whose proof can be easily adapted from [@BRR1 Proposition 3.2]. Before stating it we point out that, as usual, by a *supersolution* (resp. *subsolution*) of we mean a pair $(u,v)$ satisfying System with $\geq$ (resp. $\leq$) instead of $=$.

\[prcp1\] Let $(\un{u}, \un{v})$ and $(\ov{u},\ov{v})$ be, respectively, a subsolution bounded from above and a supersolution bounded from below of satisfying $\un{u}\leq\ov{u}$ and $\un{v}\leq\ov{v}$ at $t = 0$. Then, either $\un{u}<\ov{u}$ and $\un{v}<\ov{v}$ for all $t$, or there exists $T > 0$ such that $(\un{u}, \un{v})=(\ov{u},\ov{v})$ for $t\leq T$.

We also need the following comparison principle regarding an extended class of generalized subsolutions and which is a particular instance of [@BRR1 Proposition 3.3].

\[prcp2\] Let $E\subset {{\mathbb{R}}}\times{{\mathbb{R}}}_+$ and $F\subset \O\times{{\mathbb{R}}}_+$ be two open sets, let $(u_1, v_1)$ be a subsolution of bounded from above and satisfying $$u_1 = 0 \text{ on } (\p E) \cap ({{\mathbb{R}}}\times{{\mathbb{R}}}_+), \quad v_1= 0 \text{ on } (\p F) \cap (\O\times{{\mathbb{R}}}_+),$$ and consider $$\un{u} :=\begin{cases}
\max\{u_1, 0\} & \text{in $\ov{E}$} \\
0 & \text{otherwise,}
\end{cases} 
\qquad
\un{v} :=\begin{cases}
\max\{v_1, 0\} & \text{in $\ov{F}$} \\
0 & \text{otherwise.}
\end{cases}$$ If they satisfy $$\label{21}
\begin{split}
\un{v}(x, L, t) &\geq v_1(x, L, t) \quad \text{ for all } (x,t) 