  Thermal hysteresis in a micron-size Superconducting Quantum Interference
Device ($\mu$-SQUID), with weak links as Josephson junctions, is an obstacle
for improving its performance for magnetometery. Following the "hot-spot" model
of Skocpol et al. [J. Appl. Phys. {\bf 45}, 4054 (1974)] and by incorporating
the temperature dependence of thermal conductivity of superconductor using a
linear approximation, we find a much better agreement with the observed
temperature dependence of the retrapping current in short superconducting
Nb-based weak links and $\mu$-SQUIDs. In addition, using the temperature
dependence of the critical current, we find that above a certain temperature
hysteresis disappears. We analyze the current-voltage characteristics and the
weak link temperature variation in both the hysteretic and non-hysteretic
regimes. We also discuss the effect of the weak link geometry in order to widen
the temperature range of hysteresis-free operation.
