\frac{{\mathrm d}}{{\mathrm d}t}c_t=A(t,c_t)\quad\text{in }[0,T],\qquad  c_0=\bar c.$$ Moreover, if $c^1,\,c^2$ are the solutions starting from the initial data $\bar c^1,\,\bar c^2\in C$ respectively, we have $$\label{eq:18}
\|c^1_t-c^2_t\|_E\le \mathrm e^{Lt}\|\bar c^1-\bar c^2\|_E,\qquad\text{for every $t\in [0,T]$.}$$

For our purposes, we need the following generalization.

\[cor:Brezis\] Let hypotheses (ii) and (iii) of Theorem \[thm:Brezis\] hold for a family of operators $A(t,\cdot)\colon C\to E$, $t\in[0,T]$. Assume, in addition that

1.  for every $R>0$ there exists a constant $L_R{\geqslant}0$ such that for every $c_1,\,c_2\in C\cap B_R$ and $t\in [0,T]$ $$\label{eq:151}
    \|A(t,c_1)-A(t,c_2)\|_E\le L_R\|c_1-c_2\|_E;$$

2.  there exists $M>0$ such that for every $c\in C$, there holds $$\label{eq:152}
    \lVert A(t,c)\rVert_E{\leqslant}M(1+\lVert c\rVert_E).$$

Then for every $\bar c\in C$ there exists a unique curve $c\colon[0,T]\to C$ of class $C^1$ satisfying $c_t\in C$ for all $t\in [0,T]$ and $$\label{eq:171}
\frac{{\mathrm d}}{{\mathrm d}t}c_t=A(t,c_t)\quad\text{in }[0,T],\qquad  c_0=\bar c.$$ Moreover, if $c^1,\,c^2$ are the solutions starting from the initial data $\bar c^1,\,\bar c^2\in C\cap B_R$ respectively, there exists a constant $L=L(M,R,T)>0$ such that $$\label{eq:181}
\|c^1_t-c^2_t\|_E\le \mathrm e^{Lt}\|\bar c^1-\bar c^2\|_E,\qquad\text{for every $t\in [0,T]$.}$$

Let us fix the initial datum $\bar c\in C$, and let us choose $\bar R\coloneqq (\lVert \bar c\rVert_E+MT)e^{MT}$. Consider a smooth function with compact support $\chi\colon {\mathbb{R}^{+}}\to [0,1]$ such that $\chi(r)=1$ for every $r{\leqslant}\bar 