the flux function $f(u)$ and the far field states $u_{\pm }$ on the asymptotic behavior. It can be expected that the large-time behavior is closely related to the weak solution (“Riemann solution”) of the corresponding Riemann problem (cf. [@lax], [@smoller]) for the non-viscous hyperbolic part of (1.1): $$\begin{aligned}
 \left\{\begin{array} {ll}
 \partial _t u + \partial _x \bigl( f(u) \bigr)=0 
 \qquad &(t>0, x\in \mathbb{R}),\\[5pt]
u(0,x)=u_0 ^{\rm{R}} (x)\qquad &(x \in \mathbb{R}),
 \end{array}
  \right.\,\end{aligned}$$ where $u_0 ^{\rm{R}}$ is the Riemann data defined by $$u_0 ^{\rm{R}} (x)=u_0 ^{\rm{R}} (x\: ;\: u_- ,u_+)
          := 
          \left\{\begin{array} {ll}
          u_-  & \; (x < 0),\\[5pt]
          u_+  & \; (x > 0).
          \end{array}\right.$$ In fact, for the usual linear viscosity case: $$\begin{aligned}
 \left\{\begin{array}{ll}
  \partial_tu +\partial_x \bigl(f(u) \bigr)= \mu \, \partial_x^2 u
  \qquad &(t>0, x\in \mathbb{R}), \\[5pt]
  u(0,x) = u_0(x) \qquad &( x \in \mathbb{R} ),\\[5pt]
  \displaystyle{\lim_{x\to \pm \infty}} u(t,x) =u_{\pm}  
  \qquad &\bigl( t \ge 0 \bigr),   
 \end{array}
 \right.\,\end{aligned}$$ when the smooth flux function $f$ is genuinely nonlinear on the whole space $\mathbb{R}$, i.e., $f''(u)\ne 0\ (u\in \mathbb{R})$, Il’in-Ole[ĭ]{}nik [@ilin-oleinik] showed the following: if $f''(u) > 0\ (u\in \mathbb{R})$, that is, the Riemann solution consists of a single rarefaction wave solution, the global solution in time of the Cauchy problem 