2.1 Preliminaries and notations

The purpose of this part is to introduce some notations and recall some basic mathematical results. we denote by $\textbf {H}^1$ the Hilbert space of functions defined by

$$ \begin{align*} \textbf{H}^1= \left\lbrace u \in H^1(\Omega_1)\times H^1(\Omega_2), u=0 \ in \ \Gamma_1 \ \textrm{and} \ \Gamma_2 \right\rbrace. \end{align*} $$

We endow it with the norm

$$ \begin{align*} \Vert u \Vert_{\textbf{H}^1}=\left( \Vert u \Vert_{H^1(\Omega_1)}^2 + \Vert u \Vert_{H^1(\Omega_2)}^2 \right)^{\frac{1}{2}}. \end{align*} $$

We designate $(\cdot , \cdot )$ as the inner product in $\textbf {H}^1$ and $\left \langle \cdot , \cdot \right \rangle $ denote the duality bracket of $\textbf {H}^1$ with its dual space $(\textbf {H}^{1})^{\star }$ .

2.2 Assumptions

For further work in this paper, we formulate the following hypotheses:

For $i= 1,\ldots , m$ , we assume that

(2.1) $$ \begin{align} k_1= \cdots=k_m=k, \hspace{1cm} \varphi_i(v_i)=D_i v_i^{r_i},\ \dfrac{(d-2)^{+}}{d} <r_i < 2. \end{align} $$

For $i= 1,\ldots , m$ , $R_i:Q_T \times [0,+\infty )^m \rightarrow \mathbb {R}$ be such as

(2.2) $$ \begin{align} &\textbf{Regularity}: \nonumber\\ &\begin{cases} R_i \ \text{is measurable},\\ \forall T>0, R_i(\cdot,\cdot, 0) \in L^1(Q_T),\\ \exists K : [0,+\infty)\rightarrow [0,+\infty) \ \text{nondecreasing such that: }\\ \vert R_i(x,t, v) - R_i(x,t, \tilde{v}) \vert \leq K(M) \sum \limits_{j=1}^{m} \vert v_j - \tilde{v}_j \vert \\ \text{for all} \ M>0 \ \text{for all} \ v, \tilde{v} \in (0,M)^m \ \text{and a.e.} \ (x,t)\in Q_T. \end{cases} \end{align} $$

We assume that the nonlinearities $R_i$ satisfy the properties:

(2.3) $$ \begin{align} &\hspace{-12pt}\textbf{Quasi-positivity:} \nonumber\\ &\hspace{-12pt}\textbf{(P):} \begin{cases} R_i(t,x,v_1, \ldots, v_{i-1},0,v_{i+1}, \ldots,v_m) \geq 0 \\ \text{for all} \ v=(v_1,\ldots,v_m)\in [0,+\infty)^m \ \mathrm{a.e.} \ (t,x)\in Q_T. \end{cases} \end{align} $$

(2.4) $$ \begin{align} &\quad\ \kern1.5pt\textbf{Control of mass:} \nonumber\\ &\kern1.5pt\ \quad \textbf{(M):} \begin{cases} \text{there exists} \ (\xi_1, \ldots,\xi_m) \in (0, +\infty)^m \ \text{such as} \\ \forall v=(v_1,\cdot,v_m)\in [0,+\infty)^m, \ \text{ for a.e.} \ (t,x)\in Q_T, \ \sum \limits_{j=1}^{m} \xi_jR_j(x,t,v) \leq 0. \end{cases} \end{align} $$

(2.5) $$ \begin{align} &\hspace{-63pt}\textbf{Sub-quadratic growth:} \nonumber\\ &\hspace{-63pt}\begin{cases} \forall i = 1, \ldots, m, \forall v= (v_1,\cdot,v_m)\in [0,+\infty)^m, \\ \vert R_i(v) \vert \leq C\left( 1 + \sum \limits_{j=1}^{m} v_j^{r_i+1} \right). \end{cases} \end{align} $$

Remark 1 Note that all our given results extend immediately if (M) is replaced by

(2.6) $$ \begin{align} \textbf{(M')} \begin{cases} \forall \ v=(v_1,\ldots,v_m)\in [0,+\infty)^m, \text{for a.e.} \ (x,t)\in Q_T, \ \sum \limits_{j=1}^{m} R_j(x,t,v) \leq C\sum \limits_{j=1}^{m} v_j + h(x,t), \\ \text{for some} \ C>0 \, h \in L^1_{loc}([0,+\infty); L^2(\Omega)^+). \end{cases} \end{align} $$

The properties (P) and (M) or (M’) exist naturally in applications. In fact, evolutionary reaction–diffusion systems are mathematical models for evolutionary phenomena undergoing both spatial diffusion and (bio)chemical reactions. In these models, the unknown functions are generally densities, concentrations, and temperatures, so their nonnegativity is required. In addition, it is often necessary to control the total mass, sometimes even the preservation of the total mass is naturally guaranteed by the model. Interest in these models has grown recently, particularly for applications in biology, ecology, and population dynamics. We refer to [Reference Pierre and Schmitt29] for examples of reaction–diffusion systems with properties (P) and (M) or (M’).

We now present the notion of solution and also the main result that is the subject of our mathematical analysis in this paper.

2.3 Main result

We define our space of test functions as

$$ \begin{align*} W_T& =\left\lbrace ({}^1 \Psi,^2 \Psi)\in C^{\infty}([0,T]\times \overline{\Omega_1}) \times C^{\infty} ([0,T]\times \overline{\Omega_2}), \Psi \geq 0 ,\right. \\ &\left. \hspace{1cm} \Psi(\cdot,T)=0, \Psi =0 \ in \ \Sigma_T, \nabla {}^1\Psi.\nu_1 = \nabla {}^2\Psi.\nu_1 =k_i({}^2\Psi - {}^1\Psi) \ in \ [0,T]\times \Gamma \right\rbrace, \end{align*} $$

where $\Psi = \begin {cases} {}^1\Psi , \hspace {0,5cm} in \ \Omega _1, \\ {}^2\Psi , \hspace {0,5cm} in \ \Omega _2\end {cases}. $ We now introduce the notion of weak solution of problem (1.1) and also the existence and regularity result of this solution.

Definition 1 Given $v_{0,i}\in L^1(\Omega )\cap (\textbf {H}^1)^{\star }, v_{0,i} \geq 0$ , $i = 1,\ldots ,m$ , a global weak solution of system (1.1) is a nonnegative function $\textbf {v}=(v_1,\ldots , v_m)$ such that for all $T> 0$ and $i=1, \ldots , m$ , $v_i \in C([0,T],L^1(\Omega )),\ \varphi _i(v_i) \in L^1(0,T,W^{1,1}), \ R_i(v) \in L^1(Q_T)$ , and

(2.7) $$ \begin{align} - \int_{\Omega} v_i(\cdot,0)\Psi(\cdot,0)dx - \int_{0}^{T}\int_{\Omega} v_i\partial_t\Psi+ \varphi_i(v_i).\Delta\Psi dxdt = \int_{0}^{T}\int_{\Omega} R_i(v)\Psi dxdt \end{align} $$

for all $\Psi \in W_T$ .

3.1 The approximate reaction–diffusion system

In this subsection, we introduce an approximation of the system (1.1).

We first approximate the initial data and the reaction terms as follows:

(3.1) $$ \begin{align} v^n_{0,i}:=\inf\left\lbrace v_{0,i},n \right\rbrace \hspace{1cm} \text{and} \hspace{1cm} R^{n }_i :=\dfrac{R_i}{1 + \frac{1}{n }\sum_{1\leq j \leq m} \vert R_j \vert}. \end{align} $$

For each fixed n, $v^{n }_{0,i}(x)\in L^{\infty }(\Omega )$ , $i=1,\ldots ,m$ , and converges to $v_{i,0}$ in $L^1(\Omega )\cap (\textbf {H}^1)^{\star }$ . We consider the following regularised system:

(3.2) $$ \begin{align} &\begin{cases} \textrm{for } i=1,\ldots,m,\\ \textrm{for all } T> 0, v_i^n \in L^{\infty}(Q_T)^{+}, \varphi_i(v_i^n) \in L^{2}(0,T;H^{1}(\Omega)), \\ \partial_tv^{n }_i - \Delta \varphi_i( v^{n }_i )= R^{n }_i(v^{n }_1, \ldots, v^{n }_m ) & \text{in} \hspace{1cm} Q_T:=(0,T)\times \Omega, \\ v^{n }_i =0 & \text{in} \hspace{1cm} \Sigma_{T}:=(0,T)\times (\Gamma_1 \cup \Gamma_2), \\ \partial_{\nu_1} {{}^1\varphi_i( v^{n }_i)}=\partial_{\nu_1} {{}^2\varphi_i( v^{n }_i)}= k_i({}^2\varphi_i( v^{n }_i) - {}^1\varphi_i( v^{n }_i)) & \text{in} \hspace{1cm} \Sigma_{T,\Gamma}:=(0,T)\times \Gamma,\\ v^{n }_i(0,x)= v^{n }_{0,i}(x) \geq 0 & \text{in} \hspace{1cm} \Omega, \end{cases} \end{align} $$

where the approximate nonlinearities $R_i^n$ are essentially “truncations” of the $R_i$ ’s. More precisely, we will assume that $R^n_i$ is locally Lipschitz continuous and satisfies (2.2) with $K(\cdot )$ independent of n, and (2.3)–(2.5) with h independent of n, and is in $L^{\infty }(Q_T \times \mathbb {R}^{m})$ . Moreover, thanks to our choice, we have $\Vert R^n_i \Vert _{L^{\infty }} \leq n$ for each fixed n. Therefore, the approximate system (3.2) has a nonnegative bounded global solution (see, e.g., [Reference Laamri and Pierre20, Lemma 2.3] and [Reference Laamri18] or [Reference Vázquez35] for more details). Let us denote

(3.3) $$ \begin{align} \epsilon^n_M:=\max \limits_{1 \leq i \leq n} \sup \limits_{v \in [0,M]^m}\vert R^n_i(v) - R_i(v) \vert, \end{align} $$

where $ v=(v_1,\ldots ,v_m)$ . Then, we check that

(3.4) $$ \begin{align} \epsilon^n_M \leq \dfrac{C_M m}{n} \textrm{ and } \epsilon^n_M \longrightarrow 0 \ \textrm{in } L^1(Q_T) \text{ and a.e. in } Q_T \textrm{ as } n\rightarrow +\infty. \end{align} $$

For $i=1,\ldots ,m$ , $v^{n }_{0,i}(x)\in L^{\infty }(\Omega )$ and converges to $v_{i,0}$ in $L^1(\Omega )\cap (\textbf {H}^1)^{\star }$ .

3.2 The key estimate

Lemma 3.1 Assume that, for $1 \leq i \leq m$ , $v_{i,0} \in L^{1}(\Omega )\cap (\textbf {H}^1)^{\star }$ and $h \in L^1_{loc}([0,+\infty ); L^2(\Omega ))$ under the assumption (2.6). Then, for all nonnegative regular functions $v_i$ solution of (3.2) with $k_i=k, i= 1,\ldots , m$ , there exists $C(T)>0$ such that

(3.5) $$ \begin{align} \Vert v^n_i \Vert_{L^{r_i+1}(Q_T)} \leq C(T)\left( 1+ \Vert v_{i,0}^n \Vert_{(\textbf{H}^1)^{\star} } \right). \end{align} $$

Moreover, with the assumption (2.5), the $R_i^{n}(v^n)$ are uniformly bounded in $L^1(Q_T)$ , more precisely, for all $T>0$ , there exists a constant $C'> 0$ independent of n such that

(3.6) $$ \begin{align} \Vert R_i^{n}(v^n) \Vert_{L^1(Q_T)} \leq C'. \end{align} $$

Proof By summing the m equations of the regularized system (3.2) and then using (2.6) of Remark 1, we obtain

(3.7) $$ \begin{align} \partial_t \left( \sum \limits_{i=1}^{m} v_i^n \right) - \Delta \left( \sum \limits_{i=1}^{m} \varphi_i( v_i^n) \right) \leq C \sum \limits_{i=1}^{m} v_i^n +h. \end{align} $$

By multiplying this inequality by $e^{-Ct}$ , and observing that

$ e^{-Ct} \sum \limits _{i=1}^{m} v_i^n = \partial _t \left ( e^{-Ct} \sum \limits _{i=1}^{m} v_i^n \right ) + C \sum \limits _{i=1}^{m} v_i^n $ , we obtain

(3.8) $$ \begin{align} \partial_t \left( e^{-Ct} \sum \limits_{i=1}^{m} v_i^n \right) - \Delta \left( e^{-Ct} \sum \limits_{i=1}^{m} \varphi_i( v_i^n) \right) \leq e^{-Ct}h \leq h. \end{align} $$

By integrating from 0 to t, we obtain

(3.9) $$ \begin{align} e^{-Ct} \sum \limits_{i=1}^{m} v_i^n - \Delta \left(\int_0^t e^{-Cs} \sum \limits_{i=1}^{m} \varphi_i( v_i^n(s,\cdot)ds \right)\leq \sum \limits_{i=1}^{m} v_{i,0}^n + \int_0^t h(s,\cdot)ds. \end{align} $$

Let us set $\widehat {U}(t)= \displaystyle e^{-Ct} \sum \limits _{i=1}^{m} v_i^n(t,\cdot )$ and $\displaystyle \widehat {V}(t)= \int _0^t e^{-Cs} \sum \limits _{i=1}^{m} \varphi _i( v_i^n(s,\cdot )ds $ , then we obtain the following problem:

(3.10) $$ \begin{align} \begin{cases} \displaystyle \widehat{U} - \Delta \widehat{V} \leq \widehat{U}_0 + \int_0^t h(s,\cdot)ds, & \text { in } Q_{T},\\ \widehat{V}=0, & \text { in } \Sigma_{T}, \\ \partial_{\boldsymbol{\nu}_{1}} {{}^{1}\widehat{V}}=\partial_{\boldsymbol{\nu}_{1}} {{}^{2}\widehat{V}}=k\left({}^{2}\widehat{V}-^{1}\widehat{V}\right), & \text { in } \Sigma_{T, \Gamma}, \\ \widehat{V}(0, x)= 0, & \text { in } \Omega,\end{cases} \end{align} $$

multiplying the inequation of (3.10) by $\partial _t \widehat {V}\geq 0 $ , then integrating over $Q_T$ , we get

(3.11) $$ \begin{align} \int_{Q_T}(\partial_t \widehat{V}) \widehat{U} - \int_{Q_T}(\partial_t \widehat{V}) \Delta\widehat{V} \leq \int_{Q_T}(\partial_t \widehat{V})\left[\widehat{U}_0 + \int_0^t h(s,\cdot)ds\right], \end{align} $$

where $\widehat {U}_0:= \widehat {U}(0)= \displaystyle \sum \limits _{i=1}^{m} v_{i,0}^n $ . Integrating by parts and majorating the right-hand side, we get

(3.12) $$ \begin{align} \int_{Q_T}(\partial_t \widehat{V}) \widehat{U} + \int_{Q_T}\nabla (\partial_t \widehat{V}) \nabla\widehat{V} - \int_0^T\int_{\partial \Omega }\partial_{\nu} \widehat{V} \partial_t \widehat{V} \leq \int_{Q_T} (\partial_t \widehat{V})\left[\widehat{U}_0 + \int_0^T h(s,\cdot)ds\right]. \end{align} $$

Now, we can see that

$$ \begin{align*} \int_0^T \int_{\partial \Omega} \partial_\nu \widehat{V} \partial_t \widehat{V} &=\int_0^T \int_{\Gamma} \partial_{\nu_1} {{}^1\widehat{V}}\partial_t {{}^1\widehat{V}} + \int_0^T \int_{\Gamma} \partial_{\nu_2} {{}^2\widehat{V}}\partial_t {{}^2\widehat{V}}\\ &= - \int_0^T\int_{\Gamma} k([\![\widehat{V}]\!]) \partial_t([\![\widehat{V}]\!])\\ &= -\frac{k}{2}\int_0^T \frac{d}{d t} \int_{\Gamma}([\![\widehat{V}]\!])^2\\ &= -\frac{k}{2} \int_{\Gamma}([\![\widehat{V}(T)]\!])^2,\end{align*} $$

since $ \widehat {V}(0)=0$ . So, from (3.12), it follows that

$$ \begin{align*} \int_{Q_T}(\partial_t \widehat{V}) \widehat{U} + \dfrac{1}{2}\int_0^T \dfrac{d}{dt} \int_{\Omega} \vert \nabla\widehat{V} \vert^2 +\frac{k}{2} \int_{\Gamma}([\![\widehat{V}(T)]\!])^2 \leq& \int_{\Omega}\left[\widehat{U}_0 + \int_0^T h(s,\cdot)ds\right]\widehat{V}(T),\\ \int_{Q_T}(\partial_t \widehat{V}) \widehat{U} + \dfrac{1}{2}\int_{\Omega} \vert \nabla\widehat{V}(T) \vert^2 +\frac{k}{2} \int_{\Gamma}([\![\widehat{V}(T)]\!])^2 \leq& \int_{\Omega}\left[\widehat{U}_0 + \int_0^T h(s,\cdot)ds\right]\widehat{V}(T). \end{align*} $$

Therefore,

(3.13) $$ \begin{align} \int_{Q_T}(\partial_t \widehat{V}) \widehat{U} \leq \int_{\Omega}\left[\widehat{U}_0 + \int_0^T h(s,\cdot)ds\right]\widehat{V}(T). \end{align} $$

Furthermore, the second term of the right-hand side of the above inequality is bounded for all $T>0$ . To see this, we introduce, thanks to the Lax–Milgram theorem, the solution of

(3.14) $$ \begin{align} \begin{cases} W \in \textbf{H}^1, \quad W \geq 0, \\ - \displaystyle \Delta W = \widehat{U}_0 + \int_0^T h(s,\cdot)ds, & \text { in } Q_{T},\\ W=0, & \text { in } \Sigma_T, \\ \partial_{\boldsymbol{\nu}_{1}} {{}^{1}\widehat{W}}=\partial_{\boldsymbol{\nu}_{1}} {{}^{2}\widehat{W}}=k\left({}^{2}\widehat{W}-^{1}\widehat{W}\right), & \text { in } \Sigma_{T, \Gamma},\\ W(0,x)=W_0(x)\geq 0 & \text { in } \Omega. \end{cases} \end{align} $$

Consider the inequality from (3.10) at time $t=T$ . Multiplying by W and using the growth of the integral, we integrate over $\Omega $ to obtain

(3.15) $$ \begin{align} \int_{\Omega} \widehat{U}(T)W - \int_{\Omega} \Delta \widehat{V}(T)W \leq \int_{\Omega} \widehat{U}_{0} W + \int_{\Omega} W \int_{0}^{T}h(s,\cdot)ds := -\int_{\Omega}W \Delta W. \end{align} $$

Using two integrations by parts on the second term of the left-hand side, and exploiting equation (3.14), we observe that

(3.16) $$ \begin{align} \int_{\Omega} \Delta \widehat{V}(T)W = \int_{\Omega} V(T)\Delta W := -\int_{\Omega} \left[ \widehat{U}_0 + \int_0^T h(s,\cdot)ds \right]\widehat{V}(T), \end{align} $$

and with an integration by parts on the right-hand side term, we have

(3.17) $$ \begin{align} -\int_{\Omega} W\Delta W = \left( \int_{\Gamma} k [\![W]\!]^2 + \int_{\Omega} |\nabla W|^2 \right). \end{align} $$

Taking W as the test function in the weak formulation of (3.14), it follows that

(3.18) $$ \begin{align} \left( \int_{\Gamma} k [\![W]\!]^2 + \int_{\Omega} |\nabla W|^2 \right)&:= \int_{\Omega} \left( \widehat{U}_0 + \int_{0}^{T}h(s,\cdot)ds \right)W \nonumber\\ &\leq \|W\|^2_{\textbf{H}^1} \left( \|\widehat{U}_0 \|^2_{(\textbf{H}^1)^{\star} } + \left \| \int_{0}^{T} h(s,\cdot)ds \right\|^2_{L^2(\Omega)} \right) \nonumber \\ &\leq \|W\|^2_{\textbf{H}^1} \left( \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } + \left \| \int_{0}^{T} h(s,\cdot)ds \right\|^2_{L^2(\Omega)} \right) \nonumber \\ &\leq \max\left\lbrace 1, \left \| \int_{0}^{T} h(s,\cdot)ds \right\|^2_{L^2(\Omega)} \right\rbrace \|W\|^2_{\textbf{H}^1} \left( 1+ \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right) \nonumber \\ &\leq C \left( 1+ \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right), \end{align} $$

where $C= \max \left \lbrace 1, \left \| \int _{0}^{T} h(s,\cdot )ds \right \|^2_{L^2(\Omega )} \right \rbrace \|W\|^2_{\textbf {H}^1} $ . From (3.15)–(3.18), we obtain

$$ \begin{align*} \int_{\Omega} \widehat{U}(T)W + \int_{\Omega} \left[ \widehat{U}_0 + \int_0^T h(s,\cdot)ds \right]\widehat{V}(T) \leq C \left( 1+ \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right). \end{align*} $$

Thanks to the positivity of W and $v^n_{i}$ , it follows that

(3.19) $$ \begin{align} \int_{\Omega} \left[ \widehat{U}_0 + \int_0^T h(s,\cdot)ds \right]\widehat{V}(T) \leq C \left( 1+ \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right). \end{align} $$

From (3.13) and (3.19), it comes that

$$ \begin{align*} \int_{Q_T}(\partial_t \widehat{V}) \widehat{U} \leq C \left( 1+ \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right), \end{align*} $$

which can be rewritten as

$$ \begin{align*} \int_{Q_T} \sum\limits_{i=1}^{m} v_{i}^{n}(t,\cdot) \sum\limits_{i=1}^{m} \varphi_{i}(v_{i}^{n}(t,\cdot))\leq {e}^{2CT}C \left( 1+ \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right). \end{align*} $$

Finally, from (3.1) and the fact that $\varphi _i(v_i)=D_i v_i^{r_i}$ , it follows that

$$ \begin{align*} \min \limits_{i} \left\lbrace D_i \right\rbrace \sum\limits_{i=1}^{m} \int_{Q_T}\vert v_i^n\vert^{r_i+1} \leq {e}^{2CT}C \left( 1+ \sum \limits_{i=1}^{m} \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right) \leq {e}^{2CT}C \sum \limits_{i=1}^{m} \left( 1+ \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right). \end{align*} $$

This implies

(3.20) $$ \begin{align} \int_{Q_T}\vert v_i^n\vert^{r_i+1} \leq \textbf{C} \left( 1+ \|v^n_{i,0} \|^2_{(\textbf{H}^1)^{\star} } \right), \quad \text{ where } \textbf{C}=\dfrac{{e}^{2CT}C}{\min \limits_{i} \left\lbrace D_i \right\rbrace}. \end{align} $$

This proves (3.5).

Noting that $\vert R_i^{n} \vert \leq \vert R_i \vert $ , then from (2.5) and (3.20), we have

(3.21) $$ \begin{align} \int_{Q_T}\vert R_i^{n}(\textbf{v}^n)\vert \leq C \left[ T\vert \Omega \vert + \sum \limits_{i=1}^{m} \int_{Q_T} \vert v_i^n \vert^{r_i+1} \right] \leq C(T, \Omega,\textbf{C},m) \left( 1+ \sum \limits_{i=1}^{m} \Vert v_{i,0}^n \Vert^2_{(\textbf{H}^1)^{\star} } \right). \end{align} $$

Hence, $\left \lbrace R_i^n(v^n)\right \rbrace $ is bounded in $L^1(Q_T)$ . This proves (3.6) and completes the proof of Lemma 3.1.

Let us now recall the main compactness properties of the solutions of (1.1). We start with the following compactness lemma (see [Reference Baras2, Reference Baras and Pierre3, Reference Bothe and Pierre5]; for the compactness of the trace, we use the continuity of the trace operator from $W^{1,1}$ into $L^{1}(\partial \Omega )$ ). Let $(w_0,F)\in L^1(\Omega )\times L^1(Q_T)$ . We consider w the solution of the problem in dimension $d \geq 2$

(3.22) $$ \begin{align} \begin{cases} \partial_tw - D\Delta w^{\alpha} = F & \text{in} \hspace{1cm} Q_T:=(0,T)\times \Omega, \\ w =0 & \text{in} \hspace{1cm} \Sigma_{T}:=(0,T)\times (\Gamma_1 \cup \Gamma_2), \\ \partial_{\nu_1} {{}^1\varphi(w)}=\partial_{\nu_1} {{}^2\varphi(w)}= k_i({}^2\varphi(w) - {{}^1\varphi(w)}) & \text{in} \hspace{1cm} \Sigma_{T,\Gamma}:=(0,T)\times \Gamma,\\ w(0,x)= w_{0}(x) \geq 0 & \text{in} \hspace{1cm} \Omega. \end{cases} \end{align} $$

• • The mapping $(w_0,f) \mapsto w$ is compact from $ L^1(\Omega )\times L^1(Q_T)$ to $L^1(Q_T)$ for all $\alpha> \dfrac {(d-2)^{+}}{d}$ .

• • The trace mapping $(w_0,f) \mapsto w_{\vert \Gamma } \in L^{1}\left ( \Gamma \right )$ is also compact.

Lemma 3.2 (see [Reference Lukkari22])

Let $(w_0,F)\in L^1(\Omega )\times L^1(Q_T)$ and $\alpha> \dfrac {(d-2)^{+}}{d}$ . We consider w the solution of the problem (3.22) in dimension $d \geq 2$ . Then,

(3.23) $$ \begin{align} \int_{Q_T} \vert w \vert^{\alpha \gamma} &\leq C \text{ for } 0<\gamma< \dfrac{2+\alpha d}{\alpha d}, \end{align} $$

(3.24) $$ \begin{align} \int_{Q_T} \vert \nabla w^{\alpha} \vert^{\beta} &\leq C \text{ for } 0<\beta< 1+ \dfrac{1}{1+ \alpha d}. \end{align} $$

The constant C depends only on $\vert Q_T \vert $ , $\Vert w_0 \Vert _{L^1(\Omega )},\Vert F \Vert _{L^1(Q_T)}, \gamma , \beta , \alpha \textrm { and of dimension } d $ .

In the rest of the demonstrations, we will need to use Vitali’s lemma, which we recall below.

3.3 Existence of global weak supersolution

Thanks to Lemma 3.1, we know that the reaction term $R^n$ is bounded in $L^1(Q_T)$ . Thus, we can assert the existence of a supersolution of the system (1.1) through the following theorem.

Theorem 2 (Existence of a supersolution)

Assume that the $L^1$ -estimate (3.6) holds for the solution $v^n$ of (3.2) with $\dfrac {(d-2)^{+}}{d}<r_i<2 $ . Assume that, for $1 \leq i \leq m$ , $R_i$ satisfy (2.5). Let $ v^n=(v^n_1,\ldots ,v^n_m)$ be a nonnegative solution of approximate system (3.2). Let us consider that $k_i= k,$ for $i=1,\ldots ,m$ . Then, up to a subsequence, $v^n$ converges in $L^1(Q_T)$ and a.e. in $Q_T$ to a supersolution $v=(v_1,\ldots ,v_m)$ of system (1.1), which means that for $i=1, \ldots ,m$ ,

(3.25) $$ \begin{align} -\int_{\Omega} v_i(\cdot,0)\Psi(\cdot,0)dx &+ \int_{0}^{T}\int_{\Omega} - v_i\partial_t\Psi + \nabla \varphi_i(v_i).\nabla\Psi dxdt \nonumber\\ &+k \int_{0}^{T}\int_{\Gamma}[\![ \Psi ]\!] [\![ \varphi_i(v_i) ]\!] \geq \int_{0}^{T}\int_{\Omega} R_i(v)\Psi dxdt \end{align} $$

for all $\Psi \in W_T$ .

Proof Thanks to Lemma 3.1, $\left \lbrace R_i^n(v^n) \right \rbrace $ is bounded in $L^1(Q_T)$ , and according to Lemma 3.2, $\left \lbrace v_i^n \right \rbrace $ is relatively compact in $L^1(Q_T)$ .Therefore, up to a subsequence,

(3.26) $$ \begin{align} v_i^n \to v_i \textrm{ in } L^1(Q_T) \textrm{ and a.e. } Q_T. \end{align} $$

Thanks to the choice of $R_i^n(v^n)$ , and the a.e. convergence of $ \left \lbrace v_i^n \right \rbrace $ (3.26), it follows that

$$ \begin{align*} R_i^n(v^n) \to R_i(v) \textrm{ a.e. in } Q_T. \end{align*} $$

By Fatou’s lemma, we have

$$ \begin{align*} \int_{Q_T} \vert R_i(v) \vert \leq \liminf_{n \to +\infty} \int_{Q_T} \vert R_i^n(v^n) \vert, \end{align*} $$

which implies

$$ \begin{align*} R_i(v) \in L^1(Q_T)^m. \end{align*} $$

Thanks to the estimation (3.6), we can apply (3.23) of Lemma 3.2 to the $ith$ equation of (3.2). Then, it comes that $\left \lbrace \varphi _i(v_i^n) \right \rbrace _n$ is bounded in $L^{\gamma }(Q_T)$ for all $1 \leq \gamma < 1+ \dfrac {2}{r_id}$ and for all $T> 0$ , i.e., there is a constant $C> 0$ such that

$$ \begin{align*} \int_{Q_T} \vert \varphi_i(v_i^n) \vert^{\gamma} \leq C, \end{align*} $$

which implies that $(\varphi _i(v_i^n))^{\gamma } \textrm {is bounded in } L^1(Q_T)$ . By arbitrarity nature of $\gamma $ in $\left [1, 1+ \dfrac {2}{r_id} \right )$ , $\left (\varphi _i(v_i^n)\right )^{\gamma }$ is even uniformly integrable. Since it also converges a.e. to $\varphi _i(v_i)$ , by Vitali’s Lemma 3.3, the convergence holds strongly in $L^{\gamma }(Q_T)$ to $\varphi _i(v_i)$ .

Next, according to the estimation (3.6), we can apply (3.24) of Lemma 3.2 to the ith equation of (3.2). Then, it comes that $\left \lbrace \varphi _i(v_i^n) \right \rbrace _n$ is bounded in $L^{\beta }(0,T; W^{1,\beta }(\Omega ))$ for all $1 \leq \beta < 1+ \dfrac {1}{1+r_id}$ . These space being reflexive (for $\beta> 1$ ), it follows that $\varphi _i(v_i)$ also belongs to these same spaces.

Since $v_i^n$ satisfies (3.2), then for all $i=1,\ldots , m$ , we have

(3.27) $$ \begin{align} - \int_{\Omega} v_i^n(\cdot,0)\Psi(\cdot,0)dx &+ \int_{0}^{T}\int_{\Omega} - v_i^n\partial_t\Psi+ \nabla \varphi_i(v_i^n).\nabla\Psi dxdt \nonumber\\ &+ k \int_{0}^{T}\int_{\Gamma}[\![ \Psi ]\!] [\![ \varphi_i(v_i^n) ]\!]= \int_{0}^{T}\int_{\Omega} R_i^n(v_i^n)\Psi dxdt \end{align} $$

for all $\Psi \in W_T$ .

Now, we want to take the limit as $ n \to \infty $ in (3.27). But first, let us make the following remark.

3.4 Truncation method

Here, the main idea is that if $v_i^n$ is a weak solution of (3.2), then the regular approximation of the truncation function $T_b(v_i)$ (with $T_b$ defined below (3.28) and $v_i$ being the limit of $v_i^n$ as $n \rightarrow +\infty $ ) constitutes a supersolution of (1.1). To demonstrate this, we let $n \rightarrow +\infty $ in the inequality satisfied by an appropriate approximation of $T_b(v_i^n)$ (specifically (3.45) in Lemma 3.7).

In the semilinear case, the method consisted of writing, for each index i, the inequality satisfied by $T_b(v_i^n + \eta \sum _{j \neq i} v_j^n)$ with $\eta> 0$ , then letting $n \rightarrow +\infty $ for fixed $\eta $ and b, then $\eta \rightarrow 0$ , and finally $b \rightarrow +\infty $ (see [Reference Ciavolella and Perthame10, Reference Pierre27]).

Here, the ideas remain the same but need to be adapted to nonlinear diffusions. Therefore, to prepare the proof of existence of supersolution, let us introduce the truncating functions $T_b :[0, +\infty ) \to [0, +\infty )$ of class $C^3$ which satisfy the following for all $b \geq 1$ :

(3.28) $$ \begin{align} \begin{cases} T_b(\sigma)=\sigma \textrm{ if } \sigma \in [0,b-1],\\ T_b(\sigma) \leq b, \textrm{ for all } \sigma \geq 0. \\ T^{\prime}_b(\sigma) =0 \textrm{ if } \sigma \geq b,\\ 0\leq T^{\prime}_b(\sigma)\leq 1 \textrm{ and } -1\leq T^{\prime \prime}_b(\sigma)\leq 0 \textrm{ for all } \sigma \geq 0. \end{cases} \end{align} $$

For example, we may choose $T_b$ as

(3.29) $$ \begin{align} T_b(\sigma)= \begin{cases} \sigma, &\textrm{ if } \sigma \in [0,b-1], \\ \dfrac{\left( \sigma -b+1 \right)^{4}}{2} -\left( \sigma -b+1 \right)^{3} + \sigma, &\textrm{ if } \sigma \in [b-1,b], \\ b-\dfrac{1}{2}, &\textrm{ if } \sigma \in ]b,+\infty). \end{cases} \end{align} $$

Next, for all $i=1,\ldots , m$ , and for all $(n,\eta ,b) \in \mathbb {N}^{\star }\times (0,1)\times [1,+\infty )$ , we introduce

(3.30) $$ \begin{align} A^{n}_{i,\eta,b}= \partial_t \left[ T_b(v_i^n)T^{\prime}_b(\eta V_{i}^{n}) \right] - \nabla.\left[T^{\prime}_b (v_i^n)T^{\prime}_b(\eta V_i^n)\nabla \varphi_i(v_i^n)\right], \end{align} $$

where $T_b(v_i^n)$ is such that $\vert v_i^n \vert = \min \limits _{ j } \left \lbrace \vert vj \vert \right \rbrace $ and $V_i^n= \sum \limits _{j \neq i}v_j^n$ , $j \in \left \lbrace 1, \ldots ,m \right \rbrace $ .

Note that the operator $A^{n}_{i,\eta ,b} \to \partial _tv_i^n - \nabla .(\nabla \varphi _i(v_i^n))= \partial _tv_i^n - \Delta \varphi _i(v_i^n) $ as $\eta \to 1 $ and $b \to +\infty $ .

Using a computation, we can prove that

(3.31) $$ \begin{align} A^{n}_{i,\eta,b}= T^{\prime}_b(v_i^n)T_b(\eta V_i^n)R_i^n(v^n) + A_i^n + B_i^n, \end{align} $$

where

$$ \begin{align*} A_i^n= - \nabla \varphi_i(v_i^n).\nabla [T^{\prime}_b(v_i^n)T^{\prime}_b(\eta V_i^n)] \ \textrm{ and }\hspace{3cm}\\ B_i^n= \eta T_b(v_i^n)T^{\prime\prime}_b(\eta V_i^n)\partial_t V_i^n = \eta T_b(v_i^n)T^{\prime\prime}_b(\eta V_i^n)\sum \limits_{j \neq i}\left( \Delta \varphi_j(v_j^n) + R_j^n(v^n) \right). \end{align*} $$

By multiplying $A^{n}_{i,\eta ,b}$ by $\Psi \in W_T$ and then integrating over $Q_T$ , we obtain

(3.32) $$ \begin{align} \int_{Q_T} A^{n}_{i,\eta,b} \Psi = \int_{Q_T} T^{\prime}_b(v_i^n)T_b(\eta V_i^n)R_i^n(v^n) \Psi + \int_{Q_T} A_i^n \Psi + \int_{Q_T} B_i^n \Psi. \end{align} $$

Now we are going to bound the last two terms of (3.32). To arrive at this, we need the following lemma.

Lemma 3.5 Let $F \in L^{1}\left (Q_{T}\right )^{+}, w_{0} \in L^{1}(\Omega )^{+}$ . Then w the solution of (3.22) satisfies the following: there exists $C=C\left (\int _{Q_{T}} F, \int _{\Omega } w_{0}\right )$ such that, for all nondecreasing $\theta :(0,+\infty ) \rightarrow (0,+\infty )$ of class $\mathcal {C}^{1}$ and with $\theta \left (0^{+}\right )=0$ ,

(3.33) $$ \begin{align} \int_{[\theta(w) \leq b]}|\nabla \theta(w)||\nabla \varphi(w)|=\int_{[\theta(w) \leq b]} \nabla \theta(w) \nabla \varphi(w) \leq C b. \end{align} $$

In particular,

(3.34) $$ \begin{align} \int_{[\varphi(w) \leq b]}|\nabla \varphi(w)|^{2} \leq C b, \int_{[w \leq b]}|\nabla w|^{2} \leq C b^{2-r_i} \end{align} $$

with $r_{i}<2$ .

Proof of Lemma 3.5

As usual, we make the computations for regular enough solutions and they are preserved by approximation for all semigroup solutions.

Multiplying the equation $\partial _{t} w-\Delta \varphi (w)=F$ by $T_{b+1}(\theta (w))$ . We obtain

(3.35) $$ \begin{align} \partial_{t} w T_{b+1}(\theta(w))-\Delta \varphi(w)T_{b+1}(\theta(w))=FT_{b+1}(\theta(w)). \end{align} $$

Let us set $j_b(w)=\displaystyle \int _0^w T_{b+1}(\theta (s))ds $ , then $J_{b}(0)=0$ and $\partial _t J_{b}(w)=\partial _t w T_{b+1}(\theta (w)).$ Integrating (3.35) over $Q_T$ , we obtain

$$ \begin{align*} \int_{\Omega} J_{b}(w)(T)&+ k\int_{0}^{T}\int_{\Gamma} [\![T_{b+1}(\theta(w))]\!][\![\varphi(w)]\!]+\int_{Q_{T}} T_{b+1}^{\prime}(\theta(w)) \nabla \theta(w) \nabla \varphi(w)\\ &\quad =\int_{Q_{T}} T_{b+1}(\theta(w)) F+\int_{\Omega} J_{b}\left(w_{0}\right). \end{align*} $$

Exploiting the increase in $T_{b+1}$ , $\theta $ and $\varphi $ , we find that $[\![T_{b+1}(\theta (w))]\!][\![\varphi (w)]\!] \geq 0$ . Since $T_{b+1} \leq b+1$ , we have $0\leq J_{b}(r) \leq (b+1) r$ for all $r \geq 0$ so that

$$ \begin{align*}\int_{[\theta(w) \leq b]}|\nabla \theta(w)||\nabla \varphi(w)| \leq(b+1)\left(\int_{Q_{T}} F+\int_{\Omega} w_{0}\right) \leq C (b). \end{align*} $$

Choosing $\theta :=\varphi $ gives the first estimate of (3.34). If $r_{i}<2$ in the expression of $\varphi $ (2.1), we choose $\theta (w):=w^{2-r_{i}}$ to obtain

$$ \begin{align*}d_{i}\left(2-r_{i}\right) r_{i} \int_{\left[w^{(2-r_{i})} \leq b\right]}|\nabla w|^{2} \leq C_1 b, \end{align*} $$

and by substituting b for $b^{2-r_{i}}$ , we get

$$ \begin{align*}\int_{\left[w \leq b\right]}|\nabla w|^{2} \leq C b^{2-r_{i}}, \end{align*} $$

which gives the second estimate of (3.34).

Thanks to the results of the previous lemma (Lemma 3.5), we are able to state the following results.

Lemma 3.6 There exist $\delta>0, C \geq 0$ independent of n and $\eta $ such that, for all $i=1, \ldots , m$ and for all $\Psi \in W_{T}$ ,

(3.36) $$ \begin{align} \int_{Q_{T}} \Psi A_{i}^{n} \geq -\eta^{\delta} C D(\Psi) \end{align} $$

and

(3.37) $$ \begin{align} \int_{Q_{T}} \Psi B_{i}^{n} \geq -\eta^{\delta} CD(\Psi), \end{align} $$

where $D(\Psi )=\|\Psi \|_{L^{\infty }\left (Q_{T}\right )}+\|\nabla \Psi \|_{L^{\infty }\left (Q_{T}\right )}$ .

Proof We have

$$ \begin{align*}\begin{aligned} \int_{Q_{T}} \Psi A_{i}^{n} & =-\int_{Q_{T}} \Psi \nabla \varphi_{i}\left(v_{i}^{n}\right) \nabla\left[T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right)\right] \\ & =-\int_{Q_{T}} \Psi \nabla \varphi_{i}\left(v_{i}^{n}\right).\left[ T_{b}^{\prime \prime}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right)\nabla v_{i}^{n} + \eta T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla V_{i}^{n}\right] \\ & =-\int_{Q_{T}} \Psi \varphi^{\prime}_{i}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right)\vert \nabla v_{i}^{n} \vert^{2} - \eta\int_{Q_{T}} \Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right).\nabla V_{i}^{n} \\ & \geq-\eta \int_{Q_{T}} \Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right). \nabla V_{i}^{n}, \end{aligned} \end{align*} $$

the last inequality is obtained thanks to the positivity of $\varphi _{i}^{\prime }, \Psi $ and conditions on $T_b$ in (3.28). We can see that apart from $ \left \lbrace (x,t) \in Q_T, v_{i}^{n} \leq b \, \ \eta V_{i}^{n} \leq b \right \rbrace $ , $ T_{b}^{\prime }\left (v_{i}^{n}\right ) T_{b}^{\prime } \left (\eta V_{i}^{n}\right )=0$ .

To simplify notation, let us write $ \left [v_{i}^{n} \leq b\right ] \cap \left [\eta V_{i}^{n} \leq b\right ] $ instead of $ \left \lbrace (x,t) \in Q_T, v_{i}^{n} \leq b \, \ \eta V_{i}^{n} \leq b \right \rbrace $ . From this observation, it follows that

$$ \begin{align*}\begin{aligned} \int_{Q_{T}}\left|\Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right) \nabla V_{i}^{n}\right| & := \int_{\left[v_{i}^{n} \leq b\right] \cap \left[\eta V_{i}^{n} \leq b\right] }\left|\Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right) \nabla V_{i}^{n}\right|. \end{aligned} \end{align*} $$

By the Schwarz inequality and conditions on $T_b$ in (3.28), there exists a constant C depending on b, and we have the following result:

$$ \begin{align*}\begin{aligned} \int_{Q_{T}}&\left|\Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right) \nabla V_{i}^{n}\right| \\ & \leq C\|\Psi\|_{L^{\infty}\left(Q_{T}\right)}\left(\int_{\left[v_{i}^{n} \leq b\right] \cap \left[\eta V_{i}^{n} \leq b\right] }\left|\nabla \varphi_{i}\left(v_{i}^{n}\right)\right|{}^{2}\right)^{1 / 2}\left(\int_{\left[v_{i}^{n} \leq b\right] \cap \left[\eta V_{i}^{n} \leq b\right] }\left|\nabla V_{i}^{n}\right|{}^{2}\right)^{1 / 2} \\ & \leq C\|\Psi\|_{L^{\infty}\left(Q_{T}\right)}\left(\int_{\left[v_{i}^{n} \leq b\right]}\left|\nabla \varphi_{i}\left(v_{i}^{n}\right)\right|{}^{2}\right)^{1 / 2}\left(\int_{\left[\eta V_{i}^{n} \leq b\right]}\left|\nabla V_{i}^{n}\right|{}^{2}\right)^{1 / 2} \\ & \leq C\|\Psi\|_{L^{\infty}\left(Q_{T}\right)}\left(\int_{\left[\varphi_i(v_{i}^{n}) \leq \varphi_i(b)\right]}\left|\nabla \varphi_{i}\left(v_{i}^{n}\right)\right|{}^{2}\right)^{1 / 2}\left(\int_{\left[V_{i}^{n} \leq \frac{b}{\eta}\right]}\left|\nabla V_{i}^{n}\right|{}^{2}\right)^{1 / 2}. \end{aligned} \end{align*} $$

By Lemma 3.5, we obtain

$$ \begin{align*}\begin{aligned} \int_{Q_{T}}\left|\Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right) \nabla V_{i}^{n}\right| &\leq C\|\Psi\|_{L^{\infty}\left(Q_{T}\right)} \sqrt{\varphi_{i}(b)}[b / \eta]^{1-r_i / 2}\\ &\leq C\|\Psi\|_{L^{\infty}\left(Q_{T}\right)} \sqrt{\varphi_{i}(b)}[b / \eta]^{1-M / 2};\\ &\quad M:=\max \left\{1, \max \limits_{i} r_{i}\right\}\\ &\leq C\|\Psi\|_{L^{\infty}\left(Q_{T}\right)} b^{1+\frac{(r_i-M)}{2}} \eta^{\frac{M}{2}-1} \\ &\leq C(b) D(\Psi) \eta^{\frac{M}{2}-1}, \end{aligned} \end{align*} $$

where $C(b)= b $ and $D(\Psi )= \Vert \Psi \Vert _{L^{\infty }\left (Q_{T}\right )} + \Vert \nabla \Psi \Vert _{L^{\infty }\left (Q_{T}\right )}$ . This implies

$$ \begin{align*}\begin{aligned} \eta \int_{Q_{T}}\Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right) \nabla V_{i}^{n} \leq C(b) D(\Psi) \eta^{\frac{M}{2}}. \end{aligned} \end{align*} $$

Therefore,

(3.38) $$ \begin{align} \int_{Q_{T}} \Psi A_{i}^{n} \geq-C D(\Psi) \eta^{\delta} \end{align} $$

for some $C=C(b)$ and $\delta =M / 2$ , hence the result (3.36) of Lemma 3.6. To prove the second estimate of Lemma 3.6, we will proceed in several steps. We have

$$ \begin{align*} \int_{Q_{T}}\Psi B_{i}^{n}= \sum \limits_{j \neq i}\left(\eta \int_{Q_{T}}\Psi T_b(v_i^n)T^{\prime\prime}_b(\eta V_i^n) \Delta \varphi_j(v_j^n) + \eta\int_{Q_{T}}\Psi T_b(v_i^n)T^{\prime\prime}_b(\eta V_i^n) R_j^n(v^n) \right). \end{align*} $$

Let us put $I_n= \eta \displaystyle \int _{Q_{T}}\Psi T_b(v_i^n)T^{\prime \prime }_b(\eta V_i^n) \Delta \varphi _j(v_j^n) $ and $J_n= \eta \displaystyle \int _{Q_{T}}\Psi T_b(v_i^n) T^{\prime \prime }_b(\eta V_i^n) R_j^n(v^n)$ .

• • Let us bound $J_n$ . We have:

using the fact that $-1 \leq T^{\prime \prime }_b(\sigma ) \leq 0$ , $0 \leq T_b(\sigma ) \leq b$ for all $\sigma \geq 0$ , and the bound $L^{1}$ on $R_{i}^{n}$ , we obtain

(3.39) $$ \begin{align} J_n \geq -\eta C(b)\Vert \Psi \Vert_{L^{\infty}(Q_{T})}.\\[-24pt]\nonumber \end{align} $$

• • Let us bound $I_n$ . We have:

$$ \begin{align*} I_n &= \eta\int_{Q_{T}} \Psi T_{b}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \Delta \varphi_{j}\left(v_{j}^{n}\right)\\ &= -\eta\int_{0}^{T}\int_{\Gamma} k[\![\Psi T_{b}(v_{i}^{n}) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right)]\!] [\![ \varphi_{j}(v_{j}^{n})]\!] \\ &\quad - \eta\int_{Q_{T}} \nabla\left(\Psi T_{b}(v_{i}^{n}) T_{b}^{\prime \prime}(\eta V_{i}^{n}) \right).\nabla \varphi_{j}(v_{j}^{n}) \\ &= -K_{n} - I_{1,n}-I_{2,n}-I_{3,n}, \end{align*} $$

where

$$ \begin{align*} &K_{n}= \eta\int_{0}^{T}\int_{\Gamma} k[\![\Psi T_{b}(v_{i}^{n}) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right)]\!] [\![ \varphi_{j}(v_{j}^{n})]\!], \\ &I_{1,n}= \eta\int_{Q_T} T_{b}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{j}\left(v_{j}^{n}\right).\nabla \Psi,\\ & I_{2,n}= \eta\int_{Q_T} \Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{j}(v_{j}^{n}).\nabla v_{i}^{n}, \\ & I_{3,n}=\eta^{2} \int_{Q_T} \Psi T_{b}\left(v_{i}^{n}\right) T_{b}^{\prime \prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{j}(v_{j}^{n}).\nabla V_{i}^{n}.\\[-18pt] \end{align*} $$

• • Let us bound $K_{n}$ .

(3.40) $$ \begin{align} \vert K_n \vert =& \eta\left\vert \int_{0}^{T}\int_{\Gamma} k\left[{}^2\Psi T_{b}({}^2v_{i}^{n}) T_{b}^{\prime \prime}\left(\eta {{}^2V}_{i}^{n}\right) - {{}^1\Psi} T_{b}({}^1v_{i}^{n}) T_{b}^{\prime \prime}\left(\eta {{}^1V}_{i}^{n}\right) \right] \left[ \varphi_{j}({}^2v_{j}^{n}) - \varphi_{j}({}^1v_{j}^{n})\right] \right\vert \nonumber\\ &\leq 4b\eta \Vert \Psi \Vert_{L^{\infty}(Q_T)}\int_{\left[v_{i}^{n} \leq b\right] \cap\Gamma} \vert v_i^n \vert \vert \varphi_j(v_j^n) \vert \nonumber\\ &\leq 4b\eta D \Vert \Psi \Vert_{L^{\infty}(Q_T)}\int_{\left[v_{i}^{n} \leq b\right] \cap\Gamma} \vert v_j^n \vert^{rj+1} \nonumber\\ &\leq C(b,T)\eta\Vert \Psi \Vert_{L^{\infty}(Q_T)}. \end{align} $$

With $D=\max \limits _{j}\left \lbrace Dj \right \rbrace $ , we have used Lemma 3.1.

• • Let us bound $I_{1,n}$ . By (3.24) (with $\beta =1$ ) and Lemma 3.5, we obtain

(3.41) $$ \begin{align} \vert I_{1,n}\vert \leq b \eta \Vert \nabla \Psi \Vert_{L^{\infty}(Q_{T})} \int_{Q_{T}}\left|\nabla \varphi_{j}\left(v_{j}^{n}\right)\right| \leq C(b)\eta \Vert \nabla \Psi \Vert_{L^{\infty}(Q_{T})}.\\[-28pt]\nonumber \end{align} $$

• • Let us bound $I_{2, n}$ .

$$ \begin{align*} \vert I_{2,n} \vert&= \eta\left \vert \int_{Q_T} \Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{j}(v_{j}^{n}).\nabla v_{i}^{n} \right\vert \\ &=\eta \left \vert \int_{\left[\eta V_{i}^{n} \leq b\right] \cap\left[v_{i}^{n} \leq b\right]} \Psi T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime \prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{j}(v_{j}^{n}).\nabla v_{i}^{n} \right\vert \\ &\leq \eta\int_{\left[\eta V_{i}^{n} \leq b\right] \cap\left[v_{i}^{n} \leq b\right]} \Psi T_{b}^{'}(v_{i}^{n})\vert T_{b}^{"}(\eta V_{i}^{n})\vert \vert \nabla \varphi_{j}(v_{j}^{n})\vert \vert \nabla v_{i}^{n}. \vert \end{align*} $$

By noticing that $\left \lbrace \eta V_{i}^{n} \leq b \right \rbrace \subset \left \lbrace \eta v_{j}^{n} \leq b \right \rbrace $ and by Schwarz’s inequality, it follows that

(3.42) $$ \begin{align} \vert I_{2,n} \vert &\leq \eta \int_{\left[\eta v_{j}^{n} \leq b\right] \cap\left[v_{i}^{n} \leq b\right]} \Psi T_{b}^{'}(v_{i}^{n})\vert T_{b}^{"}(\eta V_{i}^{n})\vert \vert \nabla \varphi_{j}(v_{j}^{n})\vert \vert \nabla v_{i}^{n} \vert \nonumber\\ & \leq \eta \Vert \Psi \Vert_{L^{\infty}(Q_{T})} \left(\int_{\left[\eta v_{j}^{n} \leq b\right]}\vert \nabla \varphi_{j}(v_{j}^{n})\vert^{2}\right)^{1 / 2}\left(\int_{\left[v_{i}^{n} \leq b\right]}\vert \nabla v_{i}^{n}\vert^{2}\right)^{1 / 2} \nonumber\\ & \leq \eta \Vert \Psi \Vert_{L^{\infty}(Q_{T})} \left(\int_{\left[\varphi_j (v_{j}^{n}) \leq \varphi_j( \frac{b}{\eta})\right]}\vert \nabla \varphi_{j}(v_{j}^{n})\vert^{2}\right)^{1 / 2}\left(\int_{\left[v_{i}^{n} \leq b\right]}\vert \nabla v_{i}^{n}\vert^{2}\right)^{1 / 2} \nonumber\\ & \leq \eta C \Vert \Psi \Vert_{L^{\infty}(Q_{T})} \sqrt{\varphi_j ( \frac{b}{\eta})} b^{1- \frac{r_{i}}{2}} \nonumber\\ &\leq C(b,D) \Vert \Psi \Vert_{L^{\infty}(Q_{T})}\eta^{1- \frac{r_{j}}{2}}, \end{align} $$

with $r_j< 2$ , $D=\max \limits _{j} \left \lbrace D_j \right \rbrace $ and $C(b,D) =C b^{1+ (r_{j}-r_{i})/2} \sqrt {D}$ , $i,j=1,\ldots ,m, j\neq i$ , and we have used Lemma 3.5.

• • Let us bound $I_{3, n}$ . Using again Schwarz’s inequality, Lemma 3.5, and $\left [\eta V_{i}^{n} \leq b\right ] \subset \left [\eta v_{j}^{n} \leq b\right ]$ , we obtain

(3.43) $$ \begin{align} \vert I_{3, n}\vert & \leq \eta^{2} \left( \int_{\left[\eta V_{i}^{n} \leq b\right]} \Psi \vert T_{b}(v_{i}^{n})\vert \vert T_{b}^{\prime \prime \prime}(\eta V_{i}^{n}) \vert \vert \nabla \varphi_{j}(v_{j}^{n})\vert \vert \nabla V_{i}^{n}\vert \right) \nonumber \\ & \leq C \eta^{2} \|\Psi\|_{L^{\infty}\left(Q_{T}\right)}\left[\int_{\left[\eta v_{j}^{n} \leq b\right]}\left|\nabla \varphi_{j}\left(v_{j}^{n}\right)\right|{}^{2}\right]^{1 / 2}\left[\left.\int_{\left[\eta V_{i}^{n} \leq b\right]}\left|\nabla V_{i}^{n}\right|{}^{2}\right]^{1 / 2}\right. \nonumber \\ & \leq C \eta^{2} \|\Psi\|_{L^{\infty}\left(Q_{T}\right)}\left[\int_{\left[\varphi_j (v_{j}^{n}) \leq \varphi_j(\frac{b}{\eta})\right]}\left|\nabla \varphi_{j}\left(v_{j}^{n}\right)\right|{}^{2}\right]^{1 / 2}\left[\left.\int_{\left[\eta V_{i}^{n} \leq b\right]}\left|\nabla V_{i}^{n}\right|{}^{2}\right]^{1 / 2}\right.\nonumber \\ & \leq C \eta^{2} \Vert \Psi\Vert_{L^{\infty}\left(Q_{T}\right)} \sqrt{\varphi_{j}(b / \eta)}\left[\frac{b}{\eta}\right]^{1-M / 2} \nonumber \\ & \leq C \eta^{2} \Vert \Psi\Vert_{L^{\infty}\left(Q_{T}\right)}\sqrt{D}\left[\frac{b}{\eta}\right]^{1+\frac{r_j-M}{2}} \nonumber \\ & \leq C(b,D) \Vert \Psi\Vert_{L^{\infty}\left(Q_{T}\right)}\eta^{1+\frac{M-r_j}{2}} \nonumber\\ & \leq C(b,D) \Vert \Psi\Vert_{L^{\infty}\left(Q_{T}\right)}\eta, \end{align} $$

where $C(b,D)=C\sqrt {D}b^{1+\frac {r_j-M}{2}}$ and $M=\max \limits _{j}\left \lbrace 1,\max \limits _{j}\left \lbrace r_j \right \rbrace \right \rbrace , j=1,\ldots ,m, \ j\neq i. $

From (3.40)–(3.43), we obtain

(3.44) $$ \begin{align} I_{n} \geq C(b,T,D)\eta^{\delta}D(\Psi), \end{align} $$

where $ \delta = 1-\frac {M}{2}, $ and $ D(\Psi )= \Vert \Psi \Vert _{L^{\infty }\left (Q_{T}\right )} + \Vert \nabla \Psi \Vert _{L^{\infty }\left (Q_{T}\right )} $ .

By (3.39) and (3.44), we obtain the estimation (3.37) of Lemma 3.6.

This proves Lemma 3.6.

Now, thanks to Lemma 3.6, we have the following result.

Lemma 3.7 There exist $\delta>0, C>0$ independent of n and $\eta $ such that, for all $i=1, \ldots $ , m and for all $\Psi \in W_{T}$ ,

(3.45) $$ \begin{align} \int_{Q_{T}} A_{i, \eta, b}^{n} \Psi \geq \int_{Q_{T}} T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right) R_{i}^{n}\left(v^{n}\right) \Psi-C D(\Psi) \eta^{\delta}, \end{align} $$

where $ A_{i, \eta , b}^{n}=\partial _{t}\left (T_{b}\left (v_{i}^{n}\right ) T_{b}^{\prime }\left (\eta V_{i}^{n}\right )\right )-\nabla \cdot \left (T_{b}^{\prime }\left (v_{i}^{n}\right ) T_{b}^{\prime }\left (\eta V_{i}^{n}\right ) \nabla \varphi _{i}\left (v_{i}^{n}\right )\right ), \quad V_{i}^{n}=\sum \limits _{j \neq i} v_{j}^{n} $ and $D(\Psi )=\|\Psi \|_{L^{\infty }\left (Q_{T}\right )}+\|\nabla \Psi \|_{L^{\infty }\left (Q_{T}\right )}$ .

Note also that

(3.46) $$ \begin{align} \int_{Q_{T}} A_{i, n, b}^{n} \Psi=&-\int_{\Omega} T_{b}\left(v_{i 0}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}(0)\right) \Psi(0) + k\int_{0}^{T}\int_{\Gamma} [\![ \Psi T^{\prime}_{b}(v_{i}^{n})T^{\prime}_{b}(\eta V_{i}^{n}) ]\!] [\![ \varphi_{i}(v_{i}^{n}) ]\!]\nonumber \\ &+\int_{Q_{T}}-T_{b}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right) \partial_{t} \Psi+T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right) \nabla \varphi_{i}\left(v_{i}^{n}\right) \nabla \Psi. \end{align} $$

Our aim now is to pass to the limit between (3.45) and (3.46). We do it in the following order: first $n \rightarrow +\infty $ , then $\eta \rightarrow 0$ , finally $b \rightarrow +\infty $ .

• • We make $n \rightarrow +\infty $ along the subsequence introduced in Lemma 3.4 ( $\eta $ and b are fixed). Since $v_{i 0}^{n} \rightarrow v_{i 0}$ in $L^{1}(\Omega )$ and $T_{b}, T_{b}^{\prime }$ are Lipschitz continuous, it follows that

$$ \begin{align*}\int_{\Omega} T_{b}\left(v_{i 0}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}(0)\right) \Psi(0) \rightarrow \int_{\Omega} T_{b}\left(v_{i 0}\right) T_{b}^{\prime}\left(\eta V_{i}(0)\right) \Psi(0), \end{align*} $$

and thanks to Lemmas 3.2 and 3.4, it follows that

$$ \begin{align*}\int_{0}^{T}\int_{\Gamma} [\![ \Psi T^{\prime}_{b}(v_{i}^{n})T^{\prime}_{b}(\eta V_{i}^{n}) ]\!] [\![ \varphi_{i}(v_{i}^{n}) ]\!] \to \int_{0}^{T}\int_{\Gamma} [\![ \Psi T^{\prime}_{b}(v_{i})T^{\prime}_{b}(\eta V_{i}) ]\!] [\![ \varphi_{i}(v_{i}) ]\!]. \end{align*} $$

Concerning the last integral in (3.46), since, for all $j=1, \ldots , m, v_{j}^{n}$ converges in $L^{1}\left (Q_{T}\right )$ and a.e. to $v_{j}$ , we have

$$ \begin{align*}T_{b}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right) \rightarrow T_{b}\left(v_{i}\right) T_{b}^{\prime}\left(\eta V_{i}\right) \text{ in } L^{1}\left(Q_{T}\right), \end{align*} $$

where we set $V_{i}:=\sum \limits _{j \neq i} v_{j}$ . It also follows that

$$ \begin{align*}T_{b}^{\prime}\left(\eta V_{i}^{n}\right) \to T_{b}^{\prime}\left(\eta V_{i}\right) \text{ in } L^{2}\left(Q_{T}\right). \end{align*} $$

Next, $T_{b}^{\prime }\left (v_{i}^{n}\right ) \nabla \varphi \left (v_{i}^{n}\right )$ is bounded in $L^{2}\left (Q_{T}\right )$ by (3.33) in Lemma 3.5. Therefore,

$$ \begin{align*}T_{b}^{\prime}\left(v_{i}^{n}\right) \nabla \varphi\left(v_{i}^{n}\right) \to T_{b}^{\prime}\left(v_{i}\right) \nabla \varphi\left(v_{i}\right) \textrm{ weakly in } L^{2}\left(Q_{T}\right). \end{align*} $$

Indeed, let us set $ S_{b}(r):=\displaystyle \int _{0}^{r} T_{b}^{\prime }(s) \varphi _{i}^{\prime }(s) d s$ , then $ \nabla S_{b}\left (v_{i}^{n}\right )=T_{b}^{\prime }\left (v_{i}^{n}\right ) \nabla \varphi \left (v_{i}^{n}\right )$ . Since $S_{b}\left (v_{i}^{n}\right )$ converges a.e. to $S_{b}\left (v_{i}\right )$ and is bounded, the convergence holds in the sense of distributions. Therefore, the distribution limit of $\nabla S_{b}\left (v_{i}^{n}\right )$ is $\nabla S_{b}\left (v_{i}\right )=T_{b}^{\prime }\left (v_{i}\right ) \nabla \varphi _{i}\left (v_{i}\right )$ .

This ends the proof of the passing to the limit in (3.46).

Now we will go to the limit in the right-hand side of the inequality (3.45). Let us put

$$ \begin{align*}W_{n}:=T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right) R_{i}^{n}\left(v^{n}\right), \quad W : =T_{b}^{\prime}\left(v_{i}\right) T_{b}^{\prime}\left(\eta V_{i}\right) R_{i}(v) \end{align*} $$

and show that $W_{n}$ converges to W in $L^{1}\left (Q_{T}\right )$ . Since $W_{n}=0$ outside the set

$\left [v_{i}^{n} \leq b\right ] \cup \left [V_{i}^{n} \leq b / \eta \right ]$ , if $M: =\max \{b, b / \eta \}$ , then, from the definition of $\epsilon _{M}^{n}$ in (3.3), the regularity property (2.2), and the fact that $\left |T_{b}^{\prime }\right | \leq 1$ , we obtain

$$ \begin{align*}\left|W_{n}\right| \leq\left|R_{i}^{n}\left( v^{n}\right)\right| \leq\left|R_{i}(0)\right|+\epsilon_{M}^{n}+K(M)|| v^{n}(t, x)||, \textrm{ where } ||r||:= \sum \limits_{i=1}^{m} \vert r_{i} \vert. \end{align*} $$

By assumption (see (3.4)), as $n \rightarrow +\infty , \epsilon _{M}^{n}$ tends to 0 in $L^{1}\left (Q_{T}\right )$ . Moreover, $v^{n}$ converges in $L^{1}\left (Q_{T}\right )^{m}$ to v. Therefore, to prove the convergence of $W_{n}$ in $L^{1}\left (Q_{T}\right )$ , it is sufficient to prove that it converges a.e. We know that, for all $j, v_{j}^{n} \to v_{j}$ a.e. in $Q_T$ . Therefore, from the continuity of $T^{\prime }_b$ , it follows that

$$ \begin{align*}T_{b}^{\prime}\left(v_{i}^{n}\right) T_{b}^{\prime}\left(\eta V_{i}^{n}\right) \to T_{b}^{\prime}\left(v_{i}\right) T_{b}^{\prime}\left(\eta V_{i}\right) \textrm{ a.e. in } Q_T. \end{align*} $$

It remains to check that

(3.47) $$ \begin{align} R_{i}^{n}\left(v^{n}(t, x)\right) \to R_{i}( v(t, x)) \textrm{ a.e. in } Q_T. \end{align} $$

Let D be the subset of $Q_{T}$ such that, at the same time, $v^{n}(t, x)$ converges to $v(t, x)$ with $\|v(t, x)\|<+\infty $ and $\epsilon _{p}^{n}(t, x)$ converges to 0 for all positive integer p as $n \rightarrow +\infty $ along the subsequence introduced in Lemma 3.4. We know that $Q_{T}\setminus D$ is of zero Lebesgue measure. Now let $(t, x) \in D$ and let $p>\|v(t, x)\|$ . For n large enough, $\left \|v^{n}(t, x)\right \|<p$ and we may write for all $i=1, \ldots , m$ (using the definition de $\epsilon _{p}^{n}$ (3.3) and property (2.2)):

(3.48) $$ \begin{align} \left|R_{i}^{n}\left(v^{n}(t, x)\right)-R_{i}(v(t, x))\right| &\leq \epsilon_{p}(t, x)+\left|R_{i}\left(v^{n}(t, x)\right)-R_{i}(v(t, x))\right| \nonumber \\ & \leq \epsilon_{p}(t, x)+K(p) \|v^{n}(t, x)-v(t, x)||. \end{align} $$

The right-hand side of this inequality tends to 0 by definition of D.

According to the above analysis, we can pass to the limit as $n \rightarrow +\infty $ in (3.45) and (3.46) and we obtain that

(3.49) $$ \begin{align} -\int_{\Omega} T_{b}\left(v_{i 0}\right) T_{b}^{\prime}\left(\eta V_{i}(0)\right) \Psi(0) \,{+}\, & k\int_{0}^{T}\!\int_{\Gamma} [\![ \Psi T^{\prime}_{b}(v_{i})T^{\prime}_{b}(\eta V_{i}) ]\!] [\![ \varphi_{i}(v_{i}) ]\!]\nonumber \\ {+}\, &\int_{Q_{T}}\!\kern-1.3pt -T_{b}\left(v_{i}\right) T_{b}^{\prime}\left(\eta V_{i}\right) \partial_{t} \Psi+T_{b}^{\prime}\left(v_{i}\right) T_{b}^{\prime}\left(\eta V_{i}\right) \nabla \varphi_{i}\left(v_{i}\right) \nabla \Psi \nonumber \\ &\geq \int_{Q_{T}} T_{b}^{\prime}\left(v_{i}\right) T_{b}^{\prime}\left(\eta V_{i}\right) R_{i}(v) \Psi-C D(\Psi) \eta^{\delta}.\\[-24pt]\nonumber \end{align} $$

• • Now we make $\eta \rightarrow 0$ for b fixed in (3.49). Since $R_{i}^{n}\left (v^{n}\right )$ converges a.e. to $R_{i}(v)$ (see (3.47) and is bounded in $L^{1}\left (Q_{T}\right )$ , Fatou’s lemma implies that $R_{i}(v) \in L^{1}\left (Q_{T}\right )$ . We can observe that $T_b^{\prime }(r)= \chi _{[0,b-1]}(r)$ . Therefore, by continuity of $T_b^{\prime }$ (recall that $T_b \in C^3 $ ), when $\eta \rightarrow 0$ , $T_b^{\prime }\left (\eta V_i\right ) \rightarrow \chi _{[0,b-1]}(0):=1$ a.e. in $Q_T$ and remains bounded by 1, then by dominated convergence and thanks to the positivity of $\delta $ , we obtain

(3.50) $$ \begin{align} -\int_{\Omega} T_{b}\left(v_{i 0}\right) \Psi(0)&+ k\int_{0}^{T}\int_{\Gamma} [\![ \Psi T^{\prime}_{b}(v_{i})]\!] [\![ \varphi_{i}(v_{i}) ]\!] \nonumber\\ &+\int_{Q_{T}}-T_{b}\left(v_{i}\right) \partial_{t} \Psi+T_{b}^{\prime}\left(v_{i}\right) \nabla \varphi_{i}\left(v_{i}\right) \nabla \Psi \geq \int_{Q_{T}} T_{b}^{\prime}\left(v_{i}\right) R_{i}(v) \Psi. \end{align} $$

• • Finally, we let $b \rightarrow +\infty $ in this inequality (3.50). Then $T_{b}\left (v_{i}\right )$ increases to $v_{i}$ and $T_{b}^{\prime }\left (v_{i}\right )$ increases to $1, \nabla \varphi _{i}\left (v_{i}\right )$ is at least in $L^{1}\left (Q_{T}\right )$ (see (3.24) of Lemma 3.2 and $R_{i}(v) \in L^{1}\left (Q_{T}\right )$ . Therefore, we easily pass to the limit in (3.50) to obtain

(3.51) $$ \begin{align} -\int_{\Omega} v_{i 0} \Psi(0)+ k\int_{0}^{T}\int_{\Gamma} [\![ \Psi]\!] [\![ \varphi_{i}(v_{i}) ]\!] + \int_{Q_{T}}-v_{i} \partial_{t} \Psi+\nabla \varphi_{i}\left(v_{i}\right) \nabla \Psi \geq \int_{Q_{T}} R_{i}(v) \Psi. \end{align} $$

And this ends the proof of Theorem 2.

3.5 Global existence of weak solution

In this section, we finalize the proof of Theorem 1. We use the approximate system constructed earlier in Section 3.1. From Theorem 2, we already know that the limit v is a supersolution. In order to conclude the proof of Theorem 1, it is necessary to show that this supersolution is also a subsolution. To do this, we use the mass control structure property (2.6).

Proof of Theorem 1

By Theorem 2, up to subsequence, the approximate solution $v_{i}^{n}$ of the system (3.2) converges to a weak supersolution. We will show using the property of the mass control structure (2.6) that the inverse inequality of (3.51) is satisfied for the sum of its m expressions, i.e.,

(3.52) $$ \begin{align} -\int_{\Omega}\left[\sum_{i} v_{i 0}\right] \Psi(0) &+ k\int_{0}^{T}\int_{\Gamma} \left[ \sum_{i}[\![ \varphi_{i}(v_{i}) ]\!] \right][\![ \Psi]\!] \nonumber \\ &+\int_{Q_{T}}-\left[\sum_{i} v_{i}\right] \partial_{t} \Psi+\left[\sum_{i} \nabla \varphi_{i}\left(v_{i}\right)\right] \nabla \Psi \leq \int_{Q_{T}}\left[\sum_{i} R_{i}(v)\right] \Psi. \end{align} $$

This will imply that equality holds in each of the inequalities (3.51).

First, we recall some convergence results obtained above in the previous subsection. When $n \to + \infty $ , we have

(3.53) $$ \begin{align} \begin{cases} v_{i}^{n} \to v_{i} \textrm{ in } L^{1}(Q_{T}) \textrm{ and a.e. in } Q_{T},\\ \nabla \varphi_{i}(v_{i}^{n}) \rightharpoonup \nabla \varphi_{i}(v_{i}) \textrm{ weakly in } L^{2}(Q_{T}),\\ {v_{i}^{n}}_{\vert \Gamma} \to {v_{i}}_{\vert \Gamma} \textrm{ in } L^{1}(\Gamma). \end{cases} \end{align} $$

Let us look again at the approximate system (3.2) and add up the m equations. We then obtain that, for all $\Psi \in W_T$ ,

(3.54) $$ \begin{align} -\int_{\Omega}\left[\sum_{i} v_{i 0}^{n}\right] \Psi(0) &+ k\int_{0}^{T}\int_{\Gamma} \left[ \sum_{i}[\![ \varphi_{i}(v_{i}^{n}) ]\!] \right][\![ \Psi]\!] \nonumber\\ &+\int_{Q_{T}}-\left[\sum_{i} v_{i}^{n}\right] \partial_{t} \Psi+\left[\sum_{i} \nabla \varphi_{i}\left(v_{i}^{n}\right)\right] \nabla \Psi=\int_{Q_{T}}\left[\sum_{i} R_{i}^{n}\left(v^{n}\right)\right] \Psi. \end{align} $$

For the right-hand side of (3.54), thanks to the hypothesis (2.6) on $R_{i}^{n}$ , it follows that

$$ \begin{align*}C\left\|v^{n}\right\|+h-\sum_{i} R_{i}^{n}\left(v^{n}\right) \geq 0. \end{align*} $$

By a.e. convergence of all function $R_i, i=1,\ldots ,m$ (see 3.47), by $L^{1}(Q_{T})$ -convergence of $v^{n}$ and by Fatou’s lemma, we have

$$ \begin{align*}\int_{Q_{T}}\left[C\|v\|+h-\sum_{i} R_{i}(v)\right] \Psi \leq \int_{Q_{T}}(C\|v\|+h) \Psi+\liminf _{n \rightarrow+\infty} \int_{Q_{T}}-\left[\sum_{i} R_{i}^{n}\left(v^{n}\right)\right] \Psi. \end{align*} $$

Therefore,

$$ \begin{align*}\liminf _{n \rightarrow \infty} \int_{Q_{T}}\left[\sum_{i} R_{i}^{n}\left(v^{n}\right)\right] \Psi \leq \int_{Q_{T}}\left[\sum_{i} R_{i}(v)\right] \Psi. \end{align*} $$

Thus, by passing to the limit in equation (3.54) and using the convergence results (3.53), we arrive at the inequality (3.52). And, as explained above, this implies that the equality holds in (3.51), i.e.,

(3.55) $$ \begin{align} -\int_{\Omega} v_{i 0} \Psi(0)+ k\int_{0}^{T}\int_{\Gamma} [\![ \Psi]\!] [\![ \varphi_{i}(v_{i}) ]\!] + \int_{Q_{T}}-v_{i} \partial_{t} \Psi+\nabla \varphi_{i}\left(v_{i}\right) \nabla \Psi = \int_{Q_{T}} R_{i}(v) \Psi. \end{align} $$

And finally, thanks to an integration from, we arrive at the following formulation:

(3.56) $$ \begin{align} -\int_{\Omega} v_{i 0} \Psi(0) - \int_{Q_{T}} v_{i} \partial_{t} \Psi + \varphi_{i}\left(v_{i}\right) \Delta \Psi=\int_{Q_{T}} R_{i}(v) \Psi.\\[-36pt]\nonumber \end{align} $$