  In this paper, we investigate the global structure of solutions to the Cauchy
problem for the scalar viscous conservation law where the far field states are
prescribed. Especially, we deal with the case when the viscous/diffusive flux
$\sigma(v) \sim |\, v \,|^p$ is of non-Newtonian type (i.e., $p>0$), including
a pseudo-plastic case (i.e., $p<0$). When the corresponding Riemann problem for
the hyperbolic part admits a Riemann solution which consists of single
rarefaction wave, under a condition on nonlinearity of the viscosity, it has
been recently proved by Matsumura-Yoshida \cite{matsumura-yoshida'} that the
solution of the Cauchy problem tends toward the rarefaction wave as time goes
to infinity for the case $p>3/7$ without any smallness conditions. The new
ingredients we obtained are the extension to the stability results in
\cite{matsumura-yoshida'} to the case $p > 1/3$ (also without any smallness
conditions), and furthermore their precise time-decay estimates.
