rarefaction wave, viscous contact wave

35K55, 35B40, 35L65

Introduction and main theorem
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In this paper, we shall consider the asymptotic behavior of solutions for one-dimensional scalar conservation law with a nonlinearly degenerate viscosity ($p$-Laplacian type viscosity with $p>1$) $$\begin{aligned}
 \left\{\begin{array}{ll}
  \partial_tu +\partial_x \bigl(f(u) \bigr)
  = \mu \, 
    \partial_x \left( \, 
    \left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
    \right)
  \qquad &(t>0, x\in \mathbb{R}), \\[5pt]
  u(0,x) = u_0(x) \qquad &( x \in \mathbb{R} ),\\[5pt]
  \displaystyle{\lim_{x\to \pm \infty}} u(t,x) =u_{\pm}  
  \qquad &\bigl( t \ge 0 \bigr).   
 \end{array}
 \right.\,\end{aligned}$$ Here, $u=u(t,x)$ denotes the unknown function of $t>0$ and $x\in \mathbb{R}$, the so-called conserved quantity, $f=f(u)$ is the flux function depending only on $u$, $\mu$ is the viscosity coefficient, $u_0$ is the given initial data, and constants $u_{\pm } \in \mathbb{R}$ are the prescribed far field states. We suppose the given flux $f=f(u)$ is a $C^1$-function satisfying $f(0)=f'(0)=0$, $\mu$ is a positive constant and far field states $u_{\pm }$ satisfy $u_-<u_+$ without loss of generality.

We are interested in the asymptotic behavior and its precise estimates in time of the global solution to the Cauchy problem (1.1). Especially, one of the keys of the study is to investigate the influence of the shape of 