= Solution
Let $w$ be the difference between a $C^2$ solution and the formula in part (ii). It satisfies $w_{tt}-\Delta w=0$ and zero initial data. Fix $(T,x_0)$ and $\varepsilon>0$, and put $B_t=B(x_0,T+\varepsilon-t)$. Define the local energy
$$
E(t)=\frac12\int_{B_t}(w_t^2+|\nabla w|^2)\,dx.
$$
Differentiating on the shrinking ball and integrating by parts gives
$$
E'(t)=\int_{\partial B_t}\left[w_t\partial_nw-\tfrac12(w_t^2+|\nabla w|^2)\right]dS\leq0,
$$
since $2w_t\partial_nw\leq w_t^2+(\partial_nw)^2\leq w_t^2+|\nabla w|^2$. Initially $E=0$, hence $w_t=\nabla w=0$ inside each ball. In particular $w(t,x_0)$ is constant from its initial value zero, so $w(T,x_0)=0$. Since the point was arbitrary, \b[every $C^2$ solution equals the Kirchhoff solution]. This <shrinking cone energy argument> uses only energy on compact sets, so it does not assume global decay of the competing solution.
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