Write down the formula for the solution $u=u(t, x)$ for $t>0$ of the initial value problem for the $n$-dimensional heat equation

$$\begin{gathered} \frac{\partial u}{\partial t}-\Delta u=0 \\ u(0, x)=g(x) \end{gathered}$$

for $g: \mathbb{R}^{n} \rightarrow \mathbb{C}$ a given smooth bounded function.

State and prove the Duhamel principle giving the solution $v(t, x)$ for $t>0$ to the inhomogeneous initial value problem

$$\begin{aligned} &\frac{\partial v}{\partial t}-\Delta v=f \\ &v(0, x)=g(x) \end{aligned}$$

for $f=f(t, x)$ a given smooth bounded function.

For the case $n=4$ and when $f=f(x)$ is a fixed Schwartz function (independent of $t)$, find $v(t, x)$ and show that $w(x)=\lim _{t \rightarrow+\infty} v(t, x)$ is a solution of

$$-\Delta w=f \text {. }$$

[Hint: you may use without proof the fact that the fundamental solution of the Laplacian on $\mathbb{R}^{4}$ is $\left.-1 /\left(4 \pi^{2}|x|^{2}\right) .\right]$