Write down the formula for the solution $u=u(t, x)$ for $t>0$ of the initial value problem for the heat equation in one space dimension

$$\begin{aligned} &\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0 \\ &u(0, x)=g(x) \end{aligned}$$

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

Define the distributional derivative of a tempered distribution $T \in \mathcal{S}^{\prime}(\mathbb{R})$. Define a fundamental solution of a constant-coefficient linear differential operator $P$, and show that the distribution defined by the function $\frac{1}{2} e^{-|x|}$ is a fundamental solution for the operator $P=-\frac{d^{2}}{d x^{2}}+1$.

For the equation

$$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=e^{t} \phi(x)$$

where $\phi \in \mathcal{S}(\mathbb{R})$, prove that there is a unique solution of the form $e^{t} v(x)$ with $v \in \mathcal{S}(\mathbb{R})$. Hence write down the solution of $(*)$ with general initial data $u(0, x)=f(x)$ and describe the large time behaviour.