There is something wrong with your code, which I have modified to get a solution. I have no idea about your initial/boundary conditions. At least the code below produces plots and shows what you could do
sol = NDSolve[
{
D[go[v, t], t] + D[D[w[v, t], t], v]/(v^3) + (-3*v^(-4))*D[w[v, t], t] == 0,
go[v, 0] == f0[v],
D[w[v, t], t] == Pi/eps*(v^2)*D[go[v, t], v]*w[v, t],
w[v, 0] == 0
},
{go, w},
{v, u1, u2}, {t, 0, 20}] // Flatten
fgo = go /. sol
fw = w /. sol
Plot3D[{fgo[x, y], fw[x, y]}, {x, 6, 10}, {y, 0, 20}]
Plot[fw[8, t], {t, 0, 10}, PlotStyle -> {Red, Thick}]