  The broadband SEDs of blazars show two distinct components which in leptonic
models are associated with synchrotron and SSC emission of highly relativistic
electrons. In some sources the SSC component dominates the synchrotron peak by
one or more orders of magnitude implying that the electrons mainly cool by
inverse Compton collisions with their self-made synchrotron photons. Therefore,
the linear synchrotron loss of electrons, which is normally invoked in emission
models, has to be replaced by a nonlinear loss rate depending on an energy
integral of the electron distribution. This modified electron cooling changes
significantly the emerging radiation spectra. It is the purpose of this work to
apply this new cooling scenario to relativistic power-law distributed
electrons, which are injected instantaneously into the jet. We will first solve
the differential equation of the volume-averaged differential number density of
the electrons, and then discuss their temporal evolution. Since any non-linear
cooling will turn into linear cooling after some time, we also calculated the
electron number density for a combined cooling scenario consisting of both the
linear and non-linear cooling. For all cases, we will also calculate
analytically the emerging optically thin synchrotron fluence spectrum which
will be compared to a numerical solution. For small normalized frequencies f <
1 the fluence spectra show constant spectral indices. We find for linear
cooling a_SYN = 1/2, and for non-linear cooling a_SSC = 3/2. In the combined
cooling scenario we obtain for the small injection parameter b_1 = 1/2, and for
the large injection parameter b_2 = 3/2, which becomes b_1 = 1/2 for very small
frequencies, again. This is the same behaviour as for monoenergetically
injected electrons.
