Write down a formula for the solution $u=u(t, x)$, for $t>0$ and $x \in \mathbb{R}^{n}$, of the initial value problem for the heat equation:

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

for $f$ a bounded continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. State (without proof) a theorem which ensures that this formula is the unique solution in some class of functions (which should be explicitly described).

By writing $u=e^{t} v$, or otherwise, solve the initial value problem

$$\tag{†} \frac{\partial v}{\partial t}+v-\Delta v=0, \quad v(0, x)=g(x)$$

for $g$ a bounded continuous function $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and give a class of functions in which your solution is the unique one.

Hence, or otherwise, prove that for all $t>0$ :

$$\sup _{x \in \mathbb{R}^{n}} v(t, x) \leqslant \sup _{x \in \mathbb{R}^{n}} g(x)$$

and deduce that the solutions $v_{1}(t, x)$ and $v_{2}(t, x)$ of $(†)$ corresponding to initial values $g_{1}(x)$ and $g_{2}(x)$ satisfy, for $t>0$,

$$\sup _{x \in \mathbb{R}^{n}}\left|v_{1}(t, x)-v_{2}(t, x)\right| \leqslant \sup _{x \in \mathbb{R}^{n}}\left|g_{1}(x)-g_{2}(x)\right| .$$