq i,j \leq NS}$ and we sum the three equations, so we obtain \begin{align*} \frac{\partial }{\partial t}\Big(\sum_{j=1}^{i}q_{ij}\omega _{j,n}\Big) &=\Big(\sum_{j=1}^{i}q_{ij}d_{j}\Delta \omega _{j,n}\Big) +\operatorname{div}\Big(\Big(\sum_{j=1}^{i}q_{ij}m_{j}\omega _{j,n}\Big)\nabla \phi _n\Big) \\ &\quad +\sum_{j=1}^{i}q_{ij}S_{j}^{n}(\omega _n)\,. \end{align*} Then, we multiply by $j'(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\omega _{j,n})$ and integrate over $Q_T$ to obtain \begin{equation} \label{e4.14} \begin{aligned} &\int_{\Omega }j\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij} \omega _{j,n}\big)(T)+\int_{Q_T}j'\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij} \omega _{j,n}\Big)\sum_{i=1}^{NS}T_{i,n}\\ &=J_1+J_2 +\int_{\Omega }j\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\mu _{j}^{n}\Big) \end{aligned} \end{equation} where \[ J_1 =\int_{Q_T}j'\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij} \omega _{j,n}\Big)\Big[\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}d_{j} \Delta \omega _{j,n}\Big) +\operatorname{div}((\sum_{i=1}^{NS} \sum_{j=1}^{i} q_{ij}m_{j}\omega _{j,n})\nabla \phi _n)\Big] \] Concerning the term $J_2$, we have \[ J_2=\int_{Q_T}j'\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij} \omega _{j,n}\Big)\Big[\sum_{i=1}^{NS}M_i+\sum_{i=1}^{NS}L_i \Big(\sum_{l=1}^{NS}\omega _{l,n}\Big)\Big] \] To estimate this term, we use the inequality \begin{equation} \label{e4.15} j'(r).s\leq J(s)+J^{\ast }(j'(r))\leq J(s)+r, \end{equation} so that \begin{align*} J_2 &\leq \int_{Q_T}\Big[J\Big(\sum_{i=1}^{NS}M_i\Big) +\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\omega _{j,n}\Big)\Big] \\ &\quad +\int_{Q_T}\sum_{i=1}^{NS}L_i\Big[J\Big(\sum_{l=1}^{NS}\omega _{l,n}\Big) +\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i} q_{ij}\omega _{j,n}\Big)\Big] \end{align*} By using the Lemma \ref{lem3.1}, the choice of $J_2$ and \eqref{e4.9}, we see that $J_2$ is bounded independently of $n$. On the other hand, if we can control the term $J_1$. Then, from \eqref{e4.9}, \eqref{e4.14} and the estimate on $J_2$, we obtain the desired result. By definition of $T_{i,n}$ we have \begin{align*} T_n=M+L\Big(\sum_{l=1}^{NS}\omega_{l,n}\Big)-QS^n(\omega_n), \end{align*} where $T_n=(T_{1,n},\dots ,T_{NS,n})$, $L=(L_1,\dots ,L_{NS})$, $M=(M_1,\dots ,M_{NS})$ and $S^n(\omega_n)=(S_1^n(\omega_n),\dots ,S_{NS}^n(\omega_n))$. Since $Q$ is an invertible matrix, we obtain \begin{equation} \label{e4.16} S^n(\omega_n)=Q^{-1}M+\Big(\sum_{l=1}^{NS}\omega_{l,n}\Big)Q^{-1}L-Q^{-1}T_n \end{equation} which gives an estimation on the sum of nonlinearities $S_i$ for all $i=1,\dots ,NS$. For $J_1$ we have the estimate \begin{align*} J_1 &=\int_{Q_T}j'\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i}q_{ij}\omega _{j,n}\Big) \Big(\sum_{i=1}^{NS}\sum_{j=1}^{i}d_{j}q_{ij}\Delta \omega _{j,n}\Big)\\ &\quad +\int_{Q_T}j'\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i}q_{ij}\omega _{j,n}\Big) \operatorname{div}\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i}m_{j}q_{ij} \omega _{j,n}\nabla \phi _n\Big) \\ &=-\int_{Q_T}j''\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i}q_{ij} \omega _{j,n})\nabla\Big(\sum_{i=1}^{NS}\sum_{j=1}^{i}q_{ij} \omega _{j,n}\Big)[\sum_{i=1}^{NS}\sum_{j=1}^{i}d_{j}q_{ij}\nabla \omega _{j,n}\\ &\quad +\sum_{i=1}^{NS}\sum_{j=1}^{i}m_j q_{ij}\omega_{j,n}\nabla \phi _n]. \end{align*} We set $W_n=\sum_{i=1}^{NS}\sum_{j=1}^{i}q_{ij}\omega _{j,n}$ and integrate by parts to have \[ J_1 =-\int_{Q_T}j''(W_n)\nabla W_n\Big[\sum_{i=1}^{NS}\sum_{j=1}^{i}d_{j}q_{ij}\nabla \omega _{j,n} +\sum_{i=1}^{NS}\sum_{j=1}^{i}m_j q_{ij}\omega_{j,n}\nabla \phi _n\Big] \] Next, we use H\"{o}lder's and Young's inequalities to obtain \begin{equation} \label{e4.17} J_1\leq \tilde{C}\int_{Q_T}j''(W_n)\Big(\sum_{i=1}^{NS}|\omega _{i,n}|^2+\sum_{i=1}^{NS}|\nabla\omega _{i,n}|^2\Big), \end{equation} where the constant $\tilde{C}$ depends on $\max_{1\leq j\leq i}(d_{j})$, $\max_{1\leq j\leq i}(m_{j})$, $q_{ij}$, $\|\nabla \phi_n\|_{L^{\infty }(Q_T)}$, and $1\leq j\leq i\leq NS$. Since $j'$ is concave, $j''(r)\leq \frac{j'(r)}{r}$ and we have \[ \int_{Q_T}j''(W_n)\Big(\sum_{i=1}^{NS}\omega _{i,n}^2\Big) \leq \int_{Q_T}j'(W_n)\Big(\sum_{i=1}^{NS}\omega _{i,n}\Big) \] hence \[ \int_{Q_T}j''(W_n)\Big(\sum_{i=1}^{NS}\omega _{i,n}^2\Big) \leq \int_{Q_T}J\Big(\sum_{i=1}^{NS}\omega _{i,n}Big)+W_n \] which allows to say that this term is uniformly bounded. It remains to show that the first and the second term in \eqref{e4.17} are also uniformly bounded. First, we have \[ \frac{\partial \omega _{1,n}}{\partial t}-d_1\Delta \omega _{1,n}-m_1 \operatorname{div}(\omega _{1,n}\nabla \phi _n)=S_1^{n}(\omega _n)\,. \] Then we multiply by $q_{11}$ to obtain \begin{equation} \label{e4.18} \frac{\partial (q_{11}\omega _{1,n})}{\partial t}-d_1\Delta (q_{11}\omega _{1,n}) -m_1 \operatorname{div}(q_{11}\omega _{1,n}\nabla \phi _n)] =q_{11}S_1^{n}(\omega _n). \end{equation} Now, we multiply by $j'(q_{11}\omega _{1,n})$ and integrate over $Q_T$ to obtain \begin{align*} &\int_{Q_T}\frac{\partial (j(q_{11}\omega _{1,n}))}{\partial t} +d_1\int_{Q_T}j''(q_{11}\omega _{1,n})|\nabla (q_{11}\omega_{1,n})|^2\hspace{4cm} \\ &+m_1\int_{Q_T}j''(q_{11}\omega _{1,n})(q_{11}\omega _{1,n})\nabla \phi _n\nabla (q_{11}\omega _{1,n}) \\ &=\int_{Q_T}j'(q_{11}\omega _{1,n})q_{11}S_1^{n}(\omega _n), \end{align*} which implies \begin{align*} &\int_{\Omega }j(q_{11}\omega _{1,n})(T) +d_1\int_{Q_T}j''(q_{11}\omega _{1,n})|\nabla (q_{11}\omega _{1,n})|^2 \\ &\leq \int_{\Omega }j(q_{11}\mu _1^{n}) -m_1\int_{Q_T}j''(q_{11}\omega_{1,n})(q_{11} \omega _{1,n})\nabla \phi _n\nabla( q_{11}\omega _{1,n}) \\ &\quad +\int_{Q_T}j'(q_{11}\omega_{1,n})\Big[M_1 +L_1\Big(\sum_{l=1}^{NS}\omega _{l,n}\Big)\Big]\,. \end{align*} Using again H\"{o}lder's and Young's inequalities, we obtain \begin{align*} &\int_{\Omega }j(q_{11}\omega _{1,n})(T)+d_1\int_{Q_T}j''(q_{11} \omega_{1,n})|\nabla (q_{11}\omega _{1,n})|^2 \\ &\leq \int_{\Omega }j(q_{11}\mu _1^{n}) + C_{\varepsilon_1}\int_{Q_T}j''(q_{11}\omega _{1,n})(q_{11} \omega _{1,n})^2\\ &\quad +\varepsilon _1\int_{Q_T}j''(q_{11} \omega _{1,n}) |\nabla (q_{11}\omega _{1,n})|^2 +\int_{Q_T}j'(q_{11}\omega_{1,n})\Big[M_1+L_1\Big(\sum_{l=1}^{NS} \omega _{l,n}\Big)\Big]; \end{align*} therefore, \begin{align*} &\int_{\Omega }j(q_{11}\omega _{1,n})(T)+(d_1-\varepsilon _1)\int_{Q_T}j''(q_{11}\omega _{1,n})|\nabla (q_{11}\omega _{1,n})|^2 \\ & \leq \int_{\Omega }j(q_{11}\mu _1^{n}) +C_{\varepsilon _1}\int_{Q_T}j''(q_{11}\omega _{1,n})(q_{11}\omega_{1,n})^2 \\ &\quad +\int_{Q_T}j'(q_{11}\omega _{1,n})\Big[M_1+L_1\Big(\sum_{l=1}^{NS}\omega_{l,n}\Big)\Big]\,. \end{align*} Then we have \begin{align*} \int_{Q_T}j''(q_{11}\omega _{1,n})(q_{11}\omega _{1,n})^2 &\leq \int_{Q_T}j'(q_{11}\omega _{1,n})(q_{11}\omega _{1,n}) \\ &\leq \int_{Q_T}J(q_{11}\omega _{1,n})+q_{11}\omega _{1,n}\,. \end{align*} Finally, we add the condition $j''(q_{11}\omega _{1,n})\geq j''(W_n)$. 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