Write down a formula for the solution $u=u(t, x)$ of the $n$-dimensional heat equation

$$w_{t}(t, x)-\Delta w=0, \quad w(0, x)=g(x),$$

for $g: \mathbb{R}^{n} \rightarrow \mathbb{C}$ a given Schwartz function; here $w_{t}=\partial_{t} w$ and $\Delta$ is taken in the variables $x \in \mathbb{R}^{n}$. Show that

$$w(t, x) \leqslant \frac{\int|g(x)| d x}{(4 \pi t)^{n / 2}}$$

Consider the equation

$$u_{t}-\Delta u=e^{i t} f(x),$$

where $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$ is a given Schwartz function. Show that $(*)$ has a solution of the form

$$u(t, x)=e^{i t} v(x),$$

where $v$ is a Schwartz function.

Prove that the solution $u(t, x)$ of the initial value problem for $(*)$ with initial data $u(0, x)=g(x)$ satisfies

$$\lim _{t \rightarrow+\infty}\left|u(t, x)-e^{i t} v(x)\right|=0 .$$