Is the analytical solution for scalar coefficient $c$ as $u(x,t)=u(x-ct,0)$ for $u_t+cu_x=0$ work same manner for the time-dependent coefficient case $u_t+v(t)u_x=0$ with the method of characteristics? or is there a different method?
Want to solve \begin{equation} u_t+v(t)u_x = 0 \end{equation} with a given Initial condition \begin{equation} u(x,0) = f(x) \end{equation} and $v$ comes from a Solution to an ODE in time as \begin{equation} v_t = -\frac{1}{\int_0^1 u dx}\Bigg[ \frac{1}{3A}\Bigg(\int_0^1(\rho-1)dx\Bigg)+B v|v|\int_0^1 u dx\Bigg] \end{equation} with $v(0)=1$ and $A,B$ real constants