The local conservation law need not integrate to a finite conserved energy

The profile solves the equation and its density is a one-variable function: by [F2], $u_t(x,t)=-c{F}^{\mathrm{\prime }}(x_0-ct)$, ${\mathrm{\partial }}_{0}u(x,t)=F^{\mathrm{\prime }}(x_0-ct)$ and ${\mathrm{\partial }}_{i}u(x,t)=0$ for $i\mathrm{\ne }0$, so $u_{tt}=c^2{F}^{\mathrm{\prime }\mathrm{\prime }}(x_0-ct)$, $\mathrm{\Delta }u=F^{\mathrm{\prime }\mathrm{\prime }}(x_0-ct)$ and ${\mathrm{\square }}_{c}u=0$; hence [F1] holds, $u_t^2=c^2{F}^{\mathrm{\prime }}(x_0-ct{)}^2$, $\mathrm{\mid }Du{\mathrm{\mid }}^2=F^{\mathrm{\prime }}(x_0-ct{)}^2$ and $e(x,t)=c^2{F}^{\mathrm{\prime }}(x_0-ct{)}^2$, a nonnegative function of the single variable $x_0-ct$.

A positive lower bound on a slab: since $F^{\mathrm{\prime }}$ is continuous and not identically zero, there are $a<b$ and $m>0$ with $F^{\mathrm{\prime }}(s{)}^2\ge m$ for $s\in [a,b]$; consequently, for every $t$ and every $x^{\mathrm{\prime }}=(x_1,\dots ,x_{n-1})$, the density satisfies $e(x_0,x^{\mathrm{\prime }},t)=c^2{F}^{\mathrm{\prime }}(x_0-ct{)}^2\ge c^{2}m$ whenever $x_0\in [a+ct,b+ct]$.

The total energy diverges: fix $t$ and $R>0$ large enough that $[a+ct,b+ct]\subseteq [-R,R]$; Tonelli [F3] applied to the nonnegative product-measurable function $e$ (written in the product coordinates $(x_0,x^{\mathrm{\prime }})$, with $d{\lambda }_n$ the measure of [F6]; Borel product measurability and agreement of the product with Euclidean Lebesgue measure follow from The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n} and On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}) gives ${\int }_{(-R,R{)}^n}ed{\lambda }_n={\int }_{(-R,R{)}^{n-1}}({\int }_{-R}^R{c}^2{F}^{\mathrm{\prime }}(x_0-ct{)}^{2}d{x}_0)d{x}^{\mathrm{\prime }}$, and by step 2.1 the inner integral is at least $c^{2}m(b-a)>0$ for every $x^{\mathrm{\prime }}$; since the box $(-R,R{)}^{n-1}$ has measure $(2R{)}^{n-1}$ in each coordinate, this integral is at least $c^{2}m(b-a)(2R{)}^{n-1}$; as $R\to \mathrm{\infty }$ the right-hand side tends to $+\mathrm{\infty }$, while for each fixed $R$ monotonicity [F4] bounds ${\int }_{(-R,R{)}^n}ed{\lambda }_n\le {\int }_{{\mathbb{R}}^n}ed{\lambda }_n$; hence ${\int }_{{\mathbb{R}}^n}ed{\lambda }_n=+\mathrm{\infty }$. (Equivalently, the iterated integral in the $x_0$-variable alone has value $c^2{\int }_{\mathbb{R}}{F}^{\mathrm{\prime }}(s{)}^{2}ds>0$ and the remaining $(n-1)$-fold integral of the constant $1$ is $+\mathrm{\infty }$, which [F3] turns into the same conclusion.)

Failure of a finite-energy conclusion: since ${\int }_{{\mathbb{R}}^n}e(x,t)dx=+\mathrm{\infty }$ for every $t$, the total energy of Wave energy density, energy flux and total energy is not a finite conserved quantity for this solution, so the local differential law [F1] holds pointwise while no finite global identity follows; in dimension $n=1$ and for $F(s)=s$ one has $e=c^2$, whose integral over $[-R,R]$ is $2R{c}^2\to +\mathrm{\infty }$ as $R\to \mathrm{\infty }$, so the whole-line integral is infinite by [F4]; thus finite energy is not implied by the local identity. This witness does not show that every sufficient hypothesis of the conservation theorem is necessary, and its extended total energy is the constant $+\mathrm{\infty }$.