Let $A$ be the closure of the Laplacian operator acting on the space $H^2(\Omega)$ for $\Omega \subset \mathbb{R}^n$ a domain. Given the standard homogeneous wave equation:
\begin{aligned}
u''(t) - Au(t) = 0 &\;\; \text{in } \Omega \times (0,T)\\
u(0) = x_1, u'(0) = x_2 &\;\; x_1 \in H^2(\Omega) \cap H_0^1(\Omega), x_2 \in H_0^1(\Omega),
\end{aligned}
we can reduce this to a first order Cauchy problem:
\begin{align*}
y'(t) &= By(t)\\
y(0) &= y_0.
\end{align*}
In the above, $y(t) = (u(t),u'(t))^T,\; y_0 = (x_1,x_2)^T,$ and
\begin{equation}
B = \begin{pmatrix}
0 & id\\
A & 0
\end{pmatrix}.
\end{equation}
If $Y = H^1_0(\Omega) \times L^2(\Omega)$, then $B$ generates a strongly continuous semigroup on $Y$.
My question is: does there exist an explicit formula for the action of this semigroup on $Y$, and does it relate to the Green's function for the wave equation? If so, can we then combine this with the variation of parameters formula used for Volterra equations to arrive at a mild solution for the inhomogeneous wave equation?