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Time-reversed energy uniqueness from final data

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $c > 0$ , $T > 0$ and let $u, v \in C^{2} (R^{n} \times [0, T])$ solve $□_{c} w = 0$ , with $u$ and $v$ lying in a class of homogeneous solutions that is closed under taking differences and in which the total energy is conserved on $[0, T]$ in one of the whole-space senses (a) or (b) of Conservation of total wave energy in three admissible settings (for instance both have fixed compact spatial support inside a common compact set). If $u (\cdot, T) = v (\cdot, T)$ and $u_{t} (\cdot, T) = v_{t} (\cdot, T)$ , then $u = v$ on $R^{n} \times [0, T]$ .

Thus equal terminal displacement and velocity determine the same finite-energy homogeneous solution backward in time; reversibility is a consequence of energy uniqueness together with time-reversal symmetry, not of any representation formula. The time-reversal item Time reversal of the homogeneous wave equation itself belongs to the preceding pair and is consumed here, not reproved.

Facts & Assumptions

Given: ${A C}_{ω}$ ; two homogeneous solutions $u, v$ in the stated class with equal Cauchy data at time $T$ ; the difference $w = u - v$ lies in the class, so its energy is conserved on $[0, T]$ .

[F1]

Apply the time-reversal theorem to $w$ on the open interval $I = (0, T)$ with $τ = T / 2$ : $\tilde{w} (x, s) : = w (x, T - s)$ is $C^{2}$ and homogeneous in the interior. Continuity of $w$ and its derivatives to the endpoints gives the reflected extension on $[0, T]$ , with $\tilde{w} (\cdot, 0) = w (\cdot, T)$ and $\partial_{s} \tilde{w} (\cdot, 0) = - w_{t} (\cdot, T)$ . (Time reversal of the homogeneous wave equation)

[F2]

Energy uniqueness for the Cauchy problem: a homogeneous solution whose total energy is conserved on the time interval and whose Cauchy data vanish at the initial time is identically zero; two conserved solutions with equal initial data agree. (Energy uniqueness for the wave Cauchy problem)

[F3]

The energy density of the time-reversed solution at time $s$ equals the energy density of the original solution at time $T - s$ : the spatial gradient is unchanged and $\partial_{s} \tilde{w} = - \partial_{t} w$ . (Wave energy density, energy flux and total energy)

Proof

1.1givenF1

The difference and its reversal: $w : = u - v$ is $C^{2}$ and homogeneous by linearity, and it lies in the stated class, so its total energy is conserved on $[0, T]$ ; by hypothesis $w (\cdot, T) = 0$ and $w_{t} (\cdot, T) = 0$ ; define $\tilde{w} (x, s) : = w (x, T - s)$ for $s \in [0, T]$ .

2.1givenstep 1.1F1F3algebra

The reversal is a conserved homogeneous solution with zero initial data: by [F1], $\tilde{w}$ is a $C^{2}$ homogeneous solution on $R^{n} \times [0, T]$ with $\tilde{w} (\cdot, 0) = w (\cdot, T) = 0$ and $\partial_{s} \tilde{w} (\cdot, 0) = - w_{t} (\cdot, T) = 0$ ; by [F3] the energy density of $\tilde{w}$ at time $s$ equals that of $w$ at time $T - s$ , so $E_{\tilde{w}} (s) = E_{w} (T - s)$ and conservation of $E_{w}$ transfers to $\tilde{w}$ .

3.1step 2.1F2∎

Conclusion: [F2] applied to the conserved solution $\tilde{w}$ with vanishing Cauchy data gives $\tilde{w} \equiv 0$ ; hence $w \equiv 0$ , that is, $u = v$ on $R^{n} \times [0, T]$ .

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)