plot – Solution of the Burgers equation with some initial data

Consider the Burgers equation
$$ partial_t u + partial_x left (u ^ 2/2 right) = 0, $$

eq = D[u[t, x], t]+ D[u[t,x]^ 2/2[t, x], X]== 0

How to use Mathematica to get the solution containing the initial data $ u (0, cdot) $ Cantor stair function?

u0 (x) = F_C (x)