\displaystyle{\lim_{x\to \pm \infty}} v(t,x) =0  
  \qquad &\bigl( t \ge -1 \bigr),  
 \end{array}
 \right.\,\end{aligned}$$ where $\delta(x)$ is the Dirac $\delta$-distribution. They obtained the Barenblatt-Kompanceec-Zel’dovi[č]{} solution $$v(t,x)
:= \frac{1}{(1+t)^{\frac{1}{p+1}}}\, 
   {\mathopen{{\hbox{$\left}\vbox to 23.5pt{}\right.\n@space$}}}( \left( 
   A-B \left(\frac{x}{(1+t)^{\frac{1}{p+1}}} \right)^2 \, 
   \right)\vee 0 {\mathclose{{\hbox{$\left}\vbox to 23.5pt{}\right.\n@space$}}})^{\frac{1}{p-1}},$$ $$\begin{aligned}
\left\{\begin{array}{ll}
\displaystyle{
A=A_{p,\mu,u_{\pm}}
:= \left( \, 
   \frac{(p-1)\, \left( \, u_{+} - u_{-} \, \right)}
   {8\, \mu\, p(p+1) 
   \biggl( \, \displaystyle{\int^{\frac{\pi}{2}}_{0} 
   \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta }
   \biggr)^2 }
   \, \right)^{\frac{p-1}{p+1}}
}, \\[25pt]
\displaystyle{ 
B=B_{p,\mu}:= \frac{p-1}{2\, \mu \, p(p+1)} 
}, \\[10pt] 
\displaystyle{ 
2A^{\frac{p+1}{2(p-1)}}B^{-\frac{1}{2}}
      \int^{\frac{\pi}{2}}_{0} 
      \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta 
      = u_{+} - u_{-}},
\end{array}
\right.\,\end{aligned}$$ where the symbol “$\vee $” is defined as $
a\vee b:= \max \{a,b\}. 
$ Thus, when we define by using the solution (1.12) as $$\begin{aligned}
\begin{aligned}
U\left(\, \frac{x}{(1+t)^{\frac{1}{p+1}}}\: ;\: u_- ,u_+ \right)
&:= u_- + 
    \int^{x}_{-\infty} v(t,y)\, \mathrm{d}y \\
&=u_- + \displaystyle{
                {\mathlarger{\int}}^{\frac{\mathlarger{x}}{ \mathlarger{(1+t)}^{{\scriptscriptstyle \frac{1}{p+1}}} }}_{-\infty}
                     \Bigl( \left( \, 
                A-B \xi^2 \, 
                \right)\vee 0 \, \Bigr)^{\frac{1}{p-1}}
                \, \mathrm{d}\xi } 
             \\
&\quad \, (t>-1, x\in \mathbb{R}), 
\end{aligned}\end{aligned}$$ and change the variable as $1+t \mapsto t > 0$ and $x \mapsto x-\tilde{\lambda} \, t$ in this order, we have a desired canditate of the asymptotic state as $$\begin{aligned}
U\left(\, \frac{x-\tilde{\lambda} \, t}
          {t^{\frac{1}{p+1}}}\: ;\: u_- ,u_+ \right) 
