(a) State and prove the Duhamel principle for the wave equation.

(b) Let $u \in C^{2}\left([0, T] \times \mathbb{R}^{n}\right)$ be a solution of

$$u_{t t}+u_{t}-\Delta u+u=0$$

where $\Delta$ is taken in the variables $x \in \mathbb{R}^{n}$ and $u_{t}=\partial_{t} u$ etc.

Using an 'energy method', or otherwise, show that, if $u=u_{t}=0$ on the set $\left\{t=0,\left|x-x_{0}\right| \leqslant t_{0}\right\}$ for some $\left(t_{0}, x_{0}\right) \in[0, T] \times \mathbb{R}^{n}$, then $u$ vanishes on the region $K(t, x)=\left\{(t, x): 0 \leqslant t \leqslant t_{0},\left|x-x_{0}\right| \leqslant t_{0}-t\right\}$. Hence deduce uniqueness for the Cauchy problem for the above PDE with Schwartz initial data.