, then there exists $\zeta_{\epsilon} \in C([0,\infty)) \cap C^1( (0,\infty))$ such that ${n_1}_{\epsilon} $ and ${n_2}_{\epsilon} $ verify 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 .

Considering ${n_1}_\epsilon$ and ${n_2}_\epsilon$ defined previously, we have for $i=1,2$ $$\partial_t  {n_i}_\epsilon= \partial_t  {n}_{\epsilon} (t,x)\mathbf{1}_{x \leq  \zeta_{\epsilon}(t)} + {n}_{\epsilon} \zeta_{\epsilon}'(t)\delta_{x=\zeta_{\epsilon}(t)} .$$ Given , for all $\varphi \in C_c^{\infty} (-L,L)$ we compute, for $i=1, 2$ $$\begin{aligned}
\int_{\RR} \partial_t {n_i}_{\epsilon} \varphi ~dx = & \int_{-\infty}^{\zeta_{\epsilon}(t)} \partial_t {n}_{\epsilon} \varphi ~dx + {n}_{\epsilon} (t, \zeta_{\epsilon}(t)) \zeta_{\epsilon}'(t) \varphi(\zeta_{\epsilon}(t))\\
= &  \int_{-L}^{\zeta_{\epsilon}(t)} \partial_x ({n}_\epsilon \partial_x p_\epsilon) \varphi ~dx + \int_{-L}^L {n_i}_\epsilon G_i(p_{\epsilon}) \varphi ~dx + {n}_{\epsilon}(t,\zeta_{\epsilon}(t)) \zeta_{\epsilon}'(t) \varphi(\zeta_{\epsilon}(t)) \\
= &  - \int_{-L}^{\zeta_{\epsilon}(t)} {n}_\epsilon \partial_x p_\epsilon \partial_x \varphi ~dx  +  {n}_\epsilon(t, \zeta_{\epsilon}(t)) \partial_x p_\epsilon(t, \zeta_{\epsilon}(t)) \varphi( \zeta_{\epsilon}(t)) \\
& +  \int_{-L}^L {n_i}_\epsilon G_i(p_{\epsilon}) \varphi ~dx - {n}_{\epsilon}(t, \zeta_{\epsilon}(t)) \partial_x p_\epsilon(t,  \zeta_{\epsilon}(t))\varphi( \zeta_{\epsilon}(t)) \\ 
= &  \int_{-L}^L {n_i}_\epsilon  \partial_x p_\epsilon \partial_x \varphi ~ dx  +  \int_{-L}^L {n_i}_\epsilon G_i(p_{\epsilon}) \varphi ~dx \\
= &  \int_{-L}^L(\partial_x ({n_i}_\epsilon  \partial_x p_\epsilon) + {n_i}_\epsilon G_i(p_{\epsilon}) )\varphi ~ dx .  \end{aligned}$$ Hence ${n_1}_\epsilon$ and ${n_2}_\epsilon$ are weak solutions to and on $(-L,L)$ respectively. This result 