We studied the effects caused by Joule heating in GaAs/AlAs superlattices. We measured the current-voltage characteristics of superlattice mesas of different cross sections. Each current-voltage characteristic showed a corresponding peak current followed by a region of negative differential conductivity, according to miniband transport in the superlattice. We found that Joule heating in the mesa reduced the peak-current density. Current jumps in the region of negative differential conductivity disappeared for large cross-section mesas indicating a suppression of electron domains. The results are of significance for the development of high-power superlattice oscillators for submillimeter radiation. (C) 2002 American Institute of Physics.