  A fourth-order and a second-order nonlinear diffusion models in spectral
space are proposed to describe gravitational wave turbulence in the
approximation of strongly local interactions. We show analytically that the
model equations satisfy the conservation of energy and wave action, and
reproduce the power law solutions previously derived from the kinetic equations
with a direct cascade of energy and an explosive inverse cascade of wave
action. In the latter case, we show numerically by computing the second-order
diffusion model that the non-stationary regime exhibits an anomalous scaling
which is understood as a self-similar solution of the second kind with a front
propagation following the law $k_f \sim (t_*-t)^{3.296}$, with $t<t_*$. These
results are relevant to better understand the dynamics of the primordial
universe where potent sources of gravitational waves may produce space-time
turbulence.
