Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 30C / iii / 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 $\epsilon >0$, and put $B_t=B(x_0,T+\epsilon -t)$. Define the local energyDifferentiating on the shrinking ball and integrating by parts givessince $2w_t{\partial }_{n}w\le w_t^2+({\partial }_{n}w{)}^2\le w_t^2+|\nabla w{|}^2$. 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, 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.