Energy uniqueness for the wave Cauchy problem

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $c>0$, $T>0$ and let $u$ be a classical solution of ${\mathrm{\square }}_{c}u=0$ on ${\mathbb{R}}^n\times (0,T)$ with Cauchy data $(u_0,u_1)$ and differentiable displacement $u_0$, so $D{u}_0$ in the initial-energy hypothesis is defined, and assume the total energy has a finite initial value and is conserved in the sharp form $E_{{\mathbb{R}}^n}(t)=E_{{\mathbb{R}}^n}(0)$ for every $t\in (0,T)$, where $E_{{\mathbb{R}}^n}(0)=\frac{1}{2}{\int }_{{\mathbb{R}}^n}(u_1^2+c^2\mathrm{\mid }D{u}_0{\mathrm{\mid }}^2)dx$. This holds, for instance, when the solution has fixed compact spatial support (by Conservation of total wave energy in three admissible settings(a) together with continuity of $e$ up to $t=0$ on the fixed support with its value given by the data density), or when the integrability hypotheses of that theorem's case (b) hold and $E_{{\mathbb{R}}^n}$ has a continuous extension to $0$ with value equal to the displayed data energy.

(i) If the Cauchy data vanish, $u_0=u_1=0$, then $E_{{\mathbb{R}}^n}(0)=0$, hence $E_{{\mathbb{R}}^n}(t)=0$ for all $t$, hence $u_t(\cdot ,t)=Du(\cdot ,t)=0$ and $u(\cdot ,t)$ is constant on ${\mathbb{R}}^n$ for every $t\in (0,T)$; the constant is the common limit of $u(\cdot ,t)$ as $t\downarrow 0$, namely $u_0=0$, so $u\equiv 0$.

(ii) More generally, if $u_1=0$ and $D{u}_0=0$ (equivalently $E_{{\mathbb{R}}^n}(0)=0$ under the conserved-solution hypotheses), then $u(\cdot ,t)\equiv u_0$ for every $t$, where the displacement datum $u_0$ is then constant: the energy sees only $(u_t,Du)$, and the displacement datum fixes the residual spatial constant. Consequently, two classical solutions with equal Cauchy data in a class closed under differences, in which each difference has the stated sharp energy conservation, agree.