= Weak energy solution of a variable-coefficient wave equation
For $u_{tt}+L(t)u=f$ with homogeneous <Dirichlet boundary conditions>, smooth uniformly elliptic principal coefficients, and initial data $(\psi_0,\psi_1)\in H_0^1(U)\times L^2(U)$, a space-time $H^1$ <weak solution> has zero lateral trace, trace $u(0)=\psi_0$, and satisfies
$$
\int_0^T\bigl[-(u_t,v_t)_{L^2}+B_t[u,v]\bigr]dt
=\int_0^T(f,v)_{L^2}dt+(\psi_1,v(0))_{L^2}
$$
for every space-time $H^1$ test function with zero lateral trace and $v(T)=0$. The velocity initial condition is encoded by the last term; an arbitrary space-time $H^1$ function need not have an $L^2$ trace of its time derivative. The equation itself gives $u_{tt}\in L^2(0,T;H^{-1}(U))$, so $u_t$ has a continuous $H^{-1}$ representative.
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