= Solution
<Free transport equation> is the measure-preserving shear $(x,v)\mapsto(x-vs,v)$. Its composition operator preserves the $L^2_{x,v}$ norm. Since $a\geq0$, the damping multiplier has modulus at most one:
$$
0\leq\exp\!\left(-\int_0^s a(r,x-v(s-r),v)\,dr\right)\leq1.
$$
Consequently \b[the <damped free-transport evolution> is contractive]:
$$
\boxed{\|F(f_0,a)(s)\|_2\leq\|f_0\|_2,\qquad
\sup_{0\leq s\leq t}\|F(f_0,a)(s)\|_2\leq\|f_0\|_2.}
$$
This needs no upper bound on $a$. Nonnegative integrals may be interpreted in the extended sense with $e^{-\infty}=0$; local integrability along characteristics additionally gives the usual continuous initial trace.
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