Consider the initial value problem

$$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=0$$

to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

$$u(0, x)=f(x), \quad \frac{\partial u}{\partial t}(0, x)=0$$

for $f$ in the Schwarz space $\mathcal{S}\left(\mathbb{R}^{n}\right)$. Use the Fourier transform in $x$ to obtain a representation for the solution in the form

$$u(t, x)=\int e^{i x \cdot \xi} A(t, \xi) \widehat{f}(\xi) d^{n} \xi$$

where $A$ should be determined explicitly. Explain carefully why your formula gives a smooth solution to (1) and why it satisfies the initial conditions (2), referring to the required properties of the Fourier transform as necessary.

Next consider the case $n=1$. Find a tempered distribution $T$ (depending on $t, x$ ) such that (3) can be written

$$u=<T, \widehat{f}>$$

and (using the definition of Fourier transform of tempered distributions) show that the formula reduces to

$$u(t, x)=\frac{1}{2}[f(x-t)+f(x+t)]$$

State and prove the Duhamel principle relating to the solution of the $n$-dimensional inhomogeneous wave equation

$$\frac{\partial^{2} u}{\partial t^{2}}-\Delta u=h$$

to be solved for $u: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$, subject to the initial conditions

$$u(0, x)=0, \quad \frac{\partial u}{\partial t}(0, x)=0$$

for $h$ a $C^{\infty}$ function. State clearly assumptions used on the solvability of the homogeneous problem.

[Hint: it may be useful to consider the Fourier transform of the tempered distribution defined by the function $\xi \mapsto e^{i \xi \cdot a}$.]