The problem under consideration is a linear convection PDE.
\(\) From Farlow’s “Partial Differential Equations for Scientists and Engineers”, chapter 15 problem 3 is $$u_t = -2u_x \quad -\infty < x < \infty \quad 0<t<\infty $$$$ u(x,0)=e^{-x^2}$$
Intuition
We see that the governing equation is a linear convection problem. The characteristic velocity is 2. We expect the solution to be a shifted initial condition $$u(x,t)=e^{-(x-2t)^{2}}$$