Write down the solution of the three-dimensional wave equation

$$u_{t t}-\Delta u=0, \quad u(0, x)=0, \quad u_{t}(0, x)=g(x),$$

for a Schwartz function $g$. Here $\Delta$ is taken in the variables $x \in \mathbb{R}^{3}$ and $u_{t}=\partial u / \partial t$ etc. State the "strong" form of Huygens principle for this solution. Using the method of descent, obtain the solution of the corresponding problem in two dimensions. State the "weak" form of Huygens principle for this solution.

Let $u \in C^{2}\left([0, T] \times \mathbb{R}^{3}\right)$ be a solution of

$$u_{t t}-\Delta u+|x|^{2} u=0, \quad u(0, x)=0, \quad u_{t}(0, x)=0$$

Show that

$$\partial_{t} e+\nabla \cdot \mathbf{p}=0$$

where

$$e=\frac{1}{2}\left(u_{t}^{2}+|\nabla u|^{2}+|x|^{2} u^{2}\right), \quad \text { and } \quad \mathbf{p}=-u_{t} \nabla u .$$

Hence deduce, by integration of $(* *)$ over the region

$$K=\left\{(t, x): 0 \leqslant t \leqslant t_{0}-a \leqslant t_{0},\left|x-x_{0}\right| \leqslant t_{0}-t\right\}$$

or otherwise, that $(*)$ satisfies the weak Huygens principle.