I understand the stability with respect to initial condition as: if $u$ and $v$ solve the above equation with the same $f$ and the same boundary condition $u(0,t )=0=v(0,t)$, then $\|u(\cdot , T)-v(\cdot , T)\|_{L^2}$ is bounded in terms of $\|u(\cdot , 0)-v(\cdot , 0)\|_{L^2}$. In this case, we don't have to worry about $f$ at all because the difference $w=u-v$ solves the homogeneous transport equation. The solution $w$ is constant on every line with slope $1/a$. Taking into account the boundary data $0$, you should be able to see that the $L^2$ norm of $w$ with respect to the space variable $x$ does not increase with $t$.