The setup is: We have a Robertson Walker spacetime $V= I \times S$, where $I=[t_0, \infty) \subset \mathbb{R}$ with metric $$ g=-dt^2+R^2 \sigma $$ where $\sigma$ is a Riemannian metric on $S$, $S$ has non zero injectivity radius and $R$ a smooth function on $I$.
The wave equation is defined as
$$ \text{tr}_g (\nabla^2 u)=0$$
where $u:V \rightarrow M$ and $(M,h)$ is a Riemannian manifold.
I want to prove local in time existence, i.e. that for initial data for some $u_0,u_1 \in H^{1}(S)$ there exists a $T$, only depending on the bounds of the initial data, s.t. there exists a solution $u:[t_0,T) \times S \rightarrow M$ for
$$ \text{tr}_g (\nabla^2 u)=0\\ u(t_0,.)=u_0 \\ \partial_t u(t_0,.)=u_1$$
Now I know how to show, that there exists a $T$ and open sets $U_i$ covering $S$ and solutions $u_i: [t_0,T) \times U_i$. My problem is how to show that we can glue these solutions together to a global in space one.
I know that for any set $\Omega \subset S$, s.t. there exists a solution on the future Cauchy development of $\Omega$, on $D^+(\Omega)$:
Now using this uniqueness, we can obtain a solution on some neighborhood of $S$ in $M$, but I still need to prove that I obtain a solution for some $T$ on $[t_0, T) \times S$ and that $T$ only depends on the bounds of the initial data.
I only have an idea how to show this, if $S$ is compact. Then I could construct solutions on sets $U_i$ covering $S$ with the following properties:
- there exist open sets $W_i \subset U_i$ covering $S$, s.t. $D^+(W_i) \subset [t_0,T) \times U_i$
- there exist open sets $V_i \subset W_i$ covering $S$ and $T^{'}_i$, s.t. $[t_0, T^{'}_i) \times V_i \subset D^+(W_i)$
Since $S$ is compact, there exist finitely many $V_1,...,V_l$ covering $S$ and taking $T^{'}$ as the minimum of the $T^{'}_1,.., T^{'}_l$, we obtain a solution on
$[t_0,T^{'} ) \times S$. Is that idea correct? And how to prove this for non-compact $S$?