  We study the relaxation of a spin density injected into a two-dimensional
electron system with generic spin-orbit interactions. Our model includes the
Rashba as well as linear and cubic Dresselhaus terms. We explicitly derive a
general spin-charge coupled diffusion equation. Spin diffusion is characterized
by just two independent dimensionless parameters which control the interplay
between different spin-orbit couplings. The real-time representation of the
diffuson matrix (Green's function of the diffusion equation) is evaluated
analytically. The diffuson describes space-time dynamics of the injected spin
distribution. We explicitly study two regimes: The first regime corresponds to
negligible spin-charge coupling and is characterized by standard charge
diffusion decoupled from the spin dynamics. It is shown that there exist
several qualitatively different dynamic behaviors of the spin density, which
correspond to various domains in the spin-orbit coupling parameter space. We
discuss in detail a few interesting phenomena such as an enhancement of the
spin relaxation times, real space oscillatory dynamics, and anisotropic
transport. In the second regime, we include the effects of spin-charge
coupling. It is shown that the spin-charge coupling leads to an enhancement of
the effective charge diffusion coefficient. We also find that in the case of
strong spin-charge coupling, the relaxation rates formally become complex and
the spin/charge dynamics is characterized by real time oscillations. These
effects are qualitatively similar to those observed in spin-grating experiments
[Weber et al., Nature 437, 1330 (2005)].
