= Solution
For the difference of two solutions the equation is $w_{tt}+w_t-\Delta w=0$ with zero initial data. The same shrinking-ball energy now obeys
$$
E'(t)=-\int_{B_t}w_t^2\,dx
+\int_{\partial B_t}\left[w_t\partial_nw-\tfrac12(w_t^2+|\nabla w|^2)\right]dS\leq0.
$$
Again $E(0)=0$ forces the difference to vanish at every chosen spacetime point. Therefore \b[the two $C^2$ solutions agree for every $t>0$]. Damping contributes an additional negative energy term and does not weaken the local uniqueness proof.
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