  We study the statistical properties of solutions to Burgers' equation, $v_t +
vv_x = \nu v_{xx}$, for large times, when the initial velocity and its
potential are stationary Gaussian processes. The initial power spectral density
at small wave numbers follows a steep power-law $E_0(k) \sim |k|^n$ where the
exponent $n$ is greater than two. We compare results of numerical simulations
with dimensional predictions, and with asymptotic analytical theory. The theory
predicts self-similarity of statistical characteristics of the turbulence, and
also leads to a logarithmic correction to the law of energy decay in comparison
with dimensional analysis. We confirm numerically the existence of
self-similarity for the power spectral density, and the existence of a
logarithmic correction to the dimensional predictions.
