I'm trying to solve the following initial value problem.
$\begin{cases} u_t + u_x - u_{xx} &= -u, \quad \text{on} \quad \mathbb R \times \mathbb R_+\\ u(x,0) &= \frac{1}{4\pi} e^\frac{-x^2}{4}, \quad x \in \mathbb R \end{cases}$
I know how to solve the heat equation $u_t = u_{xx}$, but I have no idea how to get started with this. So far -inspired by the initial value- I have tried substituting and checking various forms of $\frac{e^\frac{-x^2}{4t}}{\sqrt{4{\pi}t}}$ to no avail. Is there a general technique that can be used in this case?