  The temporal evolution of quantal Joule heating of 2D electrons in GaAs
quantum well placed in quantizing magnetic fields is studied using a difference
frequency method. The method is based on measurements of the electron
conductivity oscillating at the beat frequency $f=f_1-f_2$ between two
microwaves applied to 2D system at frequencies $f_1$ and $f_2$. The method
provides $direct$ access to the dynamical characteristics of the heating and
yields the inelastic scattering time $\tau_{in}$ of 2D electrons. The obtained
$\tau_{in}$ is strongly temperature dependent, varying from 0.13 ns at 5.5K to
1 ns at 2.4K in magnetic field $B$=0.333T. When temperature $T$ exceeds the
Landau level separation the relaxation rate $1/\tau_{in}$ is proportional to
$T^2$, indicating the electron-electron interaction as the dominant mechanism
limiting the quantal heating. At lower temperatures the rate tends to be
proportional to $T^3$, indicating considerable contribution from
electron-phonon scattering.
