Conservation of total wave energy in three admissible settings

Localisation in time and the common shape of the three cases: fix a nondegenerate compact interval $[a,b]\subseteq (0,T)$; it suffices to prove that $t\mapsto E_{\mathrm{\Omega }}(t)$ is constant on $[a,b]$, since $[a,b]$ is arbitrary and $(0,T)$ is an interval [F6]; the local balance [F1] gives ${\mathrm{\partial }}_{t}e=-\mathrm{div}q$ pointwise, while the differentiation and boundedness arguments needed to integrate this identity are supplied separately under the hypotheses of (a), (b), and (c).

Case (a): choose $R>0$ with the compact $K$ of the statement contained in $B_R(0)$; for $t\in [a,b]$ the support condition gives $e(x,t)=0$ for $x\mathrm{\notin }K$, so $E_{{\mathbb{R}}^n}(t)={\int }_{B_R(0)}e(x,t)dx$, and $u$, $u_t$ and $Du$ all vanish identically on the open complement of $K$, hence on a neighbourhood of $\mathrm{\partial }{B}_R(0)$, so $q=0$ there; [F2] applied on the fixed ball with the domination constant ${\sup }_{{\bar{B}}_R\times [a,b]}\mathrm{\mid }{\mathrm{\partial }}_{t}e\mathrm{\mid }$ gives $E_{{\mathbb{R}}^n}^{\mathrm{\prime }}(t)={\int }_{B_R(0)}{\mathrm{\partial }}_{t}e(x,t)dx=-{\int }_{B_R(0)}\mathrm{div}q(x,t)dx$ on $(a,b)$, for $n\ge 2$, [F4] on the ball gives ${\int }_{B_R(0)}\mathrm{div}qd{\lambda }_n={\int }_{\mathrm{\partial }{B}_R(0)}q\cdot \nu dS=0$ because $q$ vanishes on the boundary; for $n=1$, [F7] and [F8] instead give ${\int }_{-R}^R{\mathrm{\partial }}_{x}qdx=q(R,t)-q(-R,t)=0$; hence $E_{{\mathbb{R}}^n}^{\mathrm{\prime }}=0$ on $(a,b)$ and $E_{{\mathbb{R}}^n}$ is constant on $[a,b]$ by [F6].

Case (b): the differentiation hypothesis gives $E_{{\mathbb{R}}^n}^{\mathrm{\prime }}(t)={\int }_{{\mathbb{R}}^n}{\mathrm{\partial }}_{t}e(x,t)dx$ for every $t\in (0,T)$ (the stated sufficient Lebesgue criterion follows from [F2] on an open interval compactly contained in $(0,T)$, with the fixed dominating $L^1$ function), and [F1] makes the integrand $-\mathrm{div}q(\cdot ,t)$, which lies in $L^1({\lambda }_n)$ by hypothesis; hence $E_{{\mathbb{R}}^n}^{\mathrm{\prime }}(t)=-{\int }_{{\mathbb{R}}^n}\mathrm{div}q(x,t)dx=0$ by [F5], and [F6] makes $E_{{\mathbb{R}}^n}$ constant on $[a,b]$.

Case (c), dimension $n\ge 2$: $e$ and ${\mathrm{\partial }}_{t}e$ are continuous on the compact $\bar{U}\times [a,b]$, so [F2] gives $E_U^{\mathrm{\prime }}(t)={\int }_U{\mathrm{\partial }}_{t}e(x,t)dx=-{\int }_U\mathrm{div}q(x,t)dx$ on $(a,b)$ [F1], and [F4] gives ${\int }_U\mathrm{div}qd{\lambda }_n={\int }_{\mathrm{\partial }U}q\cdot \nu dS$; the boundary integrand vanishes: in the Dirichlet case the map $t\mapsto u(p,t)$ is identically zero at every $p\in \mathrm{\partial }U$ and differentiable with derivative ${\mathrm{\partial }}_{t}u(p,t)$ (the $C^1$ extension to $\bar{U}$ makes the difference quotient converge to the continuous extension of ${\mathrm{\partial }}_{t}u$), so ${\mathrm{\partial }}_{t}u(p,t)=0$ and hence $q(p,t)\cdot \nu (p)=-c^2{u}_t(p,t){\mathrm{\partial }}_{\nu }u(p,t)=0$; in the Neumann case ${\mathrm{\partial }}_{\nu }u=0$ on the boundary by hypothesis; either way $q\cdot \nu =0$ on $\mathrm{\partial }U\times (a,b)$, so $E_U^{\mathrm{\prime }}=0$ on $(a,b)$ and [F6] gives constancy on $[a,b]$.

Case (c), dimension $n=1$: write $U$ as the disjoint union of its finitely many bounded open intervals $({\alpha }_j,{\beta }_j)$ with pairwise disjoint closures; for each $j$ the function $x\mapsto q(x,t)$ is $C^1$ on $[{\alpha }_j,{\beta }_j]$, so on that interval [F7] gives the Darboux integral ${\int }_{{\alpha }_j}^{{\beta }_j}{\mathrm{\partial }}_{x}q(x,t)dx=q({\beta }_j,t)-q({\alpha }_j,t)$, [F8] converts this Darboux value first to the Riemann and then to the Lebesgue integral of ${\mathrm{\partial }}_{x}q(\cdot ,t)$ over the interval, and at each endpoint both boundary conditions kill $q$: $q({\beta }_j,t)=-c^2{u}_t({\beta }_j,t)u_x({\beta }_j,t)$ and $q({\alpha }_j,t)=-c^2{u}_t({\alpha }_j,t)u_x({\alpha }_j,t)$, and in the Dirichlet case $u_t=0$ at both endpoints while in the Neumann case the outward normal is $+1$ at ${\beta }_j$ and $-1$ at ${\alpha }_j$, so $u_x({\beta }_j,t)=0=u_x({\alpha }_j,t)$; hence ${\int }_U\mathrm{div}q(\cdot ,t)d{\lambda }_1={\sum }_j(q({\beta }_j,t)-q({\alpha }_j,t))=0$. Since $e$ and ${\mathrm{\partial }}_{t}e$ are continuous on $\bar{U}\times [a,b]$, [F2] gives $E_U^{\mathrm{\prime }}(t)={\int }_U{\mathrm{\partial }}_{t}ed{\lambda }_1=-{\int }_U\mathrm{div}qd{\lambda }_1=0$ on $(a,b)$, and [F6] gives constancy on $[a,b]$.

Completion: in each of the three settings, and in both dimensions of case (c), the energy $E_{\mathrm{\Omega }}$ has vanishing derivative on every nondegenerate compact subinterval of $(0,T)$, hence is constant on each such subinterval by [F6]; a function constant on every compact subinterval of an interval is constant on the interval, so $E_{\mathrm{\Omega }}$ is constant on $(0,T)$ in all three settings.