Let $$ u_{tt}-\Delta u=0\;\;\;\mbox{in}\;\;\;\mathbb{R}^n\times(0,t)\;\;\;\;\;\;(1.1)$$
$$u=f\;,u_t=g\;\;\mbox{in}\;\;\;\mathbb{R}^n\times\{t=0\}\;\;(1.2)$$
Defining $u_t=v$, the problem $(1.1)-(1.2)$ can be written $$U_t(t)=AU(t),\;\;U(0)=\left( \begin{array}{c} f \\ g \\ \end{array} \right)\;\;\;(1.3)$$ where $U=\left( \begin{array}{c} u \\ v \\ \end{array} \right)$ and $A=\left( \begin{array}{c} 0&1\\\Delta&0\\ \end{array} \right)$.
My problem is to show that $(1.3)$ is well-posedness in $H^s(\mathbb{R}^n)\times H^{s-1}(\mathbb{R}^n)$, but how to interpret $(1.3)$?