previous study in [@matsumura-yoshida] to the case where the viscosity is of $p$-Laplacian type (the related problems are studied in [@gur-mac], [@nagai-mimura1], [@nagai-mimura2] and so on). For this case, a main difficulty arises from the fact that when $u_\pm \in [\, a,b\, ]$, the asymptotic state is expected to be a self-similar type solution of a nonlinearly degenerate convective heat equation which may need the more subtle treatment than the Gaussian kernel type one (1.8) of the equation (1.7). There is only one result for the asymptotic behavior for the problem (1.1) in the case where the flux function is genuinely nonlinear on the whole space $\mathbb{R}$. Namely, Matsumura-Nishihara [@matsu-nishi2] proved the asymptotics which tends toward a single rarefaction wave by using the $L^2$ and $L^p$-energy estimates. We then consider the case where the flux function is given as (1.4) and the far field states as (1.5). We expect the asymptotic behavior of solutions to the Cauchy problem (1.1) to be similar as in [@matsumura-yoshida]. In more detail, under the conditions (1.4) and (1.5), if the far field states $u_{\pm}$ satisfy $u_\pm \in (-\infty,a\, ]$ or $u_\pm \in [\, b,\infty)$, the asymptotic state of the solutions to the Cauchy problem 