is defined in . We deduce from that $$\begin{aligned}
\pa_t (n_\epsilon - n_M) -\partial_{xx} (H(n_\epsilon)-H(n_M)) & \leq  {n_1}_\epsilon G_1(P(n_\epsilon)) - n_M  \mathbf{1}_{x\leq  \zeta_{\epsilon}(t)}  G_1(P_M) \\
& + {n_2}_\epsilon G_2(P(n_\epsilon)) - n_M \mathbf{1}_{x\geq  \zeta_{\epsilon}(t)}  G_2(P_M),\end{aligned}$$ where we use the monotonicity of $G_1$ and $G_2$ from assumption .

Notice that, since the function $H$ is nondecreasing, the sign of $n_{\epsilon}-m_{\epsilon}$ is the same as the sign of $H(n_{\epsilon})-H(m_{\epsilon})$. Moreover, $$\partial_{xx} f(y) = f''(y) |\partial_x y|^2 +f'(y) \partial_{xx} y,$$ so for $y=H(n_{\epsilon})-H(n_M)$ and $f(y)=y_+$ the positive part, the so-called Kato inequality reads $\partial_{xx} f(y) \geq f'(y) \partial_{xx} y$. Thus multiplying the latter equation by $\mathbf{1}_{n_\epsilon-n_M>0}$ and given we obtain $$\begin{aligned}
\pa_t |n_\epsilon - n_M|_+ -\partial_{xx} |H(n_\epsilon)-H(n_M)|_+ \leq
({n}_\epsilon - n_M) \mathbf{1}_{x\leq  \zeta_{\epsilon}(t)}  G_1(P(n_\epsilon)) \mathbf{1}_{n_\epsilon-n_M>0} \\
+ ({n}_\epsilon - n_M) \mathbf{1}_{x\geq  \zeta_{\epsilon}(t)}  G_2(P(n_\epsilon)) \mathbf{1}_{n_\epsilon-n_M>0} \\
+ n_M (G_1(P(n_\epsilon)) - G_1(P(n_M))  + G_2(P(n_\epsilon)) - G_2(P(n_M))) \mathbf{1}_{n_\epsilon-n_M>0}.\end{aligned}$$ Since the function $P$ is increasing and $G_1$ and $G_2$ are decreasing (see ), we deduce that the last term is nonpositive. Then, integrating on $(-L,L)$ and using $\partial_x n_\epsilon (\pm L,t) =0 $, we deduce $$\begin{aligned}
\frac{d}{dt} \int_{-L}^L |n_\epsilon - n_M|_+ \,dx \leq & \ \partial_{x} |H(n_\epsilon)-H(n_M)|_+(L,t) - \partial_{x} |H(n_\epsilon)-H(n_M)|_+(-L,t) \\
& + \int_{-L}^{ \zeta_{\epsilon}(t)} ({n}_\epsilon - n_M) 