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

Global structure of solutions toward the rarefaction waves for the Cauchy problem of the scalar conservation law with nonlinear viscosity