(1.1) should be the rarefaction wave (1.6) which connects $u_-$ and $u_+$, and if the far field states $u_{\pm}$ satisfy $u_\pm \in [\, a,b\, ]$, the one should be the “contact wave for $p$-Laplacian type viscosity” which connects $u_-$ and $u_+$, which is given by an exact solution of the following $p$-Laplacian evolution equation $$\partial _t u + \tilde{\lambda} \, \partial_x u 
  = \mu \, 
    \partial_x \left( \, 
    \left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
    \right)
  \qquad \biggl(
  \,\tilde{\lambda}:= \frac{f(b)-f(a)}{b-a},\ 
  t>0,x \in  \mathbb{R} 
  \biggr).$$ In order to look for an exact solution, especially self-similar type solution, we differentiate the evolution equation (1.9) with respect to $x$ and we have the following porous medium equation with the convection term $$\partial_t v  + \tilde{\lambda} \, \partial_x v 
= \mu \, 
  \partial_x^2 
  \left( \, 
  \left| \, v \, \right|^{p-1} v \, 
  \right),$$ where $v:=\partial_x u$. Barenblatt [@barenblatt], Zel’dovi[č]{}-Kompanceec [@zel-kom] and Pattle [@pattle] (see also [@carillo-toscani], [@huang-pan-wang], [@kamin]) introduced the following Cauchy problem of the porous medium equation $$\begin{aligned}
 \left\{\begin{array}{ll}
  \partial_t v
  =     \mu \, 
    \partial_x^2 
    \left( \, 
    \left| \, v \, \right|^{p-1} v \, 
    \right)
  \qquad &(t>-1, x\in \mathbb{R}), \\[5pt]
  v(-1,x) = (u_{+} - u_{-})\, \delta(x) 
  \qquad &( x \in \mathbb{R}\, ;\,  u_{-}<u_{+}),\\[5pt]
  