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 satisfiesfor 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.