The hydrodynamic equation for the spatial and temporal evolutions of the electron temperature T, in the breakdown of the quantum Hall effect at even-integer filling factors: ill a uniform current density j(x) is derived from the Boltzmann-type equation, which takes into account electron-electron and electron-phonon scattering processes. The derived equation has a. drift term, which is proportional to j(x) and the first spatial derivative of T-e. When applied to the spatial evolution of T-e in a sample with an abrupt change of the width along the current direction, the equation gives a distinct dependence on the current direction as well as a critical relaxation, in agreement with results of recent experiments.