The maximal solution domain is open

Let $(t_{\ast },x_{\ast })\in \mathrm{\Omega }$ and put $p:=\mathrm{\Phi }(t_{\ast },x_{\ast })$. By [L1], applied at the [L1, choose] state point $p$, there exist $\delta >0$ and an open neighbourhood $W\subseteq U$ of $p$ such that every $y\in W$ has a unique solution on $[-\delta ,\delta ]$, and these solutions vary smoothly with $y$.

By [L3], for initial states $x$ sufficiently close to $x_{\ast }$, the solution [L3, step 1.1] $\mathrm{\Phi }(\cdot ,x)$ is defined at least on a compact interval around $t_{\ast }$ and its value at time $t_{\ast }$ lies in $W$. Therefore for every such $x$ and every $\mathrm{\mid }s\mathrm{\mid }<\delta$, the solution continued from time $t_{\ast }$ by the local flow of step 1.1 is defined at time $t_{\ast }+s$. Hence all pairs $(t_{\ast }+s,x)$ with $\mathrm{\mid }s\mathrm{\mid }<\delta$ and $x$ near $x_{\ast }$ lie in $\mathrm{\Omega }$.

Step 2.1 gives an open neighbourhood of $(t_{\ast },x_{\ast })$ contained in $\mathrm{\Omega }$, [L1, L2, L3, step 2.1] so $\mathrm{\Omega }$ is open. On that neighbourhood, the evaluation map is the composite of the continuous time-$t_{\ast }$ map with the local smooth flow from step 1.1, hence is jointly continuous and smooth in the state variable. Since $(t_{\ast },x_{\ast })$ was arbitrary, the same holds on all of $\mathrm{\Omega }$.