Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 30C / iv / 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 obeysAgain $E(0)=0$ forces the difference to vanish at every chosen spacetime point. Therefore 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.