\rightarrow 0}} \, 
 \partial_x 
      \left( 
      \left( 
      \bigl( \, \partial_x u_{\epsilon} \, 
      \bigr)^2 + \epsilon \, 
      \right)^{\frac{p-1}{2}} \partial_x u_{\epsilon} \, 
      \right) 
 =\partial_x \left( \, 
  \left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
  \right)
 \in L^{2}\bigl( \, 0,T \, ;L^{2} \bigr). 
 \end{array}
 \right.\,\end{aligned}$$ We can also see that the limit function $\phi$ gives the unique global solution of (3.8) and the results in Theorem 3.3 hold. In particular, we note that the energy estimates in Theorem 3.3 are independent of $\Vert \, \partial_x \phi_0 \, \Vert_{L^2}$. For the initial data which satisfies $$\phi_0 \in L^2, 
\qquad \partial_x \phi_0 \in L^{p+1},$$ we take again the approximate sequence $\bigl\{ \phi_{0}^{\delta} \bigr\}$ which satisfies $$\phi_{0}^{\delta} \in L^2, 
\qquad \partial_x \phi_{0}^{\delta} \in L^2 \cap L^{p+1}$$ and $$\begin{aligned}
 \left\{\begin{array}{ll}
 \displaystyle{\operatorname{s-\lim}\displaylimits_{\delta \rightarrow 0}} \, 
 \phi_{0}^{\delta} = \phi_{0} 
 \in L^{2}, \\[10pt]
 \displaystyle{\operatorname{s-\lim}\displaylimits_{\delta \rightarrow 0}} \, 
 \partial_x \phi_{0}^{\delta} = \partial_x \phi_{0} 
 \in L^{p+1}.
 \end{array}
 \right.\,\end{aligned}$$ We may take the limit $\delta \rightarrow 0$ to get Theorem 3.3.

Since the energy estimates in Theorem 3.3 depend on $T$, we can not prove the asymptotics $$\Vert \, \phi (t) \, \Vert_{L^{\infty}} \rightarrow 0 
\qquad (t \rightarrow \infty).$$ In order to show the desired asymptotics, we show the following a priori estimates which are 