  In this paper, we consider the asymptotic behavior of traveling wave
solutions of the degenerate nonlinear parabolic equation:
$u_{t}=u^{p}(u_{xx}+u)-\delta u$ ($\delta = 0$ or $1$) for $\xi \equiv x - ct
\to - \infty$ with $c>0$. We give a refined one of them, which was not obtain
in the preceding work [Ichida-Sakamoto, 2020], by an appropriate asymptotic
study and properties of the Lambert $W$ function.
