$$\label{eq:segregini} 
\exists\, \zeta^0 \in \RR \mbox{ such that } {n_1}^{\mbox{\scriptsize ini}}_{\epsilon}= {n}^{\mbox{\scriptsize ini}}_{\epsilon} \mathbf{1}_{x \leq  \zeta^0} \mbox{ and } {n_2}^{\mbox{\scriptsize ini}}_{\epsilon}= {n}^{\mbox{\scriptsize ini}}_{\epsilon} \mathbf{1}_{x \geq  \zeta^0},$$ and $$\label{eq:positivity} 
{n_1}^{\mbox{\scriptsize ini}}_{\epsilon},{n_2}^{\mbox{\scriptsize ini}}_{\epsilon} \geq 0 \mbox{ and } 0 < A_0 \leq {n_1}^{\mbox{\scriptsize ini}}_{\epsilon}+{n_2}^{\mbox{\scriptsize ini}}_{\epsilon} \leq B_0$$ then there exists $\zeta_{\epsilon} \in C([0,\infty)) \cap C^1( (0,\infty))$ such that $$\label{eq:segre} 
{n_1}_{\epsilon} (t,x)= {n}_{\epsilon} (t,x)\mathbf{1}_{x \leq  \zeta_{\epsilon}(t)} \quad \mbox{ and } \quad {n_2}_{\epsilon}(t,x)= {n}_{\epsilon}(t,x) \mathbf{1}_{x \geq  \zeta_{\epsilon}(t)} ,$$ and ${n_1}_{\epsilon} $ and ${n_2}_{\epsilon} $ respectively satisfy on $\{ (t,x), x\leq \zeta_{\epsilon}(t)\}$ and on $\{ (t,x), x\geq \zeta_{\epsilon}(t)\}$. In addition ${n}_{\epsilon} = {n_1}_{\epsilon} +{n_2}_{\epsilon} $ is solution to: $$\label{eq:solBPM}
  \left\{
      \begin{aligned}
         & \partial_t {n}_{\epsilon} - \partial_x ({n}_{\epsilon} \partial_x p_{\epsilon}) = {n}_{\epsilon} G_1(p_{\epsilon}) \quad \mbox{ on } \{ (t,x), x\leq \zeta_{\epsilon}(t)\},\\
        & \partial_t {n}_{\epsilon} - \partial_x ({n}_{\epsilon} \partial_x p_{\epsilon}) = {n}_{\epsilon} G_2(p_{\epsilon}) \quad \mbox{ on } \{ (t,x), x\geq \zeta_{\epsilon}(t)\},\\
        & n_\epsilon(t,\zeta_{\epsilon}(t)^-)=n_\epsilon(t,\zeta_{\epsilon}(t)^+), \\
        & \zeta_{\epsilon}'(t) = - \partial_x p(t,\zeta_{\epsilon}(t)^-)= - \partial_x p(t,\zeta_{\epsilon}(t)^+),\\
        & \partial_x {n}_{\epsilon}(\pm L,0) = 0\mbox{ for } t>0.
      \end{aligned}
    \right.$$ In [@BPM], the reaction term is not the same than in this paper, however it is easy to see that there proof can be extended to our system under a set of assumptions 