\mathbf{1}_{n_\epsilon-n_M>0} G_1(P(n_\epsilon)) \,dx  \\
& + \int^{L}_{ \zeta_{\epsilon}(t)} ({n}_\epsilon - n_M) \mathbf{1}_{n_\epsilon-n_M>0} G_2(P(n_\epsilon)) \,dx \\
\leq &\  G_m \int_{-L}^L  |n_\epsilon - n_M|_+ \,dx.\end{aligned}$$

Then, integrating in time, we deduce $$\int_{-L}^L  |n_\epsilon - n_M|_+ \,dx \leq e^{G_m t} \int_{-L}^L  |n_\epsilon^{\mbox{\scriptsize ini}} - n_M|_+ \,dx = 0.$$

[**$L^{\infty}$ bounds**]{}.

From , we have $p_{\epsilon}^{\mbox{\scriptsize ini}} \leq P_M$. Since the function $P$ is inscreasing, we have $ n_\epsilon^{\mbox{\scriptsize ini}} \leq n_M$. With the above comparison principle, we conclude that $n_\epsilon \leq n_M$. We deduce easily with the non-negativity principle that $0 \leq p_\epsilon \leq P_M$, $ 0 \leq {n_1}_\epsilon\leq n_M$ and $ 0 \leq {n_2}_\epsilon \leq n_M$.

[**Estimates from below**]{}.

From above, we deduce that the pressure is bounded by $P_M$. Hence, using assumption we deduce $$\pa_t n_\epsilon -\partial_{xx} H(n_\epsilon) =  {n_1}_\epsilon G_1(P(n_\epsilon))
+ {n_2}_\epsilon G_2(P(n_\epsilon)) \geq - n_\epsilon g_m.$$ Let us introduce $n_m := A_0 e^{-g_m t}$. We deduce $$\pa_t (n_m-n_\epsilon) -\partial_{xx} (H(n_m) - H(n_\epsilon)) \leq -(n_m- n_\epsilon) g_m.$$ As above, for the comparison principle, we may use the positive part and the Kato inequality to deduce $$\pa_t |n_m-n_\epsilon|_+ -\partial_{xx} |H(n_m) - H(n_\epsilon)|_+ \leq - |n_m- n_\epsilon|_+ g_m.$$ Integrating in space and in time as above, we deduce that $|n_m-n_\epsilon|_+=0$.

[**$L^{1}$ bounds 