  The present work is devoted to the evolution of random solutions of the
unforced Burgers and KPZ equations in d-dimensions in the limit of vanishing
viscosity. We consider a cellular model and as initial condition assign a value
for the velocity potential chosen independently within each cell. We show that
the asymptotic behavior of the turbulence at large times is determined by the
tail of the initial potential probability distribution function. Three classes
of initial distribution leading to self-similar evolution are identified: (a)
distributions with a power-law tail, (b) compactly supported potential, (c)
stretched exponential tails. In class (c) we find that the mean potential (mean
height of the surface) increases logarithmically with time and the 'turbulence
energy' E(t) (mean square gradient of the surface) decays as 1/t times a
logarithmic correction. In classes (a) and (b) we find that the changes in the
mean potential and energy have a power-law time dependence. In class (c) the
roughness of the surface, measured by its mean--square gradient, may either
decrease or increase with time. We discuss also the influence of finite
viscosity and long range correlation on the late stage evolution of the Burgers
turbulence
