Energy well-posedness for the variable-metric wave equation

Set $c=1+\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}$, $a(u,v)=b(u,v)-c(u,v{)}_{L^2}$, and let $P_J$ be projection on $e_1,\dots ,e_J$. Solve the scalar equations $d_j^{\mathrm{\prime }\mathrm{\prime }}+({\mu }_j-c)d_j=(F,e_j)$ with the projected initial data, and put $u_J={\sum }_{j\le J}{d}_j{e}_j$. The scalar solutions exist for $L^1$ forcing by integration against the sine/hyperbolic-sine fundamental solution (at zero frequency use $t$); they have continuous first derivatives and absolutely continuous first derivatives on compact intervals. For $E_J=(\mathrm{\parallel }{u}_J^{\mathrm{\prime }}{\mathrm{\parallel }}_2^2+b(u_J,u_J))\mathrm{/}2$, multiply each equation by ${\bar{d}}_j^{\mathrm{\prime }}$ and take real parts to obtain $E_J^{\mathrm{\prime }}=\mathrm{Re}(P_{J}F+c{u}_J,u_J^{\mathrm{\prime }})\le C{E}_J+\sqrt{2E_J}\mathrm{\parallel }F{\mathrm{\parallel }}_2\mathrm{.}$ Apply this to $\sqrt{E_J+\eta }$ and let $\eta \downarrow 0$ after integrating: multiplication by $e^{-Ct}$ gives $\sqrt{E_J(t)}\le C_T(\sqrt{E_J(0)}+{\int }_0^T\mathrm{\parallel }F{\mathrm{\parallel }}_{2}dt)$. This is exactly the asserted estimate, uniformly in $J$.

For $J>K$, the same estimate for $u_J-u_K$ has data $(P_J-P_K)u_0,(P_J-P_K)u_1$ and forcing $(P_J-P_K)F$. These data tend to zero in $V\times H$ by [F1]. The forcing tends to zero in $L_t^{1}H$ because the projections converge pointwise in $H$ and are contractive, so dominated convergence applies. Hence $u_J$ is Cauchy in $C([0,T];V)$ and $u_J^{\mathrm{\prime }}$ in $C([0,T];H)$. Their limits $u,w$ satisfy $u(t)-u(s)={\int }_s^{t}w(r)dr$, so $u\in C^1([0,T];H)$ with derivative $w$ and the exact strong initial traces. Pass to the limit in the integrated finite-dimensional equation first for a finite spectral test and then for every $\varphi \in V$, using boundedness of $a$ and the form-density in [F1]. This gives precisely the displayed weak equation and energy bound. No weak-in-time limiting trace is substituted for a strong trace.

If two energy solutions have the same data and forcing, test their difference against $e_j$. The scalar coefficient is a distributional solution of $d_j^{\mathrm{\prime }\mathrm{\prime }}+({\mu }_j-c)d_j=0$ with zero strong data. Its integral equation, or the scalar fundamental solution, gives $d_j=0$. Completeness in [F1] makes the difference zero for every time. The estimate of step 1.1 passes to the strong limit, including its zero-data version. Replacing $t$ by $T-t$ proves terminal well-posedness with the same bound.

We also establish the classical regularity needed by the last assertion and the DN interfaces. Let $h$ be smooth and zero near the temporal endpoints, take the smooth lift $H$ from [F2], and solve for $w=u-H$ with zero Dirichlet and initial data and forcing $G=-L_{q}H$. This $G$ is smooth and zero near $t=0$. For every integer $\mathrm{\ell }\ge 0$, solve the same zero-data energy problem with forcing ${\mathrm{\partial }}_t^{\mathrm{\ell }}G$. Integration in time and uniqueness in step 3.1 identify these solutions successively with ${\mathrm{\partial }}_t^{\mathrm{\ell }}w$: for example integrating the solution with forcing $G_t$ gives a solution with forcing $G-G(0)=G$ and zero data. Thus all time derivatives of $w$ are continuous in $V$. At every time, $Aw=G-w_{tt}$; the right side is continuous in $L^2$, so [F1]'s elliptic estimate gives continuity of $w$ in $H^2$, by applying the estimate to time differences. Apply this argument to every time derivative. Inductively if all time derivatives are continuous in $H^k$, the identity $A{\mathrm{\partial }}_t^{\mathrm{\ell }}w={\mathrm{\partial }}_t^{\mathrm{\ell }}G-{\mathrm{\partial }}_t^{\mathrm{\ell }+2}w$ and [F1] give continuity in $H^{k+2}$. Thus all mixed derivatives have arbitrarily high spatial Sobolev regularity, and [F1]'s embedding gives a smooth representative on the closed cylinder, in particular $C^2$. Adding $H$ proves the stated smooth boundary reduction and actual classical forward solvability; time reversal gives the terminal version.