  We study the spin diffusion in a two-dimensional electron gas at the
interface of oxide heterostructure LaAlO$_3$/SrTiO$_3$ grown on multiferroic
TbMnO$_3$ at 15 K by means of the kinetic spin Bloch equation approach. The
spiral magnetic moments of Mn$^{3+}$ in TbMnO$_3$ interact with the diffusing
spins at the LaAlO$_3$/SrTiO$_3$ interface via the Heisenberg exchange
interaction. It is demonstrated that the spin diffusion length is always
finite, despite the polarization direction of the injected spins. Our study
also reveals the important role played by the Coulomb scattering, which can
effectively suppress the spin diffusion.
