Eliminate g and don't capitalize w:
a = 1/2; b = Sqrt[3]/2; ve = 1; u = 1;
f0[v_] = (a/Sqrt[Pi])*Exp[v^2/ve^2] +
(b/Sqrt[Pi])*Exp[(v - u)^2/ve^2];
u1 = Log[((1 - a^2)/a^2)^(1/2)]/(2*u) + u/2;
u2 = 3*u/2 - Log[((1 - a^2)/a^2)^(1/2)]/(2*u); 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] == 2*Pi/2*(v^2)*D[go[v, t], v]*w[v, t],
w[v, 0] == 0},
{go[v, t], w[v, t]},
{v, u1, u2}, {t, 0, 10000000000}]
