  We give a detailed treatment of the back-reaction effects on the Hawking
spectrum in the late-time expansion within the semiclassical approach to the
Hawking radiation. We find that the boundary value problem defining the action
of the modes which are regular at the horizon admits in general the presence of
caustics. We show that for radii less that a certain critical value $r_c$ no
caustic occurs for all values of the wave number and time and we give a
rigorous lower bound on such a critical value. We solve the exact system of non
linear equations defining the motion, by an iterative procedure rigorously
convergent at late times. The first two terms of such an expansion give the
$O(\omega/M)$ correction to the Hawking spectrum.
