  We perform a theoretical investigation on the time evolution of spin pulses
in an $n$-type GaAs (001) quantum well with and without external electric field
at high temperatures by constructing and numerically solving the kinetic spin
Bloch equations and the Poisson equation, with the electron-phonon,
electron-impurity and electron-electron Coulomb scattering explicitly included.
The effect of the Coulomb scattering, especially the effect of the Coulomb drag
on the spin diffusion/transport is investigated and it is shown that the spin
oscillations and spin polarization reverse along the direction of spin
diffusion in the absence of the applied magnetic field, which were originally
predicted in the absence of the Coulomb scattering by Weng and Wu [J. Appl.
Phys. {\bf 93}, 410 (2003)], can sustain the Coulomb scattering at high
temperatures ($\sim 200$ K). The results obtained are consistent with a recent
experiment in bulk GaAs but at a very low temperature (4 K) by Crooker and
Smith [Phys. Rev. Lett. {\bf 94}, 236601 (2005)].
