For $x\in \mathbb{R}$ solve using Fourier transform
$$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}-\gamma u,$$
where $k, \gamma$ are positive constants and $u(x,t)|_{t=0}=f(x).$
First generally (the result should be in a form of convolution integral), then explicitly with $f(x)=e^{-x^2}.$