= Solution
{c}
For each $t$, let
$$
B_t[u,v]=\int_U\left(\sum_{i,j}a^{ij}(x,t)u_{x_i}v_{x_j}
+\sum_i b^i(x,t)u_{x_i}v+c(x,t)uv\right)dx.
$$
A <weak energy solution of a variable-coefficient wave equation> is a function $u\in H^1(U_T)$ with zero lateral trace, equivalently $u\in L^2(0,T;H_0^1(U))$, whose time trace satisfies $u(0)=\psi_0$ in $L^2(U)$ and for which
$$
\boxed{\int_0^T\bigl[-(u_t,v_t)_{L^2}+B_t[u,v]\bigr]dt
=\int_0^T(f,v)_{L^2}dt+(\psi_1,v(0))_{L^2}}
$$
for every $v\in H^1(U_T)$ with zero lateral trace and $v(T)=0$. Here $u\in H^1(0,T;L^2(U))$ has a continuous $L^2$ representative, so the displacement trace is well defined.
The velocity condition is encoded by the boundary term in time; an arbitrary space-time $H^1$ function need not have an $L^2$ trace of $u_t$. The equation implies $u_{tt}=f-L(t)u\in L^2(0,T;H^{-1}(U))$, hence $u_t$ has a continuous $H^{-1}$ representative and $u_t(0)=\psi_1$ in that sense. This is the <weak formulation> obtained by <integration by parts> once in time and once in space.
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