(a) Define the convolution $f * g$ of two functions. Write down a formula for a solution $u:[0, \infty) \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ to the initial value problem

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

together with the boundary condition

$$u(0, x)=f(x)$$

for $f$ a bounded continuous function on $\mathbb{R}^{n}$. Comment briefly on the uniqueness of the solution.

(b) State and prove the Duhamel principle giving the solution (for $t>0$ ) to the equation

$$\frac{\partial u}{\partial t}-\Delta u=g$$

together with the boundary condition

$$u(0, x)=f(x)$$

in terms of your answer to (a).

(c) Show that if $v:[0, \infty) \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ is the solution to

$$\frac{\partial v}{\partial t}-\Delta v=G$$

together with the boundary condition

$$v(0, x)=f(x)$$

with $G(t, x) \leq g(t, x)$ for all $(t, x)$ then $v(t, x) \leq u(t, x)$ for all $(t, x) \in(0, \infty) \times \mathbb{R}^{n}$.

Finally show that if in addition there exists a point $\left(t_{0}, x_{0}\right)$ at which there is strict inequality in the assumption i.e.

$$G\left(t_{0}, x_{0}\right)<g\left(t_{0}, x_{0}\right)$$

then in fact

$$v(t, x)<u(t, x)$$

whenever $t>t_{0}$.