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.
Oscillations of dissipative resistance of two-dimensional electrons in GaAs quantum wells are observed in response to an electric current I and a strong magnetic field applied perpendicular to the two-dimensional systems. Period of the current-induce
d oscillations does not depend on the magnetic field and temperature. At a fixed current the oscillations are periodic in inverse magnetic fields with a period that does not depend on dc bias. The proposed model considers spatial variations of electron filling factor, which are induced by the electric current, as the origin of the resistance oscillations.
