  Aiming at the optimization of the spin diffusion length in (001) GaAs quantum
wells, we explore the effect of the anisotropy of the spin-orbit coupling on
the competition between the Rashba and the Dresselhaus spin-orbit couplings by
solving the kinetic spin Bloch equations with the electron-phonon and the
electron-electron scattering explicitly included. For identical strengths of
the Rashba and the Dresselhaus spin-orbit couplings, the spin diffusion length
shows strong anisotropy not only for the spin polarization direction but also
for the spin diffusion direction. Two special directions are used seeking for
the large diffusion length: ($\bar{1}$10) and (110). Without the cubic term of
the Dresselhaus spin-orbit coupling and with the identical Dresselhaus and
Rashba strengths, infinite diffusion lengths can be obtained {\em either} for
the spin diffusion/injection direction along $(\bar110)$, regardless of the
direction of spin polarization, {\em or} for the spin polarization along
$(110)$, regardless of the direction of the spin diffusion/injection.
  However, the cubic Dresselhaus term cannot be neglected, resulting into a
finite spin diffusion length which decreases with the temperature and the
electron density, and of which the anisotropy for the spin diffusion direction
and spin polarization direction is maintained. For the spin diffusion/injection
direction along ($\bar{1}$10), the spin diffusion length increases first with
the increase of the Rashba strength (from 0) which can be tuned by the external
gate voltage; when the Rashba strength is slightly smaller than (in stead of
equal to) the Dresselhaus strength, the diffusion length reaches its maximum;
and then it decreases with the Rashba strength.
